XIVth International Congress on Mathematical Physics

9T. The action-angle variables for the second symplectic form are group version of the actionangle variables for the first symplectic structure. Theorem 2.4. Let 7 S be the poles on the spectral curve of the normalized eigenvector ip of the matrix function L £ C9Q . Then the two-form wgr defined by (20) is equal to g+r—l

9r

tj =

J^ 5\nk(ls)A6z(ls).

(22)

s= l

Theorems 2.1 and 2.3 provide a framework for the existence of so-called bi-Hamiltonian structures. It was first observed by Magri that the KdV hierarchy possesses a bi-Hamiltonian structure, in the sense that all the flows of the hierarchy are Hamiltonian with respect to two different symplectic structures. If Hn is the Hamiltonian generating the n-th flow of the KdV hierarchy with respect to the first Gardner-Zakharov-Faddeev symplectic form, then the same flow is generated by the Hamiltonian Hn-\ with respect to the second LenardMagri symplectic form. Periodic chains. The two symplectic structures w and uigr are equally good in the case of a single matrix function L(z), but there is a marked difference between them when periodic chains of operators are considered (see details in [13]). Let Ln(z) € C{D) be a periodic

Algebraic versus Liouville integrability of the soliton systems chain of matrix-valued functions with a pole divisor D, Ln = Ln+x. chains is C(D)®N. The monodromy matrix Tn(z) = Ln+N-.i(z)

Ln+N-2{z)

57

The total space of such

• • • Ln(z)

(23)

is a meromorphic matrix function with poles of order Nhm at the puncture zm, i.e. Tn(z) e C{ND). For different n they are conjugated to each other. Thus the map C{D)®N .—* C(ND)/SLr

(24)

is well-defined. However, the natural attempt to obtain a symplectic structure on the space £(D)®N by pulling back the first symplectic form w on C(ND) runs immediately into obstacles. The main obstacle is that the form UJ is -only well-defined on the symplectic leaves of C{ND) consisting of matrices with fixed singular parts for the eigenvalues at the punctures. These constraints are non-local, and cannot be described in terms of constraints for each matrix Ln(z) separately. On the other hand, the second symplectic form w 9r has essentially the desired local property. Indeed, let Ln be a chain of matrices such that Ln G £(D,D_). Then the monodromy matrix defines a map f:£{D,D-)®N

^C(ND,ND^)/SLr.

The group SL^f of .^-independent matrices gn € SLr,gn

(25)

— 9n+N

actson£(D,£>_)® J V by

the gauge transformation Ln -> gn+iLng'1 (26) which is compatible with the monodromy matrix map (25). Let the space 'Pchain be defined as the corresponding quotient space of a preimage under T of a symplectic leaf CBQ C £(ND,ND-)/SLr Pchain = ( f - 1 ( P 1 C o ) ) / 5 L ^ v . (27) The dimension of this space is equal to dimPchain = N(degD)r(r

— 1) — 2r + 2.

Theorem 2.5. The pull-back by T of the second symplectic form wchain = T*{u>gT), restricted to T _ 1 (£Q>)> *S 9au9e invariant and descends to a symplectic form on Pchain- It can also be expressed by the local expression N

1 Wchain = 2 5 Z

ReS

*«

51 ^ n=l

(^nllSLn(z)

A « * „ ( * ) ) dz,

(28)

where *„+i = i „ $ n , ^n+N — ^nK, ^ = diag(fcj)5lJ. All the coefficients of the characteristic polynomial of T(z) are in involution with respect to this symplectic form. The number of independent integrals equals dimP c hain/2.

3. Periodic chains on algebraic curves The Riemann-Roch theorem implies that naive generalization of equations (1, 2) for matrix functions, which are meromorphic on an algebraic curve T of genus g > 0, leads to an overdetermined system of equations. Indeed, the dimension of r X r matrix functions of fixed

58

IGOR KRICHEVER

degree d divisor of poles in general position is r2(d — g + 1). If the divisors of L and M have degrees n and m, then the commutator [L, M] is of degree n + m. Thus the number of equations r2(n + m — g + 1) exceeds the number r2(n + m — 2g + 1) of unknown functions modulo gauge equivalence (see details in [5]). There are two ways to overcome this difficulty in defining zero curvature equations on algebraic curves. The first way is based on a choice of L with essential singularity at some point or with entries as sections of some bundle over the curve. The second way, based on a theory of high rank solutions of the Kadomtsev-Petviashvili equation, was discovered in [9]. There it was shown that if in addition to fixed poles the matrix functions L and M have rg moving poles of a special form, then the Lax equation is a well-defined system on the space of singular parts of L and M at fixed poles. In [9] it was found, that if the matrix functions L and M have moving poles with special dependence on x and t besides fixed poles, then equation (1) is a well-defined system on the space of singular parts of L and M at fixed poles. A theory of the corresponding systems was developed in [5]. In is instructive to present its discrete analog, that a theory of the discrete curvature equations (2) with the spectral parameter an a smooth algebraic curve. We begin by describing a suitable space of such functions Ln. Let T be a smooth genus g algebraic curve. According to [11], a generic stable, rank r and degree rg holomorphic vector bundle V on T is parameterized by a set of rg distinct points 7 S on T, and a set of r-dimensional vectors as = (als), considered modulo scalar factor as —> \sas and a common gauge transformation a* —»
J GLr.

In [9,12] the data (7,a) = ( 7 s , a s ) , s = l,...,rg, i = l , . . . , r , were called the Tyurin parameters. Let D± be two effective divisors on V of the same degree T>, Below, if it is not stated otherwise, it is assumed that all the points of the divisors D± = J2k=i ^ n a v e multiplicity 1, Pj^~ =fi P*i, k / m. For any sequence of the Tyurin parameters (7(n), a(n)) we introduce the space £ 7 ( n ) >a (n)(D+,D-) of meromorphic matrix functions Ln(q),q € T, such that: 1°. Ln is holomorphic except at the points 7 S , and at the points P^ of D+, where it has at most simple poles; 2°. the singular coefficient Ls(n) of the Laurent expansion of Ln at 7S Ln{z)==±M+0(l),

za = z{ya),

(29)

z — zs is a rank 1 matrix of the form La(n) = Ps(n)aJ(n)

<—> L^(n) =

fi(n)<4(n),

(30)

where /3s(n) is a vector, and z is a local coordinate in the neighborhood of 7S; 3°. the vector af(n + 1) is a left null-vector of the evaluation of Ln at 7s(^ + 1), i-e. a , ( n + l ) L n ( 7 , ( n + l ) ) = 0;

(31)

4°. the determinant of Ln(q) has simple poles at the points p£~,*ya(ri), and simple zeros at the points Pj7,7s(n + 1).

Algebraic versus Liouville integrability of the soliton systems

59

The last condition implies the following constraint for the equivalence classes of the divisors

[D+] - [D-] = J2 Mn

+ !) - 7-(n)] G JF),

(32)

s

where J(T) is the Jacobian of I\ If 2N > g(r + 1), then the Riemann-Roch theorem and simple counting of the number of the constraints (29)-(31) imply that the functional dimension of £ 7 ( n ) i a ( n )(D + ,£)_) (its dimension as the space of functions of the discrete variable n) equals 2V(r — 1) — gr2 + g + r2. The geometric interpretation of £ 7 ( n ) ) Q ( n )(D + ,D_) is as follows. In the neighborhood of js the space of local sections of the vector bundle V1^a, corresponding to (7, a), is the space Ta of meromorphic functions having a simple pole at ys of the form

/(<>= / ^ L + O C 1 ) .

A S £C.

(33)

z - z(7 s ) Therefore, if Vn is a sequence of the vector bundles on T, corresponding to the sequence of the Tyurin parameters (7(71), a(n)), then the equivalence class [Ln] of Ln modulo gauge transformation (26) can be seen as a homomorphism of the vector bundle Vn+\ to the vector bundle Vn(D+), obtained from Vn with the help of simple Hecke modification at the punctures P^", i.e. [L n ]eHom(V n + i,V n (Z? + )). (34) These homomorphisms are invertible almost everywhere. The inverse matrix-functions define the homomorphisms of the vector bundles [L-^GHomO'n.Vn+i (£>_)).

(35)

The total space £JV(-D+, DJ)\ of the chains, corresponding to all the sequences of the Tyurin parameters, is a bundle over the space of sequences of holomorphic vector bundles £(D+,D-)

^{V„}.

(36)

The fibers of this bundle are just the spaces £ 7 ( n ) )Q ,(„)(D + , £>_). Our next goal is to show algebraic integrability of the total space CN(D+, D_) of the 7Vperiodic chains, Ln — Ln+N (see details in [6]). Equation (32) implies that the periodicity of chains requires the following constraint on the equivalence classes of the divisors D±: N([D+]-[D_])

= 0GJ(T),

which will be always assumed below. The dimension of dimCN{D+,

£N(D+,D-)

£>_) = 2NV{r - 1) + Nr2 + g.

(37) equals (38)

E x a m p l e . Consider the case of 1-periodic chains, i.e. the stationary case Ln — L. Let D+ = K. be the zero-divisor of a holomorphic differential dz, and let CK be a union of the spaces Ci(lC, £>_). Then the factor-space CK/SLr is isomorphic to a phase space of the Hitchin system that is the cotangent bundle T*{M) to the moduli space of rank r stable vector ([5]).

60

Let Ln G equation

IGOR KRICHEVER

£N(D+,D-)

be a periodic chain. Then the Floque-Bloch solutions of the V>„+i = Lnipn

(39)

are solutions that are eigenfunctions for the monodromy operator Tn1pn = i>n+N = Wt/jn,

Tn = Ln+N^i

•• •L„+1L„.

(40)

The monodromy matrix Tn(q) belongs to the space of the Lax matrices introduced in [5], Tn G CND+, T~l G £ND~. The Floque-Bloch solutions are parameterized by the points Q = (w, q), q € r , of the spectral curve T defined by the characteristic equation r-l

R(w, q) = det (w • 1 - TnQ(q)) = wr + ^ r ^ w * = 0.

(41)

i=0

The coefficients ri(q) of the characteristic equation are meromorphic functions on T with the poles at the punctures Pj!~. Equation (7) defines an afhne part of the spectral curve. Let us consider its compactification over the punctures Pjf. As shown in [6], in the neighborhood of P£ one of the roots of the characteristic equation has the form w = (z-

z{P+))~N

(c+ + 0(z - z(P+))) .

(42)

The corresponding compactification point of T is smooth, and will be denoted by P£. In the general position all the other branches of w(z) are regular at P£. The coefficients ri{z) are the elementary symmetric polynomials of the branches of w(z). Hence, all of them have poles at P£ of order N. Note that the coefficient ro(z) — detT„ 0 has zero of order N at The same arguments applied to Ln x show that over the puncture Pk there is one point of f denoted by P^ in the neighborhood of which w has zero of order N, i.e., w = {z-z

(P^))N

(c~ + 0(z - Z(P-)))

.

(43)

Let us fix a normalization of the Floque-Bloch solution by the condition that the sum of coordinates ipQ of the vector ipo equals 1, Y%=i ^o = 1- Then, the corresponding FloqueBloch solution ipn(Q) is well-defined for each point Q of T. Theorem 3.1. The vector-function ipn(Q) is a meromorphic vector-function on T, such that: (i) outside the punctures P^ (which are the points of T situated on marked sheets over Pj!r) the divisor 7 of its poles % is n-independent; (ii) at the punctures P£ and P^ the vector-function ipn(Q) has poles and zeros of the order n, respectively; (Hi) in the general position, when T is smooth, the number of these poles equals g + r — 1, where g = NV(r-l)+r{g-l)

+l

(44)

is the genus ofY. Let SD+'D- be the space of the spectral curves, which can be seen as a space of the meromorphic functions r^z) on T with poles of order N at the punctures P£, and such that

Algebraic versus Liouville integrability of the soliton systems

61

r 0 has zeros of order N at the punctures Pk . The Riemann-Roch theorem implies that gD+,D- j s 0 £ dimension dim5 D +" D - = NV{r -l)-(g-

l)(r - 1) + 1.

(45)

The characteristic equation (41) defines a map CN{D+,D-) —> SD+'D-. Usual arguments show that this map on an open set is surjective. These arguments are based on solution of the inverse spectral problem, which reconstruct Ln, modulo gauge equivalence (26) from a generic set of spectral data: a smooth curve T defined by {r;} G SD+'D-, and a point of the Jacobian J(F). Theorem 3.2. The map described above Ln —> {T, 7} descends to a bijective correspondence of open sets CN(D+,D.) I GL? ^ { f e SD+-D~, [7] G J(T)}. (46) Restricted chains. Let us introduce subspaces CN+Q A~ C CN(D+,D-) with fixed equivalence classes of the divisors of Tyurin parameters

of the Lax chains

[ 7 („)] =C7 + n ([£>+]-[£>_]) e J ( r ) ,

(47)

and with fixed determinant detT = A = ro(q)- The subspace of the corresponding spectral curves will be denoted by S& G SD+'D~. The points of 5 A are sets of functions ri(q), i = 1 , . . . , r — 1, with the poles of order N at P£. For the restricted chains the equivalence class [7] G J ( f ) of the poles of the Floque-Bloch solutions belongs to the abelian subvariety Jc{?)=K1(C

+ N(r-l)[D+]/2),

IT, : J(f) — • J(T).

(48)

Corollary 3.1. The correspondence £^DA

I GLNT <-» { f G SA, [7] G Jc(t)}

(49)

is one-to-one on the open sets. Lax equations. In order to treat the zero-curvature equations (2) as a dynamical system on the space of chains, it is necessary to solve first a part of the equations and define Mn in terms of Lm. Unlike the stationary case considered above that can not be done, if Mn has fixed poles outside the punctures P^ . The singular parts of suitable matrix functions Mn at these points can be constructed locally in a way identical to the theory of discrete zero-curvature equations with the rational spectral parameter. Note that detL„ has simple pole at P£. Therefore, the residue of Ln is a matrix of rank 1, and can be written in the form /ifc(n)p^(n), where hk(n) and Pk{n) are r-dimensional vectors Lemma 3.1. Let Ln be a formal series of the form 00

Ln = /i(n)p(n) T A- 1 + £ ^(nJA* i=0

(50)

62

IGOR KRICHEVER

where h(ri),p(n) are vectors and Xi(n)

are

matrices. Then the equations

n, 4>n+lLn = 4>*n,

(51)

where n and * are r-dimensional vectors and co-vectors over the field of the Laurent series in the variable X, have (r — 1)-dimensional spaces of solutions of the form $n = £ £ ( n ) A \

*; = ££(n)A\

(52)

The equations (51) have unique formal solutions of the form

such that ( $ > „ ) = « $ n ) = 0,

(<#>„) = 1,

(54)

and normalized by the conditions i

Xo(0)=g(-l),

5^x^(0) = 0,

i>0.

(55)

For the proof of the lemma it is enough to substitute the formal series (52) or (53) in (51) and use recurrent relations for the coefficients of the Laurent series. Let us fix a point Po on T and local coordinates in the neighborhoods of the punctures P£. Then the Laurent expansion of Ln at the punctures defines with the help of the previous Lemma the formal series 4>n ,n ,
C = A"ffX(n)AM, # - = A - n f £ c - ( " ) A

(56)

From the Riemann-Roch theorem (see details in [5]) it follows that there is a unique matrix function M„ ' such that: (i) at the points 7 s (n) it has simple poles of the form: ft(n)Q (

Mn =

; f+Q(l),

Z

z.(n) = z(lt(n)),

(57)

Zs\Tl)

where /xs(n) is a vector; (ii) outside of the divisor 7 it has pole at the point P * , only, where M ^

= (z - z{P±)Yl

tk)4>^'±k) + 0(1);

(58)

(iii) MA ' is normalized by the condition MA ' (-Po) = 0. Note, that although <£„ and
Algebraic versus Liouville integrability of the soliton systems

63

Theorem 3.3. (i) The equations daLn = M£+1Ln

- LnM«,

da = d/dta,

a = (±k,l),

(59)

define a hierarchy of commuting flows on an open set of£jy(D+, -D-), which descends to the commuting hierarchy on an open set of £N(D+,D-) j GL^. (ii) The flows (59) are linearized by the spectral transform and can be explicitly solved in terms of the Riemann theta functions. In general the flows (59) do not preserve the leaves of the foliation CN+c A~ C CN{D+,D-). The linear combinations of basic flows which preserve the subspaces of the restricted chains are constructed as follows. Let / be a meromorphic function on T with poles only at the punctures P^. Then we define 6

K^E^"-

( °)

a

where c£ are the coefficients of the singular part of the Laurent expansion

f = Hi±k,i)i.z-^Pt)Tl

+O0).

(61)

df = d/Otf,

(62)

!>0

Theorem 3.4. The equations 8fLn

= M^+1Ln

- LnMl,

define a hierarchy of commuting flows on an open set of CN+£ A~, which descends to the commuting hierarchy on an open set of £N+c A~ / GL^f. Hamiltonian approach. At first glance the construction of the Hamiltonian theory for the periodic chains goes equally well on an arbitrary genus algebraic curve. The two-form Q(z) on £N(D+,D-) with values in the space of meromorphic functions on T by the formula identical to that in the genus zero case. JV-l

il(z) = J 3 Tr {V-l+l5Ln A 8^n) .

(63)

n=0

Let us fix a meromorphic differential dz on T with poles at a set of points qm. Then the formula w = --^vesgttdz, 1= {js,Pjt,qm}, (64) \ei defines a scalar-valued two-form on CN(D+,D-). This form depends on a choice of the normalization of \I>n. A change of the normalization corresponds to the transformation ^i'n = tynV, where V = V(z) is a diagonal matrix, which might depend on a point z of T. The corresponding transformation of Q has the form: n' = n + S(Tr(lnWv)),

v = SVV'1.

(65)

Let XD+,D- be asubspace of the chains £N(D+,D-) such, that the restriction of 5(\nw)dz to XD+'D- is a differential holomorphic at all the preimages on t of the punctures Pk .

64

IGOR KRICHEVER

T h e o r e m 3.5. (i) The two-form w, defined by (64) and restricted to XD+'D-, is independent of the choice of normalization of the Floque-Bloch solutions, and is gauge invariant, i. e. it descends to a form onV = XD+*D- j GL^. (ii) Let j s be the poles of the normalized Floque-Bloch solution tpn. Then g+r—l

w=

J2

Slnw(%)ASz(%).

(66)

By definition, a vector field dt on a symplectic manifold is Hamiltonian, if the contraction id,Lo(X) = uj(dt, X) of the symplectic form is an exact one-form dH{X). The function H is the Hamiltonian corresponding to the vector field dt. The proof of the following theorem is almost identical to the proof of Theorem 4.2 in [5]. Theorem 3.6. Let da be the vector fields corresponding to the Lax equations (59). the contraction of w, defined by (64) and restricted to V, equals i9auj = SHa,

Then

(67)

where H(±k,i) = res p ± (z-z

(P*))~

(lnw) dz.

(68)

The theorem implies that Lax equations (59) are Hamiltonian whenever the form w is nondegenerate. The spectral map (46) identifies an open set of £N+Q A~ / GL^ with an open set of the Jacobian bundle over <SA C SD+'D-, i.e. £%!&-/GL?

^

SA.

(69)

The fibers of this bundle are isotropic subspaces for u. Therefore, the form w can be nondegenerate only if the base and the fibers of the bundle (69), restricted to V, have the same dimension. Consider the case g > 0. Let dz be a holomorphic differential on T. Then, for each branch of w = Wi(z) the differential 5(\nw)dz is always holomorphic at P * . Hence, V = CN{D+,D-)/GL?. Recall that dimSA

= (r-l)(NV-g

+ l),

dimJ(f)

= (r-l)(NV

+ g-l)+g.

(70)

Therefore, for g > 0 the form w is degenerate on V. For 3 = 1 the space V is a Poisson manifold with the symplectic leaves, which are factor-spaces ?*=£%}&

/GL?

(71)

of the restricted chains. In that case dim S& = dim J c ( r ) = NV(r — 1). As in the genus zero case, the arguments identical to that used at the end of Section 4 in [5] prove that the form OJ is non-degenerate on V*.

Algebraic versus Liouville integrability of the soliton systems

65

Corollary 3.2. For g = 0 and g = 1 the form OJ defined by (64) descends to the symplectic form on V*, which coincides with the pull-back of the Sklyanin symplectic structure restricted to the space of the monodromy operators. The Lax equations (62) are Hamiltonian with the Hamiltonians Hf = ] T xesq(lnw)fdz. (72) The Hamiltonians Hf are in involution {Hf,Hh}

= 0.

Now we are in the position to discuss the case g > 1 mentioned in the Introduction. The space V* is equipped by g-parametric family of two-forms ivdz, parameterized by the holomorphic differentials dz on T. For all of them the fibers J c ( r ) of the spectral bundle are maximum isotropic subspaces. For each vector-field df defined by (62) the equation hf^dz = 5HfZ holds. Equation (10) implies that each of the forms u>dz is degenerate on V*. Let us describe the kernel of w<jz. According to Theorem 3.2, the tangent vectors to J c ( f ) are parameterized by the space A(T,P±) of meromorphic functions / on T with the poles at P^ modulo the following equivalence relation. The function / is equivalent to / i , if there is a meromorphic function F e A(t, P*) on T with the poles at Pj^, such that in the neighborhoods of these punctures the function ir*(f — f\) — F is regular. Let Kdz C A(T,P±) be the subspace of functions such that there is a meromorphic function F on T with poles at P^ and at the preimages n*(qs), dz(qs) = 0 of the zero-divisor of dz, and such that / — F is regular at Pj^. Then, from equations (67) and (72) it follows that: / £ Kdz '—• id,^dz = 0. Let Kdz be the factor-space of Kdz modulo the equivalence relation. Then the Riemann gap theorem implies that in the general position Kdz is of dimension 2(g — l)(r — 1), which equals the dimension of the kernel of u>dz- Therefore, the kernel of u>dz is isomorphic to K-dz- Using this isomorphism, it is easy to show that the intersection of all the kernels of the forms Udz is empty, and thus the family of these forms is non-degenerate.

4. Variable base curves Until now it has been always assumed that the base curve is fixed. Let MA he the space of smooth genus g algebraic curves T with the fixed meromorphic function A, having poles and zeros of order TV at punctures Pjr, k = 1 , . . . ,V. For simplicity, we will assume that the punctures P^ are distinct. The space M.& is of dimension dim At A = 2(D + g — 1). The total space £AT,A of all the restricted chains corresponding to these data and the trivial equivalence class C = 0 £ J(T) can be regarded as the bundle over M.& with the fibers v CN+Q A~ = J C J / O ' A - ^ ) - ^ definition, the curve T corresponding to a point (I\ A) 6 At A is equipped by the meromorphic differential dz = din A. The function A defines local coordinate everywhere on T except at zeros of its differential. Let WA be the form defined by (64), where dz = din A and the variations of Ln and \t n are taken with fixed A, i.e. 1

N 1

WA = - 2 E

res

q€l

~

* E ^ (*;rJi(A)*Ln(A) AtfMA)) din A, / = { 7 . , ^ } .

(73)

n=0

Then WA is well-defined on leaves XA of the foliation on £JV,A defined by the condition: the differential <51nw(A)dlnA restricted to XA is holomorphic at the punctures Pk . This

66

IGOR KRICHEVER

condition is equivalent to the following constraints. In the neighborhood of P fc there are (r — 1) regular branches wi of the multi-valued function w, defined by the characteristic equation (41): wf = c f + 0 ( A = F l ) , *= l,...,r-l. (74) T h e leaves X& are denned by 22?(r — 1) constraints: Scf = 0 i—• cf = c o n s t f .

(75)

Note t h a t the differential 5 In w(A)din A is regular at P * for the singular branches of w, because the coefficients c^ of the expansions (42) and (43) are also fixed due to the equation T h e factor-space P A = XA/GL™ is of dimension d i m P A = 2NV(r - 1) - 2V{r - 1) + 2(2? + g — 1). T h e space <SA C S'A of t h e corresponding spectral curves is of dimension dim<S A = (r - 1)(JVP -g + 1)- 2V(r - 1) + 2(2? + g - 1). T h e second and the third t e r m s in the last formulae are equal t o the number of the constraints (75) and the dimension of MA, respectively. For the case r = 2 the last formulae imply the match of t h e dimensions dim P A = 2 dim <SA. For r = 2 the spectral curves are two-sheet cover of the base curves, and the fiber of the spectral bundle is the P r i m variety JQ(F) = Jprim(r). T h e o r e m 4 . 1 . For r = 2 the form w ^ defined by (73), and restricted to P A is nondegenerate. Ifjs are the poles of the normalized Floque-Bloch solution ipn, then g+i—1

WA =

Yl

g+i—1 5lnw

(ls)

A(51nA(7 s ) =

S= l

^

Shiw(js)

A 6h\w(j°),

(76)

3= 1

where a : F —* T is the involution, which permutes the sheets of F over F. For every function f £ , 4 ( r , P * ) the Lax equations (62) are Hamiltonian with the Hamiltonians Hj = 2_\ res g ( / I n w) d i n A.

The Hamiltonians spectral map.

Hf are in involution.

Their common

level sets are fibers J p r i m ( r ) of the

References 1. I. Krichever, D. H. Phong, "On the integrable geometry of N = 2 supersymmetric gauge theories and soliton equations", J. Differential Geometry 45, 445-485 (1997); arXiv:hep-th/9604199. 2. I. Krichever, D. H. Phong, "Symplectic forms in the theory of solitons", in Surveys in Differential Geometry I V , edited by C.L. Terng and K. Uhlenbeck, International Press, 1998, pp. 239-313; arXiv:hep-th/9708170. 3. L. D. Faddeev, L. A. Takhtadjan, Hamiltonian methods in the theory of solitons, SpringerVerlag, Berlin, 1987. 4. A. G. Reiman, M. A. Semenov-Tian-Shansky, "Integrable systems", chapter 2 in Dynamical Systems VII, Encyclopaedia of Mathematical Sciences vol. 16, V. I. Arnold and S. P. Novikov, eds., Springer-Verlag, Berlin, 1994.

Algebraic versus Liouville integrability of the soliton systems

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5. I. Krichever, "Vector bundles and Lax equations on algebraic curves", Comm. Math. Physics 229, 229-269 (2002); arXiv:hep-th/0108110. 6. I. Krichever, "Integrable chains on algebraic curves", arXiv:hep—th/0309255 . 7. N. Hitchin, "Stable bundles and integrable systems", Duke Math. Journ. 54, 91-114 (1987). 8. V. Zakharov, A. Shabat, "Integration of non-linear equations of mathematical physics by the method of the inverse scattering problem. II", Funct. Anal, and Appl. 13, 13-22 (1979). 9. I. M. Krichever, S. P. Novikov, "Holomorphic bundles over algebraic curves and non-linear equations", Uspekhi Mat. Nauk 35, 47-68 (1980). 10. B. A.Dubrovin, I. M. Krichever, S. P. Novikov, "Integrable systems", in Dynamical Systems IV, Encyc. Math. Sciences, Springer-Verlag, Berlin, 1990. 11. A. Tyurin, "Classification of vector bundles over an algebraic curve of arbitrary genus", Amer. Math. Soc. Translat., II, Ser. 63, 245-279.(1967). 12. I. M. Krichever, "The commutative rings of ordinary differential operators", Funk. anal, i pril. 12, 20-31, (1978). 13. E. D'Hoker, I. Krichever, D. H. Phong, "Seiberg-Witten theory, symplectic forms and Hamiltonian theory of solitons", arXiv:hep-th/0212313 .

Mathematical quasicrystals: a tale of two topologies ROBERT V. MOODY (U. Alberta)

In quasicrystals, both mathematical and physical, there is an interplay of two notions of order: a local order and a long-range order manifested through diffraction. These two types of order lead to two very different notions of closeness, the local and autocorrelation topologies, which are in principle unrelated. Given a discrete point set in Rd, each one leads to a dynamical hull. It is exactly the context of the cut and project formalism that marries these concepts. In particular we give a new characterization of regular model sets in terms of a mapping relating the two dynamical hulls.

1. Introduction The distinguishing feature of physical crystals and quasicrystals is their point-like diffraction. This phenomenon is a measure of their internal long-range order. It is particularly remarkable for quasicrystals in as much as they do not seem to be based on repetition of a fundamental cell. By contrast, the physical properties and processes of formation of quasicrystals seem to be largely dependent on their local structure and one can suppose that similar local environments must result in similar physics. These two forms of order — long-range and local — can be put into a setting in which they are seen as a manifestation of two very different topologies. In quasicrystals, physical and mathematical, they are married in a remarkable way. This paper is about these two topologies and what is means for them to live in harmonious co-existence. From the very beginning of quasicrystals, it was observed that indexing the Bragg peaks required more than the expected number of parameters: for example, in the icosahedral cases, six rather than the familiar three for crystals. This suggested that the atomic positions of these substances might be modeled by projecting from a lattice lying in a higher dimensional space. Thus arose the cut and project formalism, which has been a mainstay in attempts to understand and model quasicrystals. The very same mechanism has been seen to be an alternative way to describe many of the well-known tiling models and has played an important role in the development of the mathematics of diffraction in aperiodic sets. In all these cut and project systems two things happen: the two topologies are in coexistence and the resulting structures satisfy the Meyer condition. a In this paper we will see that in fact these two properties characterize the cut and project formalism. In more detail, beginning with a discrete subset A in real Euclidean space M.d we construct two dynamical hulls: X(vl) and A(/l), derived from the local and long-range topologies respectively, each of which carries an R d action. The connection between them, when it exists, is the existence of a continuous mapping f3 : X(yl) —> A(A). In the case that A is Meyer, this mapping exists and is 1-1 almost everywhere if and only if A is a regular model a

I n the case of modulated model sets, the Meyer property may be destroyed.

68

M a t h e m a t i c a l quasicrystals: a tale of two topologies

69

set (see Propositions 4.2, 6.1 for the precise statements). In this case A is also pure point diffractive. Perhaps the most remarkable point about this is the intricacy of the almost 1-1-ness. Except for crystals — in which case (3 is an isomorphism — the set of points at which the map is not 1-1 is dense in A(A).

2. Uniformly discrete sets and two topologies We start by assembling the basic definitions for discrete point sets. We work in Rd with the standard metric. b For c € Rd, we let Br(c) be the ball of radius r about c, with Br := Br(0). Definition 2.1. A subset A C Rd is locally finite if its intersection with every ball Br(c) is finite. Let r > 0. A subset A C Rd is called r-uniformly discrete if all the sets x + Br, x £ A, are mutually disjoint. We will denote by D r the set of all r-uniformly discrete subsets of Rd. This space admits an action by Rd — namely the translation action. For A G D r and x G Rd we denote the x-translate of A by x + A. A locally finite subset A C Rd has finite local complexity if for all R > 0, as c runs over Rd, there are, up to translation, only finitely many sets of the form A fl BR(c). A subset of A G D r is called Delone if it is relatively dense: that is, there is an R > 0 so that for all c e R ^ y l n BR(c) =£ 0. The uniform discreteness is the hard-shell condition for atoms, which is very reasonable. It implies local finiteness. Finite local complexity is also natural enough, though one should note that it imposes restrictions on orientational order. Finite local complexity can also be characterized by the property that A—A

is locally

finite.

(1)

A strong form of this, which we will need and discuss later, is the Meyer condition: Definition 2.2. A C Rd satisfies the Meyer condition if A—A

is uniformly discrete (for some r > 0).

(2)

It is a Meyer set if in addition it is relatively dense. We now come to the two notions of closeness of point sets that are the fundamental concepts of the paper. We will define two topologies, each defined by means of a uniformity — a description of the sets of point set pairs that are to be considered close to one another. Definition 2.3. For each pair (s,BR(c)), define

consisting of a positive real number and a ball,

U(s, BR{c)) := {(A, A') e D r x D r : (v + A) n BR(c) = A' n BR(c), b

for some v e Bs}.

(3)

Almost everything in this paper carries over to the setting of separable locally compact Abelian groups with minor changes.

70

ROBERT V. MOODY

These sets form a fundamental system for a uniform structure on D r whose topology has the sets U(s,BR(c))[A] := {A1 € D r : (A, A') € U(s,BR(c))} (4) as a neighbourhood basis of A. We call it the local topology on D r . This uniformity can also be described by a metric, though it is not particularly natural. The intuition is that two sets are close if after a small shift they agree on a large ball. One should note that the uniformity is certainly invariant under translations, but the individual entourages U(s, BR(c)) are not. The second notion of closeness is based on average matching of the two sets. Definition 2.4. Let A, A' € D r . MA MS r #{(A A A') C) Br) d(A, A') := hm sup —^ —-L '- , r—>oo

._. (5)

VO\.\JtSr)

where A is the symmetric difference operator. It is rather easy to see that d is a pseudometric on D r . We obtain a metric by defining the equivalence relation A = A' ^d{A,A')

=0

and factoring d through it: D r = := D r / =

and d : D r = x D r = —> R> 0

(6)

and note that the translation action of Rd is retained on D r = . We call the resulting topology on D r the coarse autocorrelation topology and denoted by j3 the natural mapping (which is an Remapping): /? : D r —+ D r = .

(7)

Proposition 2.1. i) D r is a complete metric space in the local topology [15,18]. ii) D r = is a complete space and the Hausdorff completion o / D r in the coarse autocorrelation topology [13]. The first of these statements is quite intuitive. The second is trickier since it is hard to see how to build up a point set from a Cauchy sequence of sets in which the matching is improving only in density and not in particular patches of space. A simple example shows why we should not expect (3 to be particularly nice with respect to the local and autocorrelation topologies. E x a m p l e 2 . 1 . Let An := Z\{—n, -n + 1 , . . . ,n}. Then {An} converges to 0 in the local topology and to Z in the autocorrelation topology. If instead An := ([—n, n ] n Z ) U { . . . , — n — 4, —n —2}U{n + 2,n + 4,...} then convergence is to Z in the local topology and the sequence fails to converge in the autocorrelation topology.

Mathematical quasicrystals: a tale of two topologies

71

The coarse autocorrelation topology needs to be modified to put it on an equal footing with the local topology. The local topology is obtained as large coincidence after a small shift. As it stands A, A' can be close in the coarse autocorrelation topology only if they have a large statistical coincidence — no small shift involved. We introduce the autocorrelation topology by defining a uniform structure on D r with the fundamental system of entourages U(s, e) := {(A, A') e D r x D r : d(v + A, A') < e for some v £ Bs} . Proposition 2.2. D r is complete in the autocorrelation topology.

3.

Dynamical hulls

In the dynamics of a physical system one considers a (compact) configuration space or phase space and the time evolution of points of this space. Of special importance is the notion of recurrence, in the sense of the time-orbit of a point returning very closely to the initial state. In the geometry of point sets and tilings a parallel construction on points sets is possible, using the translation action of the underlying space as the group action, instead of time. This provides a powerful way of understanding the internal geometry of a given point set. We begin with D r and its translation action by Rd, take a single point set of interest, A £ D r , and then construct the closure of its orbit under translation: Rd + A. Our two topologies provide two separate interpretations of the orbit closure of A e D r : X(A) = Rd + A in the local topology, A(A) = Rd + A in the autocorrelation topology, which are the local and autocorrelation hulls respectively. Local hulls have been used extensively in tilings and the study of aperiodic sets. The autocorrelation hull is the subject of [13]. It is the interplay between these two that we wish to understand. It is useful to note that both X(A) and A(A) can be interpreted as completions of our starting space Rd. In each case one simply provides Rd with a new topology by pulling back of the uniform topology on D r to Rd using the mapping x t-> x + A. In both cases, points that are usually close in Rd remain so. But now also x, x' e Rd are close if (x + A,x' + A) are close in the local (resp. autocorrelation) topology. It is interesting that A(A) has the structure of an Abelian group, a fact that arises from the group structure of Rd and the translation invariance of the autocorrelation entourages. In particular it has a unique probability measure 9A.. In general X(A) does not get a group structure in this way. We need to know what the basic open sets for this new autocorrelation topology of Rd look like. It suffices to consider open sets around 0, and for these a basis consists of the sets U'(s,e)[0) =Bs xPe where P e := {t G Rd : d{t + A,A)<e}. The elements of Pe are called the statistical e-almost periods of A.

(8)

72

ROBERT V. MOODY

Proposition 3.1. Let A G D r . i) X(yl) is compact if and only if A has finite local complexity, ii) A(yl) is compact if and only if, for all e > 0, Pe is relatively dense. Thus both X(yl) and A(A) become dynamical systems. A dynamical system which is the closure of an orbit is minimal if and only if it is uniformly recurrent (for each point x and each open neighbourhood V of x the set of t in Rd for which t.x is in V is relatively dense in Rd). In the case of X(A) this condition on A is called repetitivity. In the case of A(yl) is comes for free with the compactness, though one can see that in fact it amounts to the relative denseness of each of the sets of e-almost periods Pe.

4. Model sets Fortunately there are large families of examples in which we can get a good feel for what X(yl) and A(A) are like. A cut and project scheme is a triple (Rd,H,L) of locally compact Abelian groups in which L is a lattice in Rd x H and for which the natural projections 7TI,7T2 satisfy TTI\^ is injective, and TC2(L) is dense in H: Rd ^-RdxH

^^H.

(9)

Z We let L := ITI{L) and * : L —> H be the mapping -KI O (TTI|J;) —X- By hypothesis, L is a discrete group and the group T := (Rd x H)/L = {(t,t*) \ t G L} is compact. The obvious Md-action on T (x + (t, t*) + L H-> (x + t,t*) + L) makes it into a minimal dynamical system. A regular model set (denned by the cut and project scheme (9)) is a non-empty set of the form A = x + {t G L : t* G W} where W C H is compact and satisfies the conditions

W =W

and

eH(dW)=0

and where OH is Haar measure on H. It is possible to replace the cut and project scheme by one with a smaller H if necessary, so that for u G H, u + W = W if and only if it = 0 [17]. We will assume that this condition holds in what follows. The regular model set A is generic if dW n L* = 0. Generic model sets are repetitive. 0 It is not hard to show that model sets are always Delone sets [11,12] the uniform discreteness coming from the compactness of W and the relative denseness from its non-empty interior. In fact model sets are even Meyer sets, as can be seen by applying the previous sentence to the model set A - A = {t G L : t* G W - W}. Regular model sets have uniquely defined autocorrelations,11 so we can consider the autocorrelation group A(A). A key point is that A(A) and T are isomorphic as topological groups [13], so in fact T(A) for a regular model set has a very natural interpretation — namely the completion of the orbit of A under the autocorrelation topology. c d

Model sets were introduced by Y. Meyer [9] in his study of harmonious sets. T h e autocorrelation can be expressed, using a Weyl-type theorem for uniform distribution [12,17], as an integral over H which is dependent only on W.

Mathematical quasicrystals: a tale of two topologies

73

Proposition 4.1. [13] Let A be a regular model set of the cut and project scheme (9). Then A(A) ~ T(yl) and the isomorphism is also a G-mapping. Now using the torus parameterization of Schlottmann [18] we have a fundamental result connecting our two topologies. Proposition 4.2. [18] Let A be a regular repetitive model set. Then the mapping /3 of (7) provides a continuous surjective W1,-mapping j3 : X(A) —» A(A). This mapping @ is 1-1 almost everywhere, in the sense that the set of elements ofA(A) which there lie more than one point of\(A) is of Haar measure 0.

(10) over

It is interesting to look at the nature of the 1-1-ness condition. We have pointed out that A(A) ~ T. Now T may be viewed as a parametrization of the different sets A(x, y) := x + {t G L : t* G — y + W} that are obtainable from W by changing varying x and y: A(x,y) = A(x',y') if (x, y) = {x',y') mod L. Each point set A' in X(A) is repetitive, since A is, and if )3(A') = (x, y) mod L then x + {t<EL:t*

£-y

+ W°} C A' C x + {t G L : t* G -y + W} ,

that is, it is determined by some set — y + W between the interior of — y + W and — y + W itself. The situation is easy if (— y + dW) D L* = 0 for then there are no points of A coming from the boundary and there is only one possibility for A'. But if t* G (—y + dW) D L* then there are at least two preimages under (3, one of which has t and one which does not. As y varies over H there is a dense set of translates — y + W for which —y + dW meets L*, whence the singular cases are inextricably bound in with the non-singular cases. For the construction of the mapping X(A) see [18]. It was noted by Anderson and Putnam [1] that locally X(yl) has the structure of an open ball in Rd times a Cantor set. To see this let us suppose that 0 G A. The set BT x Z where Z is all the points A' G X(A) for which 0 G A' is an open set neighbourhood of A in X(A). Furthermore, for each finite patch S of points of A containing 0, the set of points A' of X(A) containing S is a both open and closed in the induced topology on Z, so Z is totally disconnected. Z itself is closed in X(/l), hence compact, and it is also perfect, so a Cantor set. The cut and project formalism is sometimes criticized because of its introduction of nonphysical internal dimensions. In fact these internal dimensions are of completely physical origins (that is, determined by the geometry of the model set). We have just seen that T is none other than A(A) which is the completion of R d under the topology that considers the long-range order of A. We also have a remarkably simple description of the internal space H. Consider the subgroup L of M.d generated by the set of differences A — A. Supply this with the topology of the (coarse) autocorrelation — that is the topology for which an open neighbourhood basis of each point t G L is given by the sets t + Pe, e > 0. Then H is the completion of L under this topology [3]. So the internal space is simply a reflection of the almost periodic structure of A.

74

ROBERT V. MOODY

5. Diffraction The diffraction of a distribution of density in space is usually described as the square absolute value of some suitably normalized Fourier transform of this density. Alternatively it is the Fourier transform of the volume averaged autocorrelation. For finite sets of scatterers or for crystals, both lead to the same thing. With quasicrystals one has to be careful — because of issues of convergence, only the second one makes mathematical sense. Formally the definitions are as follows: Let A £ D r be locally finite. We represent this set as a non-zero regular positive Borel measure in the form of a Dirac comb

where 5X is the unit point (or Dirac) measure located at x.e Define wR = w| B and CoR = (UR)~- Then, the measure y(X>

7

"

. .UR**R

•

l

V-

vol(B fl ) ~ vol(B fi )

g

V

is well defined, since it is the (volume averaged) convolution of two finite measures.5 The autocorrelation ~/u of u; exists and is the limit of (7L ') in the vague topology as R —+ 00, if this limit exists [7]. It is then a positive definite measure with 7W = YlteA-A vW^t where the coefficients are given by rj(t)= 'V

lim R-^oo

}

Y

Vol( BR)

^

1= lim R^oo

* VO[(BR)

.#(AD(t K

+ A)nBR). K

'

(11) '

X

'

x-y=t

The second inequality uses the fact that as R becomes large the error caused by the mismatching of BR and t + BR becomes negligible with respect to the volume of BR. The function 7? can be extended from A — A to all of Md by setting r)(t) = 0 for all other t. It is a positive definite function. What is particularly relevant here is the fact that for all t,

d(A,t + A)= lim *{{A*£?BR) R—»oo

= 2(??(0)

- *'» '

(12)

WOl\tiR)

as can be seen from A A (t + A) = (A U (t + A))\A n (t + A). Thus the metric d and the autocorrelation coefficients 77(f) are directly related, whence our choice of name for the autocorrelation topology. The autocorrelation measure 7W is a positive definite, translation bounded measure and, as such, has a well-defined Fourier transform 7^ (also defined on Md), which is a positive and also translation bounded measure [4,7]. e

In a more realistic setting these points might be weighted to represent different scattering intensities or convolved with some atomic profiles. f If / is a function on R d then / is the function / ( x ) = /(—x) and for a measure /i, the measure p, is defined

by M) = Kf) for all /.

Mathematical quasicrystals: a tale of two topologies

75

Definition 5.1. j w is the diffraction of w. The pure point part of this measure is called the Bragg spectrum. The measure w is pure point diffractive if j u is a pure point (also called discrete or atomic) measure on R d . Proposition 5.1. [6] The measure u is pure point diffractive if and only if the autocorrelation measure j u is strongly almost periodic. This key result of the theory introduces a new concept — strong almost periodicity — which by virtue of its appearance here is crucial for the fundamental understanding of pure point diffraction. However it is not particularly intuitive. Fortunately, under the assumption that A — A is uniformly discrete (the Meyer condition), we have Proposition 5.2. [3] The autocorrelation measure 7W of u> is strongly almost periodic if and only if, for all e > 0, Pe is relatively dense. Thus, for our point set A, pure pointedness of the diffraction hinges around the relative denseness of the sets Pe — in other words the compactness of A(A).

6. The main theorem Proposition 6.1. [2] Let A C Rd be a Meyer set with a well-defined autocorrelation. Suppose that j3 : X(A) —> ^(^1) is continuous and 1-1 almost everywhere. Then A differs in density 0 from a regular model set. Furthermore, if A is repetitive then i) ii) Hi) iv) v)

X(7l) is the hull of a regular model set; X(/l) is uniquely ergodic and minimal (strictly ergodic); A is pure point diffractive; X(vl) has pure point dynamical spectrum; X(A) has continuous eigenfunctions.

j3 is a bisection if and only if A is a crystal. Thus model sets are characterized as Meyer sets for which a good map from ~K(A) to A(A) exists. All of ii) through v) are consequences of being a regular model set. It is conjectured that, along with the Meyer condition, collectively they are equivalent to the existence of the existence of the continuous almost everywhere 1-1 map /3. In outline, the proof of proposition 6.1 for repetitive A may be sketched as follows. The Meyer property implies finite local complexity (see equations (1) and (2)) and hence compactness of X(yl) (see proposition 3.1). Since (3 is continuous, A(A) is compact, whence by proposition 3.1 the sets Pe of almost periods are relatively dense. This implies pure pointedness, but more importantly, it provides us with a way to create the cut and project scheme. We let L be the subgroup of Rd generated by the set A — A and give it the group topology for which the sets Pe form a neighbourhood basis at 0. This is a uniform topology and its Hausdorff completion <j> : L —> H provides us with the internal group H. The subgroup L := {(t, (f>(t)) : t E L} of Rrf x H is a lattice and R d , H, L) is the cut and project scheme.

76

ROBERT V. MOODY

Its *-map is the mapping . One proves that A(yl) is isomorphic to the group T emerging from the cut and project scheme. There is no harm is shifting A so that it contains 0, whence it also lies in L. Define W C H as the closure of A* in H. One proves that W is compact and A C {t £ L : t* £ W}. The hard part is to show that this is equality. In fact it is not true in general. But it is true that if (x, y) + L is a point at which j3 is 1-1 then over that point, the corresponding set A' is o

the model set x + {t £ L : t* £ — y + W} and W = W. The uniform distribution properties of [12] and the almost everywhere 1-1-ness can be used to show that the boundary of W has measure 0. Hence X(A) = X(yl') is the local hull of a regular model set.

7. Additional thoughts The point of the paper is to show the relationship of the nature of quasicrystals to the intertwining of topologies of long-range and local order. Prom a physical point of view such a connection seems to be a basic underlying assumption. If small samples of quasicrystalline materials are supposed to be representative of some materials that can be extended indefinitely in space, and if the long-range order is supposed to be a defining characteristic of these materials then there has to be some notion of convergence in long-range structure on the basis of local convergence of structure. The present version of this work was intended to shed light on the nature of model sets, and as such makes some assumptions that are physically unrealistic. The primary of these assumptions are the treatment of atoms as points of equal scattering intensity and the very strong assumption of the Meyer condition. It might be as well to show what this innocent looking condition implies. Suppose that A is a Meyer set. Then A — A is uniformly discrete. A consequence of this (due to J. Lagarias) is that A — A C A + F for some finite set F. A consequence of that is that A ± • • • ± A (any fixed number of terms) is also uniformly discrete. For a whole circle of equivalences one may consult [10] or the starting point for all these results [9]. Recently Strungaru [14] proved that for any Meyer set (even weighted Meyer sets) the support of Bragg spectrum is relatively dense in R d . In other words all Meyer sets are, in the general sense of a pervasive Bragg spectrum, quasicrystals. The rigidity of Meyer sets, and even finite local complexity, limits the applicability of this work in understanding physical quasicrystals. Its value is more in pointing out the way in which the tension between long and short range order can be resolved in a nice mathematical setting Future work will focus on removing the Meyer condition and relaxing the notion of the point sets under consideration to measures. Generalizing in this way forces one to generalize the local and autocorrelation topologies and will create new versions of X and A. But one expects that some version of proposition 6.1 will emerge. A multi-colour version of the characterization of model sets by dynamical systems appears in [8].

Acknowledgments Thanks to Michael Baake, Daniel Lenz, Nicolae Strungaru, all of whom have contributed substantially to this project. I would also like to thank the Natural Sciences and Engineering

Mathematical quasicrystals: a tale of two topologies

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Research Council of C a n a d a a n d t h e Banff International Research Station for supporting this research.

References 1. J. Anderson, I. Putnam, "Topological invariants for substitution tilings and their C*-algebras", Ergodic Th. and Dynam. Sys. 18, 509-537 (1998). 2. M. Baake, D. Lenz, R. V. Moody, "A characterization of model sets", in preparation. 3. M. Baake, R. V. Moody, "Weighted Dirac combs with pure point diffraction", preprint arXiv: math.MG/0203030. 4. C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, 1975. 5. N. Bourbaki, Elements of Mathematics: General Topology, Chapters 1-4 and 5-10, reprint, Springer, Berlin, 1989. 6. J. Gil de Lamadrid, L. N. Argabright, Almost Periodic Measures, Memoirs of the AMS vol. 428, AMS, Providence, RI, 1990. 7. A. Hof, "On diffraction by aperiodic structures", Comm. Math. Phys. 169, 25-43 (1995). 8. Jeong-Yup Lee and R. V. Moody, "Characterization of model multi-colour sets", Annales Inst. Henri Poincare, in press. 9. Y. Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland, Amsterdam, 1972. 10. "Meyer sets and their duals", in The Mathematics of Aperiodic Order, ed. R. V. Moody, NATO ASI vol. 489, 1997, pp. 403-441. 11. R. V. Moody, "Model sets: a survey", in From Quasicrystals to More Complex Systems, eds. F. Axel, F. Denoyer and J. P. Gazeau, EDP Sciences, Les Ulis, and Springer, Berlin, 2000, pp. 145-166; arXiv:math.MG/0002020. 12. R. V. Moody, "Uniform distribution in model sets", Can. Math. Bulletin 45, 123-130 (2002). 13. R. V. Moody, N. Strungaru, "Point sets and dynamical systems in the autocorrelation topology", preprint (available on the author's website). 14. N. Strungaru, "Almost periodic measures and long-range order in Meyer sets", Disc, and Comp. Geometry, in press. 15. C. Radin, M. Wolff, "Space tilings and local isomorphism", Geometriae Dedicata 42, 355-360 (1992). 16. W. Rudin, Fourier Analysis on Groups, Wiley, New York, 1962; reprint, 1990. 17. M. Schlottmann, "Cut-and-project sets in locally compact Abelian groups", in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs vol. 10, AMS, Providence, RI, 1998, pp. 247-264. 18. M. Schlottmann, "Generalized model sets and dynamical systems", in Directions in Mathematical Quasicrystals, eds. M. Baake and R. V. Moody, CRM Monograph Series vol. 13, AMS, Providence, RI, 2000, pp. 143-159.

Strings through the microscope V O L K E R SCHOMERUS (SPhT

Saclay)

Over the last few years, string theory has changed profoundly. Most importantly, novel duality relations have emerged which involve gauge theories of brane excitations on one side and various closed string backgrounds on the other. In this lecture, we introduce the fundamental ingredients of modern string theory and explain how they are modeled through 2D (boundary) conformal field theory. This so-called 'microscopic description' of strings and branes is an active research area with new results ranging from the classification and construction of boundary conditions to studies of 2D renormalization group flows. We shall provide an overview of such developments before concluding the lecture with an extensive outlook on some present and future research that is motivated by current problems in string theory. This includes investigations of non-rational and non-unitary conformal field theories.

1. Introduction During the last years several new elements have entered the picture of string theory and they have inspired many novel ideas in a variety of fields, including high energy physics and cosmology. The stringy image of our world (see figure 1) contains super-gravity propagating in a 10D background with some dimensions being compactified. In addition there are branes stretching out along p + 1-dimensional hyper-surfaces. Excitations of these branes give rise to gauge theory and matter that can propagate along the brane's world-volume. As inspiring as this picture has been for many recent developments in physics, it can also be misleading, especially when applied to some extreme situations in which the space becomes strongly

yi

I J

1 1

/

1

\

/ Figure 1. The picture of modern super-string theory contains closed strings which propagate in a 10D background, p-branes stretching along p + 1-dimensional surfaces and open strings whose end-points move within the branes' world-volume.

78

Strings through the microscope

79

curved or even singular. It is therefore crucial to keep in mind that such an image of the world only arises in limit of string theories in which the involved length scales are long compared to the string length and hence that an important task in string theory is to develop techniques which allow computing stringy corrections to the picture we sketched above. This goal has been a fruitful challenge for more than two decades now and it has lead to many new insights within the last years. Such developments are the main focus of this lecture. To begin with, we shall take a much closer look at the various elements of figure 1, enlarging them so that we can see the strings' finite extension. In the process we shall learn how to model the picture in mathematical physics. This will involve the whole wealth of boundary conformal field theory (CFT), a domain that is also known for its beautiful applications to critical phenomena (see e.g. the lecture of Smirnov for some recent applications). As we scan over the picture of the world, we shall gradually set up a dictionary between its various elements and concepts of 2D boundary conformal field theory. After this introductory part we shall start discussing some of the recent string inspired developments in boundary conformal field theory. Our presentation will follow the traditional devision into rational vs. non-rational conformal field theory. The former is relevant for the study of strings on compact components of the 10-dimensional world. This subject has seen an enormous boost in recent years and by now there exist very powerful techniques that are essentially universal, i.e., apply to very large classes of compact target spaces. Non-rational conformal field theory, on the other hand, is relevant for string theory in non-compact spaces. As we will discuss, such non-compact backgrounds are an essential ingredient in string theory duals of (large TV) gauge theory. They are also vital for studies of time-dependent string backgrounds. Applications to these two domains have pushed non-rational conformal field theory to the forefront of research in string theory.

2. Strings, branes and boundary conformal field theory The main purpose of this section is to zoom in on the elements of the picture we sketched above, to understand what they look like in full string theory and how they are modeled in mathematical physics. 2.1. Closed strings and bulk conformal field theory The first element we are going to enlarge is a junction at which three closed strings come together (see figure 2). Our aim here is to assign a number to this junction which we can interpret as an amplitude for the joining/splitting of closed strings. This number depends on the particular background we consider and on the states <j) of the three closed strings that participate in the process. The latter decorate the three external legs of figure 2 and they are taken from an infinite set of possible closed string modes. When we deal with strings in flat space, for example, states are characterized by a center of mass momentum and an infinite variety of different vibrational modes. To assign an amplitude to the vertex, we consider the latter as an image of a 3-punctured 2-sphere P 1 under the parametrization map (X^(z,z))n=o,i,...,9- Being the parametrization of the closed string's world-surface, XM are components of a 10-dimensional bosonic field

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Figure 2. The interaction 3-vertex of three closed string modes $ „ is obtained from a 3-point function of a 2D conformal field theory on a closed surface P 1 . The fundamental fields of the 2D conformal field theory arise through the parametrization of the closed string's world-surface.

with action SCS[X} = S[X] = ^Lj

[ dzdz (g^(X)

+ B^(X))

3X»dXv

+ •••

(1)

where the dots stand for additional terms e.g. involving Fermions. If our strings propagate in flat space then the background metric g and B-neld B are constant. This implies that the action is quadratic and computations in the resulting free field theory can be reduced to Gaussian integrals. For more general backgrounds, however, both g and B depend on the coordinates of the background and we are dealing with (a special class of scale invariant) 2D non-linear u-models. Our discussion here has brought us to the first entry in our dictionary: it is claimed that possible closed string backgrounds are associated with 2D conformal field theories on closed surfaces. Closed string modes cf> correspond to fields <J>(z, z) in the conformal field theory. The 3-point functions of such fields are determined by conformal symmetry up to some constants ( * i ( z i , z i ) $2(22,-22) $3(23,23)} =

C(l,2,3)

\z12r"

12

12 A;23

1*23 r " 2 3 |zi3

where z^ = Zi — Zj and the exponents Ay are certain linear combinations of the scaling dimensions Aj of the fields <3>j. Along with the spectrum of these scaling dimensions A;, the 3-point couplings C are known to contain all the information about the conformal field theory. They obey various consistency conditions which might be difficult to solve, starting from the seminal paper by Belavin et al. [9], many solutions have been constructed. In string theory, the 3-point couplings provide the amplitude of the 3-point vertex, i.e., they tell us how likely it is that two closed string modes <j>i, 4>2 combine into a single closed string in the mode
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81

objects are very much like extremal black holes, only that they extend in p spatial directions. Since closed string theory is considered as an interesting candidate for a consistent short distance completion of gravity, we are lead to the obvious problem of finding a description for branes in string theory. To answer this question let us now enlarge another element of our image (see figure 3). We claimed above that branes are charged and massive objects. As such, they can interact with various other objects in the bulk, e.g. through exchange of gravitons and higher closed string modes. When such a closed string mode hits the brane or is emitted from it, we obtain the picture shown in figure 3. The parametrization of the closed string world-surface now

Figure 3. The interaction of closed strings with a brane is related to the bulk 1-point function in a 2D conformal field theory on a disc. Different branes correspond to different boundary conditions imposed along the boundary of the disk.

involves a map from a disc to the 10-dimensional background such that the boundary of the disc is embedded into the world-volume of the brane. We ensure this by imposing Dirichlet boundary conditions for all components of (X11) which are associated with directions transverse to the brane. Our discussion here motivates the following general proposal that was first formulated by Polchinski [45]: branes in some closed string background correspond to conformally invariant boundary conditions of the associated conformal field theory. It is well known that boundary conditions in conformal field theory can be characterized by the 1-point functions of bulk fields $. Using once more conformal invariance and a conformal mapping from the disc to the upper half-plane, the 1-point functions are easily shown to possess the following general form

<$CM)r- =

flBUW v _

r|2AA

In other words, conformally invariant boundary conditions are uniquely determined by the 1-point couplings B((p). The latter provide a measure for how strongly a given closed string mode (j> couples to the brane and hence in particular encodes information on the mass and charge of the brane. 2.3. Open strings and boundary fields At this point we are still missing the open strings that we would expect to be around as soon as we introduce branes. This is indeed the case. To argue that open strings are indeed part

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of our present setup, a brief look at lattice spin models with boundaries may be helpful. The latter are closely related to the 2D continuum field theories we are dealing with. Moreover, it is intuitively obvious that such lattice systems contain a set of excitations which can only live along the boundary and which depend on the specific boundary condition we impose. In the 2-dimensional Ising model with free boundary conditions, for example, we can measure the boundary magnetization. Once we fix the boundary spins, however, measurements of this quantity become trivial and hence do not correspond to an observable of the model. In continuum field theory, boundary excitations are described by fields which can only be inserted at points along the boundary, i.e., to so-called boundary fields. These are exactly the objects that we need in order to model open strings. As in the case of closed strings, there exists an infinite number of open string modes tp and to these we assign boundary fields ^f(u) of the corresponding boundary conformal field theory. The spectrum of open string modes depends on the brane we consider, just as the spectrum of boundary fields depends on the boundary condition we impose along the boundary. There is one new set of couplings that comes with the vertex of three open string modes (see figure 4). The computation of such a vertex involves a disc with three fields inserted

V2 Figure 4. The interaction 3-vertex of open string modes ty„ is obtained from a 3-point function of boundary fields in a 2D conformal field theory on the disc or half-plane.

along the boundary. Once more, this amplitude is determined by conformal invariance up to a set of 3-point couplings,

2 3 7/.. tm / , „ &23 |7W l 2 T1122 |lW |Ul3 |A 2 3 23r A

where and the u, are coordinates for the boundary of the upper half-plane. Needless to say that the 3-point couplings O encode the probability of two open string modes ipi,ip2 to combine into a single open string in the mode ip^. This concludes our journey through the various elements of figure 1. Along the way we have learned to characterize boundary conformal field theories through three different sets of couplings, the 3-point couplings C of the bulk fields, the couplings B appearing in the 1-point function of bulk fields and the 3-point couplings O of boundary fields. This is a rather abstract way to think about boundary conformal field theories, especially when compared with the action (1) for closed strings that we started our discussion with. Since it is sometimes helpful to think about conformal field theories in terms of such actions, we

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83

would like to mention that in the case of open strings, S[X] gets modified to Sos[X] = S[X] + - ^ - 7 / duAll(X)duX'i(u) Iva! Jan

(2)

where the first term is the same as in formula (1) with E being replaced by a surface with boundary. The new boundary term means that open strings can couple through the velocity duX** of their endpoints to a new background field, namely to a vector field A^(X). We interpret A as a gauge field on the brane and think of the string endpoints as carrying charges. The action (2) is to be supplemented by Dirichlet boundary conditions on X for all directions transverse to the brane. 2.4. Instabilities, dynamics and RG-flows In the last three subsections, our dictionary has received quite a number of entries, but there are a few more that we would like to add in passing. To motivate these additional entries, we note that many of the existing string backgrounds are actually unstable. This applies e.g. to the 26-dimensional bosonic string theory and it is a common phenomenon in the context of branes. In fact, configurations of several stable branes tend to be unstable, the most famous example being the pair of a brane and its anti-brane. In general relativity we can easily detect instabilities of some given static solution to the classical field equations. All we need to study is the spectrum of fluctuations around the solution. Modes that are exponentially enhanced are associated with instabilities. Let us now see how we can find unstable modes in string theory. To this end we consider some static background. Its conformal field theory description involves a product of a time-like free field X and a unitary conformal field theory whose target is the 9-dimensional spatial slice of the static background. This theory possesses an exponentially growing mode if we can add the following conformally invariant terms to the action 5S~

[ dzdzee*x°iz'*)$(z,z)+ JT,

[ duee*x°(u)y(u), JdT,

(3)

where $(z,z) and $f(u) are bulk and boundary fields in the conformal field theory for the spatial slice and the coefficients e$ and e* must be real in order to have an exponential growth with time. But such exponential fields of a time-like free boson possess a positive scaling dimension so that the fields $ and \I> must have scaling dimensions A<j> < 2 and A>j, < 1 if we want SS to be scale invariant. In other words, instabilities of a closed string background and branes therein correspond to relevant bulk and boundary fields, respectively. If we wanted to control the full decay process generated by an instability, we would have to solve the theory with interaction 5S. At present, no example of such a theory has been constructed within 10D string theory. Nevertheless, some insights into aspects of decay processes have been obtained using a proposed relation with renormalization group (RG) flows. This requires, however, that we reduce our ambitions and only try to identify the final state of the decay process rather than the whole dynamical evolution. Let us consider an initial state which is encoded in a CFTi for the 9-dimensional spatial slice of the static background. Next we choose some relevant bulk or boundary field $ or * to trigger the decay. After all radiation has has escaped, we expect our system to settle down in a static

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final state that is again described by a product of a time-like free boson and a CFT2 for the spatial slice of the final stable state. Hence, the dynamics has lead us from some CFTi to CFT 2 with the help of relevant fields $ or \I>. It is obviously tempting to think that CFT 2 is the IR fixed point of the RG flow from CFTi on the RG trajectory that is generated by adding $ or \I>. Supporting evidence for this proposal is strong in the case of boundary instabilities, but the proposal seems much harder to justify when we deal unstable bulk theories. A more thorough discussion of these issues and many further references can be found in the literature [27]. 2.5. Summary: dictionary between ST and B C F T Let us pause for a moment and review all the entries in our dictionary between string theory and 2D boundary conformal field theory: Closed string background X

2D CFT Cx on closed surface E

Closed string mode <j>

Bulk field $(z, z), 2 e E, in Cx

Closed string vertex C

3-point function of bulk fields

Brane B in background

Boundary condition (BC) for Cx => BCFT Cx on open surface £

Open string mode ip

Boundary.field * ( u ) , u G <9£, in Cx

Open string vertex 0

3-point function of boundary fields

Instability of a background

Relevant bulk- or boundary-field

Initial/final state of decay

UV/IR fixed point of an RG-flow

Recall that the last line has the status of a conjecture. To test our dictionary one may form sentences in string theory (or gravity) and check that they translate into meaningful sentences of (boundary) conformal field theory.

3. Rational BCFT and strings in compact backgrounds We are now prepared to begin reviewing results on the explicit construction of boundary conformal field theories. Systematic rational conformal field theory model building usually starts with theories whose target space is a compact group manifold G. It then proceeds to cosets and orbifolds. Among them one finds all known models with interesting applications to statistical physics and string theory. Here we shall explain the most central results of the field using one special example, namely the group G =SU(2)= S3. A few comments on various generalizations and extensions are collected at the end. 3.1. Strings on the 3-sphere Before we look at strings moving on a 3-sphere, let us recall that, to leading order in a', cr-models of the form (1) are conformal invariant if the background fields satisfy

Strings through the microscope

a'Tl^(g) - jH^HJT

85

= 0(a'2).

Here, 7c is the curvature of the background metric g and H = dB. Hence, if strings move in a curved background, then a non-vanishing magnetic field B is unavoidable. With this in mind let us consider strings on a 3-sphere S3 = SU(2). Their world-surfaces are parametrized by a group valued map h : E —> SU(2) and these parametrization fields appear in an action of the form

S[h] = -£- f dzdz tr(h-ldh){h-l8h) 47T J

+—

/ V ^ / i " 1dh)*\

(4)

127T J

The second term is known as the Wess-Zumino-Witten (WZW) term. Locally, it may be rewritten in the form of the second term in equation (1). We also note that the parameter k is a measure for the size of the 3-sphere and that, in the quantum theory, k must be integer. The WZW model possesses a large symmetry, given by two commuting actions of the affine Lie algebra su(2) k . Each of these two algebras is generated by the Laurent modes J%,n G Z, of an su(2)-valued conserved current J = J^t^ with relations {JZ,JZl}=ifl/pJZ+m

+ kn6^6n^m.

(5)

The two affine Lie algebras act on the space of fields in the theory, extending the actions that are induced by the usual left and right translation of the group on itself. It turns out that a wide class of boundary conformal field theories can be written down in terms of data from the representation theory of their infinite dimensional symmetries. The most important such data are the set J of unitary representations, the so-called modular S-matrix, the Clebsch-Gordan multiplicities N of the fusion product and an infinite dimensional generalization of the 6J-symbols which is known as the Fusing matrix F. For the SU(2) affine Lie algebra, explicit formulae can be spelled out without much effort. In this case, the requirement of unitarity leaves us with just a finite number of representations. We label them through j € J = { 0 , 1 / 2 , . . . , k/2}. Their conformal weights are given by AJ = J(J + l)/(k + 2) and for the modular 5-matrix one finds S

Sln ij = \l Ik- TJ7 + 2""

TT(2» + 1 ) ( 2 J + 1 )

T k +^ 2

(6)

•

With the help of the Verlinde formula it is not difficult to compute the following fusion rules TV from the modular S'-matrix, for k-

\i- j\,...,min(i+j,

k-i-j),

otherwise. They are similar to the Clebsch-Gordan multiplicities of the Lie algebra su(2), apart from the truncation which appears whenever i + j > k/2. Formulae for the fusing matrix also exist in the literature. Since they are a bit more involved, we shall not present them here. Let us only mention one property concerning their limiting behavior as we send k —» oo,

hmM^]=U^}.

W

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This concludes our list of representation theoretic data for the affine Lie algebra. We shall see these quantities again in a moment when we write down formulae for the couplings B of closed strings to branes on S3, for their open string spectra and the 3-point vertices O of open string modes. 3.2. Branes on group manifolds Many constructions of conformal invariant boundary theories are ultimately based on fundamental observations made by J. Cardy [13]. When applied to the case at hand, they provide us with a finite set of k + 1 boundary conformal field theories which we label by J = 0 , 1 / 2 , . . . , k/2, just as we enumerate the unitary representations of the corresponding affine Lie algebra. Recall that these boundary theories can be uniquely characterized through the couplings B that appear in the 1-point functions of the bulk fields. According to Cardy's solution, these 1-point couplings are simply given by the matrix elements of the modular .S-matrix,

{ 2 J =

^^

7kw^

(9)

where Aj = j(j + l)/(k + 2). The superscripts a,b = —j,... ,j, placed at the symbol label different components within a tensor multiplet (j)j. One should think of these labels as the quantum numbers for ground states of the closed string. They are the same labels that appear on the matrix elements DJab(g) of group representations, except that the label j is cut off at j < k/2. Geometrically, these boundary theories were found [3] to describe branes that are localized along a discrete set of conjugacy classes of SU(2), i.e., along 2-spheres around the group unit e G SU(2). They come equipped with a magnetic field of the form B ~ tr h^dh

A ^d,h+1

h^dh,

(10)

where Ad denotes the adjoint action of the group on its Lie algebra. For higher groups G, such a field B is gives a non-trivial potential for the pull-back of the canonical WZW 3-form to the branes' world-volumes. Such 2-form potentials on conjugacy classes of a group G were considered in the mathematical literature. There they appear in connection with deformations of the theory of co-adjoint orbits. Our geometrical interpretation of the boundary theories with 1-point functions (9) may be justified in several different ways. The argument we want to give here is based on the observation [21] <*f (*. *)>J(k) " ~°° /

dn(g) S(ti(g) - tf0) 4>f(g),

(11)

JSV(2)

where dfi(g) denotes the Haar measure on SU(2), ^(g) ~ DJab(g) are the wave functions of the lightest closed string modes and fl is the azimuthal angle on the 3-sphere. In taking the limit, we allowed the boundary label J to depend on the level k and we defined -do := 27rlim J(k)/k. Since 0 < J < k/2, the angle ??o lies in the interval [0,n]. The appearance of the ^-function on the right hand side shows that closed string modes indeed detect a spherical object which is localized at the azimuthal angle •& = i90.

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87

Figure 5. Maximally symmetric branes on a 3-sphere are localized along conjugacy classes of SU(2), i.e., they are either point-like or they wrap a 2-sphere in S 3 . Such spherical branes exist only for a discrete set of radii.

3.3.

Open strings on group manifolds

Let us now turn to the open strings which can propagate along the k + 1 different spherical branes on 5 3 . According to our dictionary, this means that we want to obtain the set of boundary fields and the 3-point couplings O for each of the above boundary theories. Following general arguments, it is possible to show that the space of boundary fields carries the action of a single affine Lie algebra rather than two commuting actions as in the case of bulk fields. This reflects a similar reduction of the geometric symmetries: whereas there exist two commuting sets of (left and right) translations on SU(2), only a special combination, namely the adjoint action, leaves the conjugacy classes invariant. Under the action of the affine Lie algebra su(2) k , the space Hj of boundary fields decomposes as Hj = HJJ = Q.NjjiV,

(12)

where Vj, j = 0 , 1 / 2 , . . . , k/2, denote irreducible unitary representations of the affine Lie algebra su(2) k , and where NJJ3 are the associated fusion rules (see equation (7)). Note that only integer spins j appear on the right hand side of equation (12) and that in the limit k —• oo, the summation on the right hand side is truncated at j m a x = 2 J. This means that the decomposition of TLj is as close as it can be to the decomposition of Mat (2 J + 1) into su(2) multiplets. In fact, there is a correspondence between the spin j multiplets of matrices Yj £ Mat(2J + l) and ground states in V, C "Hj. It is also worth pointing out the similarity between the labeling of open string ground states and spherical harmonics Y£(ip,6). Only the cut-off at j m a x = 2 J on the spin j does not appear for functions on a 2-sphere. Boundary fields V"? associated with ground states of the open string are labeled by the representation j = 0 , 1 , . . . , j m a x and o = —j,..., j . Their 3-point vertices were found [4] using important previous work by Runkel [53] on minimal models, (^(Ul)^(W2)*cfcM)7 = K 2 r A l 2 | u 2 3 | - A - K 3 | - A - [ : ^ ] F J f c [ ^ ]

(13)

where the symbol in square brackets stands for the Clebsch-Gordan coefficients of su(2). The latter guarantee that both sides of the equation transform in the same way under the action of the zero mode algebra su(2). The non-trivial part of equation (13) therefore concerns the relation of 3-point vertices for open strings with entries of the Fusing matrix.

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We conclude this subsection with a brief remark on an interesting link to non-commutative geometry. Let us observe that the 3-point couplings simplify significantly if we send k to infinity,

<*?(uo ^(uamte))^ 0 0 = {jjjj}[:ikc]-

(14)

Here we have used the property (8) of the fusing matrix. It is straightforward to check that the numbers appearing on the right hand side of this equation arise naturally from the multiplication of ordinary (2J + 1) x (2J + 1) matrices. In fact, the same numbers appear when we rewrite the product Y* • Yjj' as a linear combination of the su(2) multiplets ycfc € Mat(2J + 1). In this sense, the 3-point vertices for open strings on branes in SU(2) provide an infinite dimensional deformation of matrix multiplication. The emergence of non-commutative matrix algebras in the context of open string theory is not surprising. As we have stressed before, the end-points of open strings behave like charged particles which can couple to the vector potential A of the magnetic field on the brane. Hence, the non-commutativity we encounter in the context of open strings is directly related to a similar phenomenon for e.g. electrons in a strong magnetic field. Relations between branes, open string and non-commutative geometry were initially discovered for branes in flat space [14,16,55] and they have been studied extensively for several years [17,59]. 3.4.

Brane dynamics on group manifolds

Now that we have an exact conformal field theory solution for spherical branes on S 3 , we would like to briefly comment on some dynamical processes. Here we shall use the conjectured relation with RG-flows in the 2D boundary conformal field theory. It turns out that the study of boundary renormalization groups flows in models with an SU(2) current algebra is a classical problem of mathematical physics that was first addressed in the context of the Kondo-model. The Kondo-model is designed to understand the effect of magnetic impurities on the lowtemperature conductance properties of a 3D conductor. The latter can have electrons in a number k of conduction bands. If the impurities are far apart, their effect may be understood within an s-wave approximation of scattering events between a conduction electron and the impurity. This allows to formulate the whole problem on a 2-dimensional world-sheet for which the coordinates (u, v) are associated with the time and the radial distance from the impurity, respectively. One can build several currents out of the basic fermionic fields. Among them is a spin current J(u,v). Its Laurent modes satisfy the relations (5) of a su(2)k current algebra. This spin current is the one that couples to the magnetic impurity of spin JM which is sitting at the boundary v = 0, oo

/

duk^{u,Q).

(15)

-oo

Here, the matrices A M ,^ = 1,2,3 form a (2JM + l)-dimensional irreducible representation of su(2) and the parameter A controls the strength of the coupling. Note that the term (15) is identical to the coupling of open string ends with velocity J^iu, 0) to a background gauge field Ap = AM (see formula (2)). Hence, AM may be interpreted as a constant non-abelian gauge field on one of our branes. For M > 1, the interaction is marginally relevant so that, according to the proposal formulated in subsection 2.4, switching on such non-abelian gauge

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89

fields represents an instability. Its effect on the brane can be understood by searching for an RG fixed-point along the RG trajectory generated by the term (15). Fortunately, a lot of techniques have been developed to deal with perturbations of the form (15), going back even to the work of Wilson. From the old analysis it is known that nontrivial fixed points are reached at a finite value A = A* of the renormalized coupling constant A if 2 JM < k (exact- or over-screening resp.). These fixed points have been identified through several different approaches and a simple rule summarizing the results of such investigations was formulated by Affleck and Ludwig [1]. This rule can be applied directly to our branes on SU(2) [22] and it shows that e.g. point-like branes carrying a constant U(M) gauge field decay into an extended spherical brane with label JM = {M — l ) / 2 .

Figure 6. Studies of RG flows in models with a SU(2) current symmetry show that point-like branes carrying a constant U(M) gauge field decay into an extended spherical brane with label JM = {M — l)/2.

For large values of k, the result we have just formulated admits an easier derivation using some of the geometric structures we have outlined above. It is widely known that massless open string modes give rise to a gauge theory on the brane to which they are attached. But due to the presence of the B-field on our spherical branes, the open strings detect a non-commutative (matrix) geometry (cf. last subsection). Hence, we are tempted to conclude that a special non-commutative gauge theory should be associated with branes on the 3-sphere. This is indeed the case and the precise form of the relevant gauge theory has been determined following the usual rules of string theory [5], at least to leading order in a'. Once the action of this gauge theory is found, one can search for instabilities and classical ground states. The outcome of such an analysis confirms very nicely the formation of spherical branes from point-like branes that we have described in our discussion of RGfixed points above. This agreement is not accidental. In fact, when k is large, the action of the non-commutative field theory is a functional on the space of boundary couplings whose stationary points approximate zeros of the /3-function. The analysis of RG-fixed points through non-commutative field equations has significant technical advantages over more traditional approaches. These are related to a very efficient book-keeping of the possible boundary couplings through non-commutative variables. 3.5. Overview: various generalizations and extensions The material presented in this section has been generalized in many different directions and we would like to list a few of these developments before we leave the rational boundary

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conformal field theories. a Not surprisingly, results similar to the ones we have reviewed here are also available for all other compact simple simply connected Lie groups. For many of the higher groups, moreover, there exist new families of maximally symmetric branes that are associated with outer automorphisms of G and hence have no analogue in the case of SU(2). Their boundary couplings B and open string spectra were first studied by Felder et al. [21] (see also [25,43]). Results on the open string couplings and dynamics have also been obtained [6]. In addition to the maximally symmetric branes, i.e., those that admit an action of the whole group G, one can impose boundary conditions which break part of this symmetry. Investigations of such branes were initiated in a paper by Moore et al. [39] and a rather systematic construction has been developed more recently [50]. It is worth mentioning that symmetry breaking branes on group manifolds possess interesting applications to defect lines in 2D conformal field theory. With the theory of strings and branes on group manifolds being under such good control, one can start to descend to orbifolds and cosets thereof. Studies of branes and open strings in curved orbifold backgrounds possess a long history [10,11,24,42,49]. The theory of branes in coset models has been treated by many authors (see e.g. [19,23,26,39] for some early contributions and references).

4. Recent progress for strings in non-compact spaces The last part of this lecture is devoted to some developments in the area of non-rational boundary conformal field theory. As indicated in the introduction, these are highly relevant for the study of dualities between string and gauge theory and for time-dependent processes in string theory. Here we shall explain these motivations in some detail and then focus mainly on one particular model, namely on the Liouville field theory. 4.1. String/gauge theory dualities According to an old observation by 't Hooft, structures found in closed string amplitudes are very reminiscent of features in the perturbative expansion of large N gauge theories. This suggests that it might be possible to compute gauge theory amplitudes from string theory. Though the idea has been around for a long time, only a single concrete example was known until 1997: The duality between the quantum mechanics of hermitian matrices and a toy model of string theory with a 2-dimensional target space (see e.g. the lectures of Klebanov [31] and references therein). After branes had entered the stage of string theory, the situation changed drastically. We may grasp the fundamental role of branes for such developments through the following short analysis of figure 7. The image shows a simple string diagram that admits two rather distinct interpretations. We can either think of a process in which a closed string mode is emitted from one brane and propagates a distance Ax through the background before being absorbed by a second brane. Alternatively, the process can be seen as a pair creation and subsequent annihilation of open strings which end on the two involved branes. Though none of these interpretations is distinguished a priori, actual computations may favor one of them, depending on the separation Ax between the branes. If they are very far apart only a

The references provided in the following paragraph are highly incomplete. A more extensive list can be found e.g. in the lecture notes [56].

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91

the massless closed string modes contribute significantly to the amplitude and hence one can safely perform the string theory computation in its super-gravity approximation. For short distances Ax between the branes, however, many massive closed string modes start to enter the computation and we better switch to a description in terms of light, i.e., mildly stretched, open strings. This reinterpretation leaves us with a one-loop computation in the gauge theory of massless open string modes. The reasoning we have just gone through hints toward an intimate relation between closed string models and gauge theory. This relation has a few interesting features. Observe, for example, that it does not preserve the underlying space-time dimensions since closed strings propagate in the 10D background while gauge bosons cannot leave the branes' p + 1-dimensional world-volume. In addition, the relation also mixes different loop orders as can be inferred from figure 7. Here, we found a closed string tree level amplitude that contains information about a gauge theory one-loop diagram.

Figure 7. Depending on the distance Ax between branes the shown string amplitude can be approximated by either a tree level computation in super-gravity or a one-loop gauge theory computation.

With all this in mind it seems no longer surprising that branes provide us with many new and concrete dualities between gauge theories and models of closed strings. The most famous example certainly is Maldacena's duality [35] between M — 4 Super-Yang-Mills theory and string theory on AdSs xS5. It involves an SU(N) gauge theory at large JV with 't Hooft coupling A = SYM-W2 on one side and an AdSs space with curvature radius R2/a' ~ y/\ on the other. Hence, if we want to study gauge theory at finite 't Hooft coupling, the curvature of AdS 5 cannot be neglected so that string effects become important. Not only does this bring us back to the main theme of the lecture, it also hides novel challenges arising from the fact that the relevant closed strings propagate on a non-compact curved background. Similar observations can be made for all the other known examples of such dualities between gauge and string theory [2]. Therefore, extending the methods and results reviewed in the previous section to non-compact target spaces becomes an important new task for conformal field theory. Here we shall restrict our attention to the example of Liouville theory. Since it appears as constituent of 2D string theory, Liouville theory enters one side of the aforementioned duality with matrix quantum mechanics. More recent developments show that even this duality, though it pre-dates the discovery of branes in string theory by several years, ultimately arises is the same way as all the modern Ad S/conformal field theory-type dualities.

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4.2. Liouville theory and its applications Liouville theory may be considered as the minimal model of non-rational conformal field theory. It describes the motion of strings in a single direction with an exponential potential SL[X] ~ f dzdz (dXdX

+ fj,e2bx).

(16)

Here, X is a free bosonic field with background charge (linear dilaton). Note that a change of the parameter fi can be re-absorbed in a constant shift of the field X. Hence, the dependence of Liouville theory on \x is rather trivial. On the other hand, the correlation functions of the model display a very non-trivial dependence on the second parameter b. If we send b to zero while rescaling x = bX and A = b2^,, we recover a model of particles in the potential V{x) = Aexp2x. The full field theoretic model contains both perturbative and non-perturbative corrections to this limit. The latter turn out to render the quantum theory self-dual with respect to the replacement b —* b _ 1 . After these remarks on the model itself, let us briefly comment on two interesting applications. It is well known that a flat space with trivial background fields must be 26-dimensional for bosonic strings to propagate in it. This restriction to 26 dimensions, however, may be circumvented through the presence of non-trivial background fields. An example realizing such a scenario is the famous 2D string theory. It is constructed from a product of a 1dimensional free bosonic field with Liouville theory at b = 1. Obviously, such a closed string theory is of little practical interest, but is provides a valuable toy model for bosonic string theory. Compared to its 26-dimensional relative, the 2-dimensional model has the advantage of being perturbatively stable. The second application is much more recent and also less well tested. It has been proposed [28] that a time-like version of Liouville theory at b = i describes the homogeneous condensation of a closed string tachyon. A short look at the classical action (3) makes this proposal seem rather plausible. Recall that actions of this form describe a tachyon decay with a closed/open string tachyon profile $ / * . If we consider a closed string tachyon which decays homogeneously, then we must choose $ = \i = const. In addition, the parameter e^, is now forced to be eM = 2 so that the interaction term (3) becomes scale invariant. After a Wick-rotation of the field X° = iX, the interaction term looks formally like the interaction term in Liouville theory, only that the we have b = i, just as it was claimed above. Though on first sight this may all seem rather convincing, there are many subtleties hidden in the relation between Liouville theory and tachyon condensation. We will be able to display them more clearly once we have presented the exact solution of Liouville theory. 4.3. The solution of Liouville field theory The solution of Liouville theory on closed and open surfaces has been a major success of the last ten years. We shall briefly spell out some of the most central formulae before we return to the applications. Let us begin with a few simple remarks on the spectrum of the model. It is rather obvious that ground states (fip of closed strings in the exponential potential can be labeled by a real number P > 0 which corresponds to the momentum of an incoming wave in the region X —> — oo where the potential vanishes. The states (j>p are stringy analogues of the particle

Strings through the microscope

93

wave functions
Here, we have also introduced the notation a = (a\ + a.i + a^)/2 and otj = a — a.j. In addition, the special function T is obtained from Barnes's double Gamma function by T(a):=rj1(a|6)6-1)rj1(Q-a|6)r1).

(18)

The solution (17) was first proposed by H. Dorn and H. J. Otto [15] and by A. and Al. Zamolodchikov [66], based on extensive earlier work by many authors (see e.g. the reviews [58,63] for references). Crossing symmetry of the conjectured 3-point function was then checked analytically in two steps by Ponsot and Teschner [46] and by Teschner [63,64]. Different conformally invariant boundary conditions were found and studied in several papers [20,62,67]. As one might suspect, there exist two different types of boundary theories, one describing point-like objects, the other corresponding to space-filling branes with an additional exponential boundary interaction that is felt by the end-points of open strings. It turns out that a point-like brane can only sit in the region where the Liouville potential V becomes large. The simplest such brane possesses a 1-point coupling of the form

Open strings on this brane possess a discrete spectrum without continuous zero modes, in agreement with our geometrical intuition (see [67] for details). Let us remark that the brane with boundary coupling (19) is only one in an infinite set which is parametrized by two positive integers (n,m). The status of the boundary conditions with (n,m) ^ (1,1) as physical branes in Liouville theory, however, is less clear. Extended branes in the Liouville background, on the other hand, obviously possess one continuous physical parameter. It enters the theory through an additional boundary interaction of the form VB ~ Ms e x P bX. Note that constant shifts of the field X have been used before to rescale the bulk parameter fi. Hence, there is no freedom left now to absorb the new boundary parameter /ig- Consequently, coupling constants of the boundary theory display a non-trivial dependence on HB as can be seen in the case of the bulk 1-point coupling, [20] B.(P) = (irw(b2))

' cos(2nsP)-±

_25/4i7rP

"•

Here, the parameter s is related to the boundary coupling ^B by the relation (i2B sin 7r62 = fi cosh2 irsb.

(20)

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VOLKER SCHOMERUS

Each of these extended branes comes with a continuous spectrum of open string modes. For imaginary values of the boundary parameter s, one may also encounter additional discrete states [62]. The 3-point vertex for open strings on the extended branes has been constructed by Ponsot and Teschner [47]. The formulae are more complicated and we refer the interested reader to the original papers. 4.4. Some applications and extensions In this subsection we would like to comment on some of the applications and possible extensions of the results we outlined above. As we pointed out earlier, Liouville theory is a building block for the string theory dual of matrix quantum mechanics, 'MQM

/? / dt '/

l

-{dtM{t)f + V{M{t)) ,

where M(t) are hermitian TV x TV matrices and V is a cubic potential. To be more precise, the duality involves taking TV and f3 to infinity while keeping their ratio K = N//3 fixed and close to some critical value KC. In this double scaling limit, the matrix model can be mapped to a system of non-interacting fermions moving through an inverse oscillator potential, one side of which has been filled up to a Fermi level at A

r~\ Figure 8. In the double scaling limit, the hermitian matrix model can be mapped to a system of noninteracting fermions moving through an inverse oscillator potential, one side of which has been filled up to a Fermi level at A/t = re — rec ~ g7 •

With a quick glance at figure 8, we conclude that the model must be non-perturbatively unstable against tunneling of Fermions from the left to the right. This instability is reflected in the asymptotic expansion of the partition sum and even quantitative predictions for the mass m ~ a/gs of the instantons were obtained. The general dependence of brane masses on the string coupling gs along with the specific form of the coupling (19) have been used recently to identify the instanton of matrix quantum mechanics with the localized brane in the Liouville model [7,32,40,41]. In this sense, branes had been seen through investigations of matrix quantum mechanics more than ten years ago, i.e., long before their central role for string theory was fully appreciated. Concerning the application of Liouville theory to the condensation of tachyons, there exist a few quite encouraging recent developments. Let us recall that the description of tachyon condensation requires sending the parameter b to b = i and evaluating the correlators for imaginary momenta P. We can at least find one quantity for which this procedure may be carried out easily. It is the coupling of closed strings to a decaying brane, i.e., the 1-point coupling B in the theory (3) with $ = 0 and # = HB- In fact, from the formula (20) we read off that

Strings through the microscope

(exp (iEX°(z, z)) ) ~ ME

-^-

95

E

.

This result may be checked directly [33,60] through perturbative computations [12,44,51]. Other quantities in the rolling tachyon background have a much more singular behavior at b = i. In fact, Barnes's double T-function T2{x\b,b~1) is a well denned analytic function as long as Re b ^ 0. If we send b —> i, on the other hand, T^ diverges. Nevertheless, it can be shown [57] that the function T and the bulk couplings (17) possess a well defined limit. These limits, however, are no longer analytic and hence cannot be Wick rotated. It turns out that there are at least two different limiting theories. The first one occurs for real momenta (Euclidean target space) and it coincides with the interacting c = 1 limit of unitary minimal models that was first discovered by Runkel and Watts [54]. A limiting bulk theory with imaginary momenta (Lorentzian target space) has also been constructed [57]. Future investigations will show whether this does provide an exact description of closed string rolling tachyons. Even though Liouville theory is the only non-rational model that has been solved completely, there exist at least partial solutions for a few other non-compact backgrounds. These include the H£ = SL(2, C)/SU(2) model, a non-compact relative of the group manifold SU(2) and Euclidean version of Ad S3 ~ SL(2,R). The bulk structure constants C of this theory were found several years ago by Teschner [61] and couplings B of closed strings to various branes have been proposed [34,48]. Open string couplings C for non-compact branes in #3", on the other hand, remain unknown. Let us point out that / 7 | model itself suffers from an imaginary magnetic field and hence is unphysical. But its solution represented a key step toward the construction of two rather important string backgrounds. The first concerns the description of strings in Ad S3. In a series of papers [36-38], amplitudes in this Lorentzian version of the H% model have been worked out and interpreted. Another interesting application arises from the coset space H3 /R. This model became famous as the Euclidean 2D black hole [65]. It is a non-compact analogue of parafermions and hence a close relative of N = 2 minimal models. Non-compact parafermions and N — 2 minimal models have been argued to be T-dual to the Sine-Liouville (as reviewed in [30]) and the N = 2 Liouville theory [29], respectively. References to the original literature and recent result on branes in these backgrounds can be found in the recent literature [18,52].

Acknowledgments I wish to thank the organizers of the ICMP 2003 for the opportunity to present this overview.

References 1. I. Affleck, A. W. W. Ludwig, "Critical theory of overscreened Kondo fixed points", Nucl. Phys. B 360, 641 (1991). 2. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, Y. Oz, "Large N field theories, string theory and gravity", Phys. Rept. 323, 183-386 (2000); arXiv:hep-th/9905111. 3. A. Yu. Alekseev, V. Schomerus, "D-branes in the WZW model", Phys. Rev. D 60, 061901 (1999); arXiv:hep-th/9812193.

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4. A. Yu. Alekseev, A. Recknagel, V. Schomerus, "Non-commutative world-volume geometries: Branes on SU(2) and fuzzy spheres", JHEP 09, 023 (1999); arXiv:hep-th/9908040. 5. A. Yu. Alekseev, A. Recknagel, V. Schomerus, "Brane dynamics in background fluxes and non-commutative geometry", JHEP 05, 010 (2000); arXiv:hep-th/0003187. 6. A. Y. Alekseev, S. Fredenhagen, T. Quella, V. Schomerus, "Non-commutative gauge theory of twisted D-branes", arXiv:hep-th/0205123. 7. S. Y. Alexandrov, V. A. Kazakov, D. Kutasov, "Non-perturbative effects in matrix models and D-branes", JHEP 0309, 057 (2003); arXiv:hep-th/0306177 . 8. E. Barnes, "The theory of the double gamma function", Philos. Trans. Roy. Soc. A 196, 265 (1901). 9. A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, "Infinite conformal symmetry in twodimensional quantum field theory", Nucl. Phys. B 241, 333-380 (1984). 10. M. Bianchi, A. Sagnotti, "On The Systematics Of Open String Theories", Phys. Lett. B 247, 517 (1990). 11. L. Birke, J. Fuchs, C. Schweigert, "Symmetry breaking boundary conditions and WZW orbifolds", Adv. Theor. Math. Phys. 3, 671-726 (1999); arXiv:hep-th/9905038. 12. C. G. . Callan, I. R. Klebanov, A. W. W. Ludwig, J. M. Maldacena, "Exact solution of a boundary conformal field theory", Nucl. Phys. B 422, 417 (1994); arXiv.hep-th/9402113. 13. J. L. Cardy, "Boundary conditions, fusion rules and the Verlinde formula", Nucl. Phys. B 324, 581 (1989). 14. C.-S. Chu, P.-M. Ho, "Noncommutative open string and D-brane", Nucl. Phys. B 550, 151 (1999); arXiv:hep-th/9812219. 15. H. Dorn, H. J. Otto, "Two and three point functions in Liouville theory", Nucl. Phys. B 429, 375-388 (1994); arXiv:hep-th/9403141. 16. M. R. Douglas, C. Hull, "D-branes and the noncommutative torus", JHEP 02, 008 (1998); arXiv:hep-th/9711165. 17. M. R. Douglas, N. A. Nekrasov, "Noncommutative field theory", Rev. Mod. Phys. 73, 977 (2001); arXiv:hep-th/0106048. 18. T. Eguchi, Y. Sugawara, "Modular bootstrap for boundary N = 2 Liouville theory", JHEP 0401, 025 (2004); arXiv:hep-th/0311141. 19. S. Elitzur, G. Sarkissian, "D-branes on a gauged WZW model", Nucl. Phys. B 625, 166-178 (2002); arXiv:hep-th/0108142. 20. V. Fateev, A. B. Zamolodchikov, A. B. Zamolodchikov, "Boundary Liouville field theory. I: Boundary state and boundary two-point function", arXiv:hep-th/0001012 . 21. G. Felder, J. Frohlich, J. Fuchs, C. Schweigert, "The geometry of WZW branes", J. Geom. Phys. 34, 162-190 (2000); arXiv:hep-th/9909030. 22. S. Fredenhagen, V. Schomerus, "Branes on group manifolds, gluon condensates, and twisted K-theory", JHEP 0104, 007 (2001); arXiv:hep-th/0012164. 23. S. Fredenhagen, V. Schomerus, "D-branes in coset models", JHEP 02, 005 (2002); arXiv: hep-th/0111189. 24. J. Fuchs, C. Schweigert, "Symmetry breaking boundaries. I: General theory", Nucl. Phys. B 558, 419 (1999); arXiv:hep-th/9902132 . 25. M. R. Gaberdiel, T. Gannon, "Boundary states for WZW models", arXiv:hep-th/0202067. 26. K. Gawedzki, "Boundary WZW, G/H, G/G and CS theories", arXiv:hep-th/0108044. 27. M. Gutperle, M. Headrick, S. Minwalla, V. Schomerus, "Space-time energy decreases under world-sheet RG flow", JHEP 0301, 073 (2003); arXiv:hep-th/0211063. 28. M. Gutperle, A. Strominger, "Timelike boundary Liouville theory", Phys. Rev. D 67, 126002 (2003); arXiv:hep-th/0301038. 29. K. Hori, A. Kapustin, "Duality of the fermionic 2d black hole and N = 2 Liouville theory as mirror symmetry", JHEP 0108, 045 (2001); arXiv:hep-th/0104202. 30. V. Kazakov, I. K. Kostov, D. Kutasov, "A matrix model for the two-dimensional black hole", Nucl. Phys. B 622, 141 (2002); arXiv:hep-th/0101011. 31. I. R. Klebanov, "String theory in two-dimensions", arXiv:hep-th/9108019.

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32. I. R. Klebanov, J. Maldacena, N. Seiberg, "D-brane decay in two-dimensional string theory", JHEP 0307, 045 (2003); arXiv:hep-th/0305159. 33. F. Larsen, A. Naqvi, S. Terashima, "Rolling tachyons and decaying branes", JHEP 0302, 039 (2003); arXiv:hep-th/0212248. 34. P. Lee, H. Ooguri, J. w. Park, "Boundary states for AdS(2) branes in AdS(3)", Nucl. Phys, B 632, 283 (2002); arXiv:hep-th/0112188. 35. J. M. Maldacena, "The large JV limit of superconformal field theories and supergravity", Adv. Theor. Math. Phys. 2, 231 (1998), [Int. J. Theor. Phys. 38, 1113 (1999)]; a r X i v : h e p - t h / 9711200. 36. J. M. Maldacena, H. Ooguri, "Strings in AdS(3) and SL(2,R) WZW model. I", J. Math. Phys. 42, 2929 (2001); arXiv:hep-th/0001053. 37. J. M. Maldacena, H. Ooguri, J. Son, "Strings in AdS(3) and the SL(2,R) WZW model. II: Euclidean black hole", J. Math. Phys. 42, 2961 (2001); arXiv:hep-th/0005183. 38. J. M. Maldacena, H. Ooguri, "Strings in AdS(3) and the SL(2,R) WZW model. Ill: Correlation functions", Phys. Rev. D 65, 106006 (2002); arXiv:hep-th/0111180. 39. J. Maldacena, G. W. Moore, N. Seiberg, "Geometrical interpretation of D-branes in gauged WZW models", JHEP 07, 046 (2001); arXiv:hep-th/0105038. 40. E. J. Martinec, "The annular report on non-critical string theory", arXiv:hep-th/0305148. 41. J. McGreevy, H. Verlinde, "Strings from tachyons: The c = 1 matrix reloaded", JHEP 0312, 054 (2003); arXiv:hep-th/0304224. 42. M. Oshikawa, I. Affleck, "Boundary conformal field theory approach to the critical twodimensional Ising model with a defect line", Nucl. Phys. B 495, 533-582 (1997); arXiv: cond-mat/9612187. 43. V. B. Petkova, J. B. Zuber, "Boundary conditions in charge conjugate sl(N) WZW theories", arXlv:hep-th/0201239. 44. J. Polchinski, L. Thorlacius, "Free Fermion Representation Of A Boundary Conformal Field Theory", Phys. Rev. D 50, 622 (1994); arXiv:hep-th/9404008. 45. J. Polchinski, "TASI lectures on D-branes", arXiv:hep-th/9611050 . 46. B. Ponsot, J. Teschner, "Liouville bootstrap via harmonic analysis on a noncompact quantum group", arXiv:hep-th/9911110. 47. B. Ponsot, J. Teschner, "Boundary Liouville field theory: Boundary three point function", Nucl. Phys. B 622, 309-327 (2002), arXiv:hep-th/0110244. 48. B. Ponsot, V. Schomerus, J. Teschner, "Branes in the Euclidean AdS(3)", JHEP 0202, 016 (2002); arXiv:hep-th/0112198. 49. G. Pradisi, A. Sagnotti, "Open String Orbifolds", Phys. Lett. B 216, 59 (1989). 50. T. Quella, V. Schomerus, "Symmetry breaking boundary states and defect lines", JHEP 06, 028 (2002); arXiv:hep-th/0203161. 51. A. Recknagel, V. Schomerus, "Boundary deformation theory and moduli spaces of D-branes", Nucl. Phys. B 545, 233 (1999); arXiv:hep-th/9811237. 52. S. Ribault, V. Schomerus, "Branes in the 2-D black hole", JHEP 0402, 019 (2004); arXiv: hep-th/0310024. 53. I. Runkel, "Boundary structure constants for the A-series Virasoro minimal models", Nucl. Phys. B 549, 563 (1999); arXiv:hep-th/9811178. 54. I. Runkel, G. M. T. Watts, "A non-rational CFT with c = 1 as a limit of minimal models", JHEP 09, 006 (2001); arXiv:hep-th/0107118. 55. V. Schomerus, "D-branes and deformation quantization", JHEP 06, 030 (1999); arXiv: hep-th/9903205. 56. V. Schomerus, "Lectures on branes in curved backgrounds", Class. Quant. Grav. 19, 5781 (2002); arXiv:hep-th/0209241. 57. V. Schomerus, "Rolling tachyons from Liouville theory", JHEP 0311, 043 (2003); arXiv: hep-th/0306026. 58. N. Seiberg, "Notes On Quantum Liouville Theory And Quantum Gravity", Prog. Theor. Phys. Suppl. 102, 319 (1990).

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59. N. Seiberg, E. Witten, "String theory and noncommutative geometry", JHEP 09, 032 (1999); arXiv:hep-th/9908142. 60. A. Sen, "Rolling tachyon", JHEP 0204, 048 (2002); arXiv:hep-th/0203211. 61. J. Teschner, "On structure constants and fusion rules in the SL(2,C)/SU(2) WZNW model", Nucl. Phys. B 546, 390 (1999); arXiv:hep-th/9712256. 62. J. Teschner, "Remarks on Liouville theory with boundary", arXiv:hep—th/0009138 . 63. J. Teschner, "Liouville theory revisited", Class. Quant. Gray. 18, R153-R222 (2001); arXiv: hep-th/0104158. 64. J. Teschner, "A lecture on the Liouville vertex operators", arXiv:hep-th/0303150 . 65. E. Witten, "On string theory and black holes", Phys. Rev. D 44, 314 (1991). 66. A. B. Zamolodchikov, A. B. Zamolodchikov, "Structure constants and conformal bootstrap in Liouville field theory", Nucl. Phys. B 477, 577-605 (1996); arXiv:hep-th/9506136. 67. A. B. Zamolodchikov, A. B. Zamolodchikov, "Liouville field theory on a pseudosphere", arXiv: hep-th/0101152.

Critical percolation and conformal invariance STANISLAV SMIRNOV

(Royal Inst, of Technology, Stockholm)

Many 2D critical lattice models are believed to have conformally invariant scaling limits. This belief allowed physicists to predict (unrigorously) many of their properties, including exact values of various dimensions and scaling exponents. We describe some of the recent progress in the mathematical understanding of these models, using critical percolation as an example.

1. Introduction For several 2D lattice models physicists were able to make a number of spectacular predictions (non-rigorous, but very convincing) about exact values of various scaling exponents and dimensions. Many methods were employed (Coulomb Gas, Conformal Field Theory, Quantum Gravity) with one underlying idea: that in some sense the model concerned has a continuum scaling limit (as mesh of the lattice goes to zero) and the latter is conformally invariant. Recently mathematicians came up with some new (and rigorous) approaches. We will describe some of the progress made, much of it due to Lawler, Schramm, and Werner. We will center on the percolation which has a simple definition, and is now fairly well understood. In Bernoulli site percolation with p G [0,1] each vertex of some graph is declared open (or colored blue) with probability p and closed (or colored yellow) with probability 1 — p, independently of each other. We are interested in graphs, which approximate geometry of the plane, especially square, triangular, and hexagonal lattices. One studies clusters, which are connected subgraphs of a given color. One can also color edges (bond percolation) or, in a planar graph, faces. Since vertices are colored independently, the model has locality property: observables for disjoint areas are independent. However, the model exhibits a complicated behavior. It is now well-known that in these models (both in site and bond cases) a phase transition occurs: there is a critical value pc E (0,1) such that when p < pc, there is no infinite cluster of open vertices, while if p > pc, there is a unique such infinite cluster. The value of pc is lattice-dependent. It was shown by Kesten that for bond percolation on the square lattice pc = 1/2, whereas for site percolation on the square lattice pc ~ 0.59 with exact value still unknown. See the textbooks [5,7] for a good introduction to the subject. The percolation probability 6(p) that the origin belongs to the infinite open cluster is positive for p > pc (but strictly smaller than 1 as the origin might be surrounded by closed vertices), see [5]. Arguments from theoretical physics predict the exact power law for 6: 9(P)-(P-Pc)5/36,

P^Pc

+.

(1)

The exponent 5/36 is supposed to be independent (unlike pc) of the planar lattice involved. Several other predictions were made, for example if CQ denotes the open cluster containing

99

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STANISLAV SMIRNOV

the origin (which might be empty), the expected number x(p) of vertices in it, provided it is finite, was predicted to behave like X(p):=E(#C0|#Co
p^pc.

(2)

Another power law was predicted for the correlation length:

«*-(£*£

#Co
P-^Pc:

(3)

where \y\ denotes the distance from the site y to the origin. All these results were first conjectured (on the basis of experiments and heuristical arguments) and then predicted (using Coulomb Gas methods, Conformal Field Theory, or Quantum Gravity) by physicists. Now they have been rigorously established for site percolation on triangular lattice by a combination of [8,11-13,18,19]: Theorem 1. Scaling laws (1), (2), (3) hold for percolation on the triangular lattice. Here we restrict ourselves to the discussion of rigorous methods, see [19] for references to the physics literature. Below we sketch some ideas which contributed to the proof. We start with conformally invariant scaling limits for percolation, and then describe their connections to SLE, or Schramm-Loewner Evolution, a way to obtain conformally invariant random curves, introduced by Schramm. We conclude with some open questions.

2. Harmonic conformal invariants For some time it appeared difficult to formulate rigorously that percolation has a conformally invariant scaling limit, cf. [1]. One simple interpretation, which attracted much attention, was suggested by Langlands, Pouliot, and Saint-Aubin in [9]. Given a topological rectangle (a simply connected domain Q, with boundary points a, b, c, d) one can superimpose a lattice with mesh 5 onto 0 and study the probability lis (fi, [a,b], [c,d\) that there is an open cluster joining the arc [a,b] to the arc [c,d] on the boundary of Q. In [9] extensive computer experiments were performed to check that there is a limit II := lim,5_o IXs, which is independent of the lattice, and depends only on the conformal modulus of the configuration CI, a, b, c, d. The latter conjecture authors attributed to Aizenman. The results were conclusively positive, leading to the conjecture that crossing probabilities have a conformally invariant scaling limit. Using Conformal Field Theory Cardy was able to derive in [4] (unrigorously) the exact value of the crossing probabilities. Assuming conformal invariance, he mapped the domain to the upper half plane C+, three boundary points to 0,1, oo, and derived a differential equation for II as a function of the fourth point. The solution turned out to be a hypergeometric function 2F1:

Critical percolation and conformal invariance

101

There are several definitions for the function 2-Fi- It can be written as an integral, leading to the following formula for F: F{U)

= r („ (1-v)) Jo

-2/3

dv

f\v( 1-v))

-2/3

dv.

Jo

Later Carleson observed that the very same hypergeometric function maps the half-plane to an equilateral triangle, and there Cardy's formula takes a particularly easy form. Namely, suppose that we map Q conformally to an equilateral triangle with side length 1, with three points going to vertices a, b, c and the fourth point to some z G ab. Then crossing probability II([a,2], [b, c}) converges to the distance from a to z:

6-+0

I.

Cardy's formula along with conformal invariance was proved in [18] for the site percolation on the triangular lattice: Theorem 2. For the critical site percolation on the triangular lattice, as mesh goes to zero crossing probabilities converge to a limit which is conformally invariant and satisfies Cardy 's formula. We will start by sketching the proof. The method is different from that of Cardy, and the hypergeometric function arises in a different way. As we will see later, SLE gives this function yet another interpretation. For this particular model it is known by results of Kesten and Wierman that pc = 1/2, so each vertex of the triangular lattice with mesh 5 (or equivalently each hexagon in the dual honeycomb lattice) is open or closed with equal probability 1/2. The method utilizes self-duality of the model (there is a horizontal yellow crossing if and only if there is no vertical blue crossing) and the fact that pc = 1/2 (which allows to change the colors to the opposite while preserving the probabilities). Like Cardy, we fix a domain Q and three boundary points a, 6, c, and study the behavior of II(fi, [ab], [cz]) as the forth point z moves. However we allow z to move inside domain D. as well, considering a new function Ha(z) :— H(z,a,b,c,Q,S) which is the probability that a yellow path from the boundary arc ab to the boundary arc ca separates z from be. On the arcs ab and ca this function becomes the crossing probability n , whereas on the arc be it vanishes. One defines functions Hb and Hc symmetrically. By Russo-Seymour-Welsh estimates (see [5,18]) one can bound their Holder norms, so as 5 —• 0 we can choose a uniformly converging subsequence.

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It turns out that these three functions are (approximatively) discrete harmonic, and so their limits are harmonic. The discrete derivative of a function daHa in the direction a is the difference Ha(z') — Ha(z) for two neighboring sites z, z' with z' — z = ad. Let Q denote the area above the lowest (i.e. closest to 6c) yellow crossing from ab to ca. Then Ha(z) = P(z G Q), and so Ha(z') - Ha(z) = F(z' e Q) - P ( z e Q) = P(z' e Q,z $ Q)-F(z

e Q,z' i Q).

The two probabilities on the right hand side have an easy combinatorial interpretation: e.g. in the first case the lowest yellow crossing of hexagons passes though the edge zz', and by duality there is a blue crossing from z to the arc be (which prevents existence of a lower yellow crossing). So the first term is the probability of three multicolored crossings from z to the three boundary arcs (whereas the second is the same for z'). A combinatorial argument, based on the fact that pc = 1/2, and employed earlier by Aizenman, Duplantier, and Aharony [3], shows that such probabilities are independent of the colors of crossings, as long as both colors are present. If we change the colors from yellow-yellow-blue to blue-yellow-yellow, we will have a similar picture, but instead of crossing a yellow path between ab and ca in the direction a we will be crossing a yellow path between be and ab in the rotated by 2TT/3 direction aexp(27ri/3). So probabilities contributing to daHa are identified with probabilities contributing to 9aexp(2iri/3)-^6In this way one arrives to Cauchy-Riemann equations (in the basis of cube roots of 1): daHa

« C>aexp(27ri/3)-£ff> ~ 9a exp(47ri/3)-^c-

(5)

This implies that the scaling limits of the functions H satisfy exactly such Cauchy-Riemann equations. Moreover, on every boundary arc we know the values of one of them: e.g. Ha(a) = 1, Ha = 0 on be. Such boundary value problem has a unique solution, given in the half-plane by a complexification of Cardy's hypergeometric function. It is even easier to see this in Carleson's form: in an equilateral triangle these three functions are linear. Note also that this problem and hence its solution are conformally invariant. Since the resulting limit is independent of subsequence chosen, we conclude that functions H have a scaling limit, which is conformally invariant and satisfies a complexification of Cardy's formula. Once Cardy's formula is established, one can prove some statements about the scaling limit, [18]. For example, consider domain 0 with three boundary points a, b, c and the lowest (closest to be) yellow path from the boundary arc ab to the boundary arc ca. By a priori estimates of Aizenman and Burchard [2] the family of the laws of the lowest crossings (for various values of the mesh) is weakly precompact, so every sequence has a weakly converging subsequence. To prove that there is a scaling limit, it is sufficient to show that this limit is independent of the chosen subsequence. But by Cardy's formula for any curve 77 we know the probability (in the limit) that the lowest crossing went completely below 77, which is exactly the crossing probability for the domain obtained by cutting a part of 0 away along ??. Such events generate (by disjoint unions and complements) the BorelCT-algebraof crossings with Holder topology, and so the limiting law is determined uniquely. Moreover, it is conformally invariant since the events involved are.

Critical percolation and conformal invariance

103

Building upon this, one reasons similarly to show the existence and conformal invariance of the scaling limit for the perimeter of a cluster, i.e. an interface between two clusters of opposite color. It turns out that it is characterized by the same properties as Schramm's SLE(6), so they coincide. Alternatively one can aim from the beginning at proving that the interface converges to SLE(6) — an approach we discuss below. Once this connection is established, one can employ SLE in calculating percolation exponents.

3. Loewner Evolution Loewner Evolution is a differential equation for a Riemann uniformization map for a domain with a growing slit. It was introduced by Loewner in [15] in his work on Bieberbach's conjecture. Loewner Evolution allows to write differentials of various functionals defined for planar domains when domain is perturbed by adding a slit, and it was successfully applied to many optimization problems. There are two standard setups: radial, when the slit is growing towards a point inside the domain, and chordal, when the slit is growing towards a point on the boundary. In both cases we choose a particular Riemann map by fixing value and derivative at the target point. We will restrict our discussion to the chordal case, though radial is equally interesting in our context. Chordal Loewner evolution describes uniformization for the upper half-plane C + with a slit growing from 0 to oo (one deals with another domain fl with boundary points a, b by mapping it to C+ so that a, b — i » 0, oo). Loewner dealt only with slits given by smooth simple curves, but more generally one allows any set which grows continuously in conformal metric when viewed from oo. We will omit the precise definition of "allowed slits" (more extensive discussion in this context can be found in [10]), only noting that all simple curves are included. The random curves which arise from lattice models (e.g. cluster perimeters or interfaces) are simple curves. Their scaling limits are not necessarily simple, they have no "transversal" self-intersections. For such a curve to be an allowed slit it is sufficient to visit no point thrice. Parameterizing the slit 7 in some way by time t, we denote by gt(z) the conformal map sending C+ \ jt to C+ normalized so that at infinity gt{z) = z + a(t)/z + 0(l/\z\2), the so called hydrodynamic normalization. It turns out that a(t) is a continuous strictly increasing function (it is a sort of capacity-type parameter for 7 t ), so one can change the time so that

gt(Z)=z+^+o(jLy

(6)

Denote by w(t) the image of the tip •fit). The family of maps gt (also called a Loewner chain) is uniquely determined by the real-valued "driving term" w(t). The general form of the Loewner's theorem can be stated as follows: L o e w n e r ' s t h e o r e m . There is a bijection between allowed slits and continuous real valued functions w(t) with w(0) = 0. This bijection is given by the differential equation 2 (7) dtgt(z) = — ^ 77T, go(z) = z gt{z)-w{t)

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4. Schramm-Loewner Evolution Loewner's theorem states that a deterministic curve 7 corresponds to a deterministic driving term w(t). Similarly a random 7 corresponds to a random w(t). One obtains S L E ( K ) by taking w(t) to be a Brownian motion with speed K: Definition. Schramm-Loewner Evolution, or SLE(K), taking w(t) = y/RBt, n e [0, 00).

is the Loewner chain one obtains by

It is shown in [16] that the resulting slit will be almost surely a Holder continuous curve. So we will also use the term SLE for the resulting random curve, i.e. a probability measure on the space of curves (to be rigorous one can think of a Borel measure on the space of curves with Holder norm). Different speeds K produce different curves, which become more "fractal" as K increases: we grow the slit with constant speed (measured by capacity), while the driving term "wiggles" faster. For example, for K < 4 the curve is a.s. simple, for 4 < K < 8 it a.s. touches itself, and for n > 8 it is a.s. space-filling (i.e. visits every point in C+). See [16] for the discussion of basic properties of SLEs and references. We follow Schramm to show that this choice of w(t) arises naturally. Schramm decided to describe the scaling limits of cluster perimeters, or interfaces for lattice models assuming their existence and conformal invariance. This naturally led to introduction of SLE in [17].

Figure 1. Critical site percolation on triangular lattice superimposed over a rectangle. Dobrushin boundary conditions produce the interface from the lower left to the upper right corner. The law of the interface converges to SLE(6) when mesh goes to zero.

The reasoning went similarly to the following: consider a simply connected domain Q with two boundary points, a and b. Superimpose a lattice with mesh 5 and consider some lattice model, say critical percolation with the Dobrushin boundary conditions, coloring the vertices blue on the boundary arc ab and yellow on the boundary arc ba. This enforces existence (besides many loop interfaces) of an interface between yellow and blue clusters running from a to b, which is illustrated by figure 1 for a rectangle with

Critical percolation and conformal invariance

105

two opposite corners as a and b. So we end up with a random simple curve (a broken line) connecting a to b inside Cl. The law of the curve depends of course on the lattice superimposed. If we believe the physicists' predictions, as mesh tends to zero, this law converges (in a weak-* topology) to some law A = A(Q, a, b) on curves from a to b inside O. Furthermore, in this setup the conformal invariance prediction can be formulated as follows: (A) Conformal invariance: The law is conformally invariant: for a conformal map of the domain O one has 4>(A(n,a,b)) = A((Q),cf>(a),(il) induces a map acting on the curves in £), which in turn induces a map on thepprobability measures on the space of such curves, which we denote by the same letter. By a conformal map we understand a bijection which locally preserves angles, i.e. is analytic or anti-analytic function (so it might change the orientation). Moreover, if we start drawing the interface from the point a, we will be walking around the yellow cluster following the left-hand rule — see figure 1. If we stop at some point a' after drawing the part 7' of the interface, we cannot distinguish the boundary of 0 from the part of the interface we have drawn: they both are colored yellow on arc a'b and blue on arc ba' of the domain Q \ 7'. So we can say that the conditional law of the interface (conditioned on it starting as 7') is the same as the law in a new domain with a slit. We expect the limit law A to have the same property: (B) Markov-type property: The law conditioned on the interface already drawn is the same as the law in the slit domain: A(fi,a,6)|7' = A ( n \ 7 , , a , , 6 ) If one wants to utilize these properties to characterize A, by (A) it is sufficient to study some reference domain (to which all others can be conformally mapped), say the upper half-plane C+ with a curve running from 0 to 00. Given (A), the second property (B) is easily seen to be equivalent to the following: (B') Conformal Markov property: The law conditioned on the interface is a conformal image of the original law. Namely, if G = Gy is a conformal map from C+ \ 7' to C + preserving 00 and sending the tip of 7' to 0, then A (C+, 0,00) |V = G - 1 (A (C+, 0,00)). To use the property (B'), we describe the random curve by the Loewner evolution with a certain random driving force w(t) (we assume that the curve is a.s. an allowed slit). If we fix the time s, the property (B') with the slit 7[0,s] and the map G(z) = gs{z) — w(s) can be rewritten for random conformal map Gt+S conditioned on Gs (which is the same as conditioning on 7') as Gt+s\Gs = Gs (Gt) • Expanding G's near infinity we obtain z-w(t

+ s)-i

\GS = (z- w{s) + • • •) o (z - w(t) + •••) = z - (w(s) + w(t)) -\

,

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STANISLAV SMIRNOV

concluding that w(t + s) - w(s)\Gs = w(t). This means that w(t) is a continuous (by Loewner's theorem) stochastic process with independent stationary symmetric (apply (A) with anti-conformal reflection <j>(u+iv) — —u+iv) increments. Thus w(t) has to be a Brownian motion with certain speed K £ [0, oo): w(t) — sfkBt. So one logically arrives at the definition of SLE, and what we call Schramm's principle. A random curve satisfies (A) and (B) if and only if it is given by SLE(K) with certain K £ [0,oo). The discussion above is essentially contained in Schramm's [17] for the radial version, when slit is growing towards a point inside and the Loewner differential equation takes a slightly different form. To make this principle a rigorous statement, one has to ask the curve to be almost surely an allowed slit. In order to use the above principle one still has to show the existence and conformal invariance of the scaling limit, and then calculate some observable to pin down the value of K. For percolation one can employ locality property or Cardy's formula to show that K = 6 (we will discuss below why SLE(6) is the only SLE matching Cardy's formula). So Schramm concluded in [17] that if percolation interface has a conformally invariant scaling limit, it must be SLE(6).

5. SLE as a scaling limit One way to show that certain random curve coincides with SLE is to determine infinitely many observables, which could prove difficult (for percolation, locality helps to create many observables from just one). Fortunately, the situation turns out to be much nicer: if one can show that just one (nontrivial) observable has a limit satisfying analogues of (A) and (B), convergence to S L E ( K ) (with K determined by the values of the observable) follows. This was demonstrated by Lawler, Schramm and Werner in [14] in establishing the convergence of two related models: of Loop Erased Random Walk to SLE(2) and of Uniform Spanning Tree to SLE(8). We describe how a modification of their ideas gives another proof that percolation perimeter converges to SLE(6). For percolation on a fixed lattice in a domain O with boundary points o, b and Dobrushin boundary conditions consider interface running from a to 6, see figure 1. Already mentioned estimates [1,2] imply that collection of interface laws on lattices with different mesh is precompact (in the weak-* topology on the space of Borel measures on Holder continuous curves). So to show that as mesh goes to zero the interface law converge to the law of SLE(6), it is sufficient to show that the limit of any converging subsequence is in fact SLE(6). Take some converging subsequence, whose limit is a random curve 7 with law A. Though 7 is a scaling limit of a simple curve, 7 itself will a.s. be non-simple. But known a priori estimates [1] show that 7 a.s. visits no point thrice and so is an allowed slit. So we can (by mapping Cl to C+ and applying Loewner's theorem) describe 7 by a Loewner evolution with a (random) driving force w(t). It remains to show that w(t) = V&Bt.

Critical percolation and conformal invariance

107

Add two more points on the boundary, making fi a topological rectangle axby and consider the crossing probability 11,5 (Q, [a,x], [b,y]) (from the arc ax to the arc by on a lattice with mesh 6). Parameterize the curve 7 in some way by time and assume t to be small enough so that 7[0, t] does not reach x,y. The crossing probability conditioned on 7(0, £] coincides with crossing probability in the slit domain Q \l[0,t], (an analogue of the property (A)) we can write by the total probability theorem Us (SI, [a,x], [b,y]) = E(U5 ( O \ 7 [ 0 , t ] , [l(t),x], [b,y])).

(8)

If enough a priori estimates are available, identity (8) also holds for the scaling limit II := lims^o 11(5 °f the crossing probabilities, which we know to exist, be conformally invariant, and satisfy Cardy's formula. Mapping to half-plane and applying conformal invariance for the map gt(z) — w(t) we write n(C+,[0,a:],[c» > y])=E(n(C + \7[0 ) t],[7(t),a:],[cx3 ) y])) = E(IL(C+,[0,gt(x)

-w(t)],[oo,gt(y)

-w{t)]))

•

(9)

By conformal invariance (under Mobius transformation z i-> (z — y)/(x — y)), and Cardy's formula n(C+,[0,a;],[oo,y])=n(c+,

y

,[oo,0]

=F

x-y

x-y

for Cardy's hypergeometric function F. Rewriting in this way both sides of the equation (9) we arrive at

x \ ^-y)

_EFfgt(x)-w(t) \9t(x) - gt(y),

This provides some information about w(t), but it is difficult to use before we get rid of gtTo that purpose we fix the ratio x/(x—y) =: 1/3 (anything not equal to 1/2 would do) and let x tend to infinity: y := — 2x, x —> +00. Using the normalization gt(z) = z + 2t/z + 0(1/z2) at infinity, writing Taylor expansion for F, and plugging in values of derivatives of F at 1/3, we obtain F{l\=EFf

*{x)-v,{t)

3J

\gt(x) = EF

gt(-2x),

x - w(t) + 2t/x + 0(l/x2) (x + 2t/x + 0(l/x2)) - ( - 2 i + 2t/(-2x) +

F(>_^I +

| J , + o(^

r(2/3) ^ E x r(l/3)r(4/3) 3 2 /325« 2

0(l/x2))

(„(,)» - 6 t ) + 0 ( i ) .

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STANISLAV SMIRNOV

Observing that coefficients by 1/x and 1/x2 on the right hand side should vanish, we conclude that Ew(t)=0, Ew(t)2-6t = 0. (10) Applying the same reasoning to domain Cl \ 7(0, t] relative to Q, \ 7(0, s] gives identities (10) for the increments of w(t). Thus w(t) is a continuous (by Loewner's theorem) process such that both w(t) and w(t)2 — 6i are local martingales so by Levy's theorem w(t) = ^/&Bt, and therefore SLE(6) is the scaling limit of the critical percolation interface. The argument will work wherever Cardy's formula and a priori estimates are available, particularly for triangular lattice, giving the following [18]: Theorem 3. Consider the site percolation on the triangular lattice in a simply connected domain Q with boundary points a and b and Dobrushin boundary conditions. Then the interface running from a to b has a conformally invariant scaling limit, which coincides with SLE(6). 6. Calculations for SLE The value K = 6 appears in the argument above because of the specific values of the conformal martingale considered. The above reasoning also indicates that very few functions can arise as conformal invariant martingales for SLEs. Indeed, this was used by Lawler, Schramm, and Werner to determine the values of various probabilities and expectations which are conformally invariant or covariant. We will sketch one of their calculations, showing that assuming convergence of the percolation interface to SLE(6) one can derive Cardy's formula from the properties of SLE. Of course, Cardy's formula was used to establish the convergence in the first place. But the argument below provides new insight into it and can also be applied in many other situations. The scaling limit of the crossing probability II (C+, [0, x], [00, y]) is equal to the probability that SLE(6) (which is the scaling limit of the percolation interface) touches the interval [a;,+00] before the interval [—00,y]. This can be seen in the figure 1: the vertical gray crossing forces the interface to touch the upper side of the rectangle before it touches the right side. We will calculate similar probability for S L E ( K ) , which is driven by w(t) = y/HBt. Denote this probability by IiK. By conformal invariance (under the Mobius transformation z >—> (z — y)/{x — t/)), it depends only on x/(x — y):

nK(c+,[o,x],[oo,z/]) = n J c + ,

X

x-y

-,1

=:f

H "fe)'

(n>

Denote Xt := gt(x) - y/nBu Yt := gt(y) - y/HBu Ut :='Xt/(Xt - Yt). The function IIK (C + , [0,x], [oo,j/]|7[0, £]) is a local martingale (with respect to the usual filtration for Bt). Applying conformal invariance for the map gt(z) — \fnBt as before, we write

n K ( c + , [0, x], [00, y ]| 7 [o, t]) = n K (c+ \ 7 [o, t], h(t), x], [«>, y\) = n K (C+, [0,gt(x) - y/UBt], [00,gt(y) - yfcBt]) = UK{C+,[0,Xt],[oo,Yt])

=

FK(Ut).

Critical percolation and conformal invariance

109

Thus FK(Ut) is a local martingale. Let us calculate its differential. By Loewner equation, 2 dXt = -^-dt - y/iidBu

2 dYt = —dt -

VndBt.

So by Ito's formula JTT

J

du

d

^t

_

2

Xt+Yt

y/R

~ x7=YtdBt

* = xT=Yt = xtYt(xt-Ytf = 2{Xi

+Y

^-Yt)ds

•X-tit

- yfcdB. = 2j^^ds u t/tU

- yfcdB.,

(12)

~ t)

where we changed the time so that ds = dt/(Xt — Yt)2. Applying Ito's formula to FK (one also has to prove that it is smooth) we get dFK(Ut) = F'(Ut) Uy~™*ds

- V^dB^j

= (2ut^lUut)F'(Ut)

1

+

-^Kds

+ ^ " ( ^ ) ) ds -

yfcF'(Ut)dBs.

Since FK(Ut) is a martingale, the drift term has to vanish, leading to the following differential equation for FK (which is similar to the Cardy's equation):

This equation can be integrated by writing .,

_ , , ,.,

FZ(u)

4 l-2u

( 4,

V

which leads to FK(u) =d+C2

[U (v(l - v)YA/K Jo

dv.

From obvious boundary conditions FK(0) = 0, FK(1) = 1 we derive the values of constants, arriving at FK(u)=

fU (v(l-v)yi/Kdv

I f

{v(l-v)yi/Kdv.

For K = 6 (and only for this value of K) solution coincides with Cardy's hypergeometric function. For K < 4 there is no solution, reflecting the fact that SLE is then a.s. a simple curve which a.s. doesn't touch the boundary.

7. Scaling exponents Calculations for SLE like the one above and knowledge about percolation can be combined to establish the values of mentioned scaling exponents. The method is explained in [19], and proceeds as follows.

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STANISLAV SMIRNOV

For a variety of percolation models Kesten has reduced in [8] most of the predictions about behavior near pc to establishing the so called one- and four-arm exponents for critical percolation. Namely, one has to prove that on the lattice with mesh 5 the probability of a yellow path extending distance > 1 from the J-neighborhood of the origin is x 55/is, while similar probability for two yellow and two blue paths simultaneously is x <54/3, 6 —• 0. Kesten's proof is quite involved and based on differential inequalities. These probabilities can be expressed in terms of the percolation interfaces, and with some technical work, it can be shown that they have the same asymptotics as similar probabilities for the interface scaling limit, which is SLE(6) by [18]. The required 5/48 and 4/3 exponents for SLE(6) are established by Lawler, Schramm, and Werner in [11-13].

8. Conclusions We described advances related to percolation on triangular lattice in the plane. It is interesting that the mathematical progress did not come along the lines of theoretical physics arguments, but from the different new directions. But as the physicists' approaches before, all of the described techniques use conformal invariance in an essential way and therefore are two-dimensional in nature. So unfortunately the progress made does not seem to have implications in higher dimensions. As for the other planar models, the Schramm-Loewner Evolution describes all possible conformally invariant scaling limits for interfaces. Calculation for other values of K is not much different from the case K = 6: most problems can be reduced via stochastic analysis to solving ordinary or partial differential equations. Of course, some of the resulting PDEs might be non-integrable. But it seems that more problems can be solved in this way than by theoretical physics' methods. For example, the so called "backbone exponent" for percolation which was out of reach for physicists was proven by Lawler, Schramm, and Werner [13] to coincide with the main eigenvalue of a certain PDE. Unfortunately, this eigenvalue does not seem to admit a nice formula. However some difficulties might arise with technical points in calculations: SLE(6) shares the locality property of percolation, so writing estimates is easier. Moreover, the relation between various scaling exponents, which was rigorously established for percolation by Kesten is mostly missing for other models, where it might be technically more difficult. The remaining part, establishing that other models have conformally invariant scaling limits, seems even more challenging. Lawler, Schramm, and Werner [14] have shown that the Loop Erased Random Walk converges to SLE(2), while a related model, perimeter of the Uniform Spanning Tree, converges to SLE(8). Recently Schramm and Sheffield defined a new random curve, Harmonic Explorer, which converges to SLE(4). In all these cases, like in percolation, some observables were shown to be discrete harmonic functions with various boundary conditions implying existence of a conformally invariant harmonic limit. There are two differences though: for these models discrete observables turn out to be exactly harmonic (for percolation only approximatively), and the proofs (similar to one sketched above for percolation) rely heavily on Loewner Evolution, whereas for percolation one can construct the scaling limit without appealing to SLE. Despite extensive predictions, similar problems are open for other critical lattice models in

Critical percolation and conformal invariance

111

t h e plane. P r o b a b l y there one should also look for discrete harmonic or analytic observables. A few cases which stand out are: — Other critical percolation models: the interface should also converge to SLE(6). Bond percolation on square lattice and Voronoi percolation share some duality properties of the site percolation on triangular lattice, so one might start with those. — Ising model at critical temperature: the interface should converge t o SLE(3). Here Kenyon's approach to dimer models provides some insight, but technical estimates are still missing. — Self Avoiding R a n d o m Walk: it should converge t o SLE(8/3), which also describes Brownian Frontier or percolation lowest crossing. Detailed study of SLE(8/3) by Lawler, Schramm, and Werner is discussed in Lawler's paper in this volume. There are also conjectures for t h e full spectrum of Q-state P o t t s models a n d FortuinKasteleyn models. We refer t h e reader t o expository works [6,10,20] for more details.

Acknowledgments T h e author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the K n u t and Alice Wallenberg Foundation. He was also supported by the Goran Gustafssons Foundation. Many of the discussed results are due to Greg Lawler, Oded Schramm, and Wendelin Werner.

References 1. M. Aizenman, "The geometry of critical percolation and conformal invariance", in STATPHYS 19 (Xiamen, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 104-120. 2. M. Aizenman, A. Burchard, "Holder regularity and dimension bounds for random curves", Duke Math. J. 99, 419-453 (1999). 3. M. Aizenman, B. Duplantier, A. Aharony, "Path crossing exponents and the external perimeter in 2D percolation", Phys. Rev. Let. 83, 1359-1362 (1999). 4. J. L. Cardy, "Critical percolation in finite geometries", J. Phys. A 25, L201-L206 (1992). 5. G. Grimmett, Percolation, Springer-Verlag, Berlin, second edition, 1999. 6. W. Kager, B. Nienhuis, "A guide to stochastic Lowner evolution and its applications", J. of Statistical Physics, to appear; arXiv:math-ph/0312056 . 7. H. Kesten, Percolation theory for mathematicians, Birkhaiiser, Boston, 1982. 8. H. Kesten, "Scaling relations for 2D-percolation", Comm. Math. Phys. 109, 109-156 (1987). 9. R. Langlands, Ph. Pouliot, Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. (N.S.) 30, 1-61 (1994). 10. G. F. Lawler, Conformally Invariant Processes in the Plane, book in progress, available from http://www.math.Cornell.edu/~lawler/books.html. 11. G. F. Lawler, O. Schramm, W. Werner, "Values of Brownian intersection exponents I: Halfplane exponents", Acta Mathematica 187 237-273 (2001). 12. G. F. Lawler, O. Schramm, W. Werner, "Values of Brownian intersection exponents II: Plane exponents", Acta Mathematica 187, 275-308 (2001). 13. G. F. Lawler, O. Schramm, W. Werner, "One-arm exponent for 2D critical percolation", Electron. J. Probab. 7, 13 pages (2001). 14. G. F. Lawler, O. Schramm, W. Werner, "Conformal invariance of planar loop-erased random walks and uniform spanning trees", Ann. Prob., to appear; arXiv:math.PR/0112234 .

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15. K. Lowner, "Untersuchungen iiber schlichte konforme Abbildung des Einheitskreises, I", Math. Ann. 89, 103-121 (1923). 16. S. Rohde, O. Schramm, "Basic properties of SLE", Ann. Math., to appear; arXiv:math.PR/ 0106036(2001). 17. O. Schramm, "Scaling limits of loop-erased random walks and uniform spanning trees", Israel J. Math. 118, 221-288 (2000). 18. S. Smirnov, "Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits", C. R. Acad. Sci. Paris 333, 239-244 (2001). 19. S. Smirnov, W. Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8, 729-744 (2001). 20. W. Werner, "Random planar curves and Schramm-Lowner Evolutions", in Lecture notes from the 2002 Saint-Flour Summer School, LNM, Springer, to appear; arXiv:math.PR/0303354 .

The energy of charged matter* JAN PHILIP SoLOVEjt

(Princeton)

In this talk I will discuss some of the techniques that have been developed over the past 35 years to estimate the energy of charged matter. These techniques have been used to solve stability of (fermionic) matter in different contexts, and to control the instability of charged bosonic matter. The final goal will be to indicate how these techniques with certain improvements can be used to prove Dyson's 1967 conjecture for the energy of a charged Bose gas — the sharp N7/5 law.

1. Introduction It is my aim in this contribution to review the main techniques developed to study the problem of stability of matter or rather stability or instability of ordinary matter. By ordinary matter I mean a macroscopic collection of charged particles (nuclei and electrons) interacting solely through electrostatic or electromagnetic forces. The problem of stability of (ordinary) matter interacting through electrostatic Coulomb forces was first formulated by Fisher and Ruelle [18] and proved in the seminal papers of Dyson and Lenard [12,13], although, of course the problem of stability of individual atoms goes back to the origin of quantum mechanics. An important assumption needed is that either the negatively or the positively charged particles are fermions, i.e., obey the Pauli exclusion principle. Without the fermionic assumption there is no stability as proved by Dyson in [11]. The importance of the Pauli exclusion principle for stability had been pointed out in the celebrated work of Chandrasekhar [5] on gravitational collapse and stability of white dwarfs. It was, however, not until the work of Dyson and Lenard that the importance of the exclusion principle for the stability of ordinary matter was emphasized. In [11] Dyson makes a very precise conjecture regarding the nature of the instability without the exclusion principle. It is my ultimate goal here to discuss the proof of this conjecture, which for the main part, is joint work with E. H. Lieb. Since the work of Dyson and Lenard there has been a lot of activity in the area of stability of matter. Most celebrated is the work of Lieb and Thirring [31] giving an elegant proof with a bound of the correct order of magnitude. Several variations of the model have been studied. Relativistic effects and magnetic interactions have been included. I shall review some of these results below. In recent years the attention has turned to studying the effect of quantizing the electromagnetic field, what is often referred to as non-relativistic quantum electrodynamics. I will not get into this recent development here. •Work partially supported by NSF grant DMS-0111298, by EU grant HPRN-CT-2002-00277, by MaPhySto — A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council. © 2003 World Scientific. This article may be reproduced in its entirety for non-commercial purposes. tOn leave from Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, DENMARK.

113

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JAN PHILIP SOLOVEJ

2. Charged m a t t e r in q u a n t u m mechanics The systems we discuss here are all given by iV-particle Hamiltonians (N being a positive integer) of the form

j=l

l
'

l

J

'

3

Here Zj £ R is the charge of particle j , Xj € M is the coordinate of particle j , and Tj is the kinetic energy operator for particle j . We shall here consider the kinetic operator to be of one of the following four types. (1) The standard non-relativistic kinetic energy T

3 = 3

-7T—&3-

2mj

3

(2) The relativistic kinetic energy

Tfel = ^/-c2Aj + my - mjC2. (3) The magnetic kinetic energy

(4) The magnetic Pauli kinetic energy

Above, rrij > 0 refers to the mass of particle j , A : R3 —> R3 is the vector potential for the magnetic field, and
The Pauli kinetic energy acts on spinor valued functions (spin 1/2 particles), see the discussion of the relevant Hilbert spaces below. The term U in (1) represents the magnetic field energy

U = ^- f |Vx A{x)\2dx. °7T JR3

We are using units in which Planck's constant h = 1 and the fundamental unit of charge e = 1. In these units the physical value of the speed of light c is approximately 137 or more precisely the reciprocal of the fine structure constant a. Many objections may of course be raised about the Hamiltonian HN- Even with the relativistic kinetic energy operator the Hamiltonian is not truly relativistically invariant. For spin 1/2 fermions the appropriate kinetic energy operator would rather be the Dirac operator. Also the magnetic field is treated classically as opposed to as a quantized field.

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115

Unfortunately, we do not today know how to describe a mathematically consistent fully relativistic model in which to even formulate the problem of stability. As is well known the Dirac operator is not bounded from below and introducing quantized fields requires a renormalization scheme. These questions are being actively studied at present, but we shall, as already mentioned, not discuss them here (see [4,16,25]). One approach to the Dirac operator would be to restrict to positive energy solutions for the free Dirac operator. For the non-magnetic case this would lead to a problem essentially identical to the problem with the relativistic kinetic energy T R e l . In the magnetic field case this is somewhat more difficult (see [27]). We now turn to the important question of which Hilbert Space the operator HN acts on. The operator will be unbounded, but we shall always consider the Priedrichs's extension of the restriction to C°° functions with compact support. Since the problem of stability is a question of lower bounds, the Priedrichs's extension is the correct setting. We begin by discussing the Hilbert spaces for just one particle, the one-particle space. We consider, in general the one particle spaces to be the square integrable functions corresponding to particles with q internal states, i.e., H\ = L 2 (R 3 ;C 9 ), where the non-negative integer q may be different for the different particles. In terms of spin, this corresponds to particles of spin (q — l ) / 2 . The many particle space of interest is then wphysica1=

( /\Hl)®[ V /

« l ' J. \j=N-K+l J

(2)

where K is a positive integer. In order for the Hamiltonian HN to act on the above Hilbert space we must require that all the masses and charges of the particles j = 1 , . . . , TV — K are the same. Put differently, these particles are identical fermions. For simplicity we assume that rrij = 1 and Zj = — 1, for all j = 1 , . . . , N — K. (Except for the sign of charge this simply amounts to a choice of units). Thus the fermions are negatively charged. We assume the remaining particles to be positively charged, i.e., Zj > 0 for j = N — K + 1 , . . . ,7V. Moreover, all the fermions must be described by the same kinetic energy operator. The positively charged particles could, in principle, have any kinetic energy operators and even simply vanishing kinetic energy (equivalent to infinite mass). Of course, the Pauli kinetic energy operator requires the spin of the particle to be 1/2, i.e., q = 2. Since fermions have half-integer spin one would maybe like to restrict the negatively charged fermions accordingly, i.e., the corresponding q to be even. For the discussion here this restriction, however, plays no role. The reader may ask why one does not consider the situation when all or some of the positively charged particles are fermions or bosons. In fact, stability in this case is a simple consequence of the situation discussed above, since considering fermions or bosons simply means restricting to certain subspaces. To study what happens when we ignore Fermi statistics all together we shall consider all particles to be described by the standard non-relativistic kinetic energy T and all to have mass rrij = 1. Moreover, we assume that all the charges Zj are either +1 or —1. We indeed consider the charge of each particle to be a variable. Put differently, we consider the TV-particle Hilbert space

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JAN PHILIP SOLOVEJ

N

HN =
(3)

x{l,-l»,

where the set { 1 , - 1 } refers to the charge variable. We define the ground state energy as the bottom of the spectrum of the Hamiltonian. We will not here address the question of whether this is and eigenvalue or not, i.e., whether there really is a ground state. When a magnetic field is present we consider the vector potential as a dynamic variable and we also minimize over it. Thus -^physical

= inf inf speo^phys C*\HN

— inf inf < — '

_,—

feffnw

physical JV

\{0}

(4)

and EN = inf specWjv HN = mil

* 6 C 0 ° ° n ^ \ {0}

}•

(5)

Here CQ° refers to smooth functions of compact support. In (4) we minimize over all smooth vector potentials for which |V x A\ € L 2 (R 3 ). Of course the minimization over the vector potential is only relevant if the kinetic energy really depends on A otherwise the minimum simply occurs for A = 0. The energy E^ on the space where we consider the charges as variables is precisely the same as we would get if we calculate the energy for any fixed choice of charges and afterward minimize over this choice. Note, in particular, that the charge variable commutes with the Hamiltonian H^. The reason for including the charges as variables is that the Hamiltonian is then fully symmetric in all N particles and not just in the positively or negatively charged particles separately. Strictly speaking the energies in (4) and (5) are only defined as the bottom of the spectrum if the rightmost expressions are finite (i.e., not —oo). It is in this case that we can define the operators as Priedrichs's extensions. The property that the ground state energies are bounded below is often referred to as stability of the first kind. It holds except in the cases when the fermions are described by the kinetic energy operators T R e l or Tp&uh. In these two cases stability of the first kind requires that max{zj}/c = max.j{zj}a is small enough in the case of T R e l or that q vaaxj {ZJ} / c2 = qmax.j{zj}a2 is small enough in the case of ^Pauli

3. Stability and instability of matter Stability of matter is a stronger statement than stability of the first kind. It means that there exists a constant C G R such that ^physical >

_CN

(6)

for all N, i.e., that the total binding energy per particle is bounded. Theorem 3.1 (Stability of Matter). On the space W^ hysical stability of matter (6) holds with a constant C that depends only on q and max{zj} if any one of the following situations hold:

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117

— the ferrnions are described by the standard non-relativistic kinetic energy T or by the magnetic kinetic energy T M a s ; — the ferrnions are described by the relativistic kinetic energy T R e l and qa(— qc~1) is small enough and ma,Xj{zj}a < 2/n; — the ferrnions are described by the Magnetic Pauli kinetic energy T P a u l i and qa{= qc~l) and qmaxj{zj}a2 are small enough. The case of the standard non-relativistic kinetic energy is the situation first settled by Dyson and Lenard [13] and later by Lieb and Thirring [31] with a constant of the correct order of magnitude. There was also a proof by Federbush [14]. Stability for the magnetic kinetic energy is an immediate consequence of the non-magnetic case and the diamagnetic inequality. In fact, the ground state energy is achieved without a magnetic field. Stability for the relativistic kinetic energy was first solved by Conlon [6] and then improved by Fefferman and de La Llave [17]. The version formulated here (which is sharp with respect to the bound 2/n) is due to Lieb and Yau [32]. The case of one electron and one nucleus had been studied previously by Herbst [22] and Weder [36] and the case of one electron and several nuclei by Lieb and Daubechies [10]. A proof of stability for the magnetic Pauli kinetic energy was first published by Lieb, Loss, and Solovej [26], but had been previously announced, (although with weaker bounds) by Fefferman (only later published in [15]). The result as formulated here is optimal in the sense that if either qa or qva.ax.j{zj)al are large then stability does not hold. This was realized in a series of papers [20,24,33]. In the case without Fermi statistics Dyson [11] proved that there is no stability and he, in fact, conjectured the following result. Theorem 3.2 (The asymptotic energy of a charged Bose gas). For the energy EN defined in (5) we have the asymptotics

j^= i " f {5./i v ^-*'/>h o '/* 2 = 1 }'

(7)

where

The reader may wonder why the theorem refers to a charged Bose gas when, in fact, no statistics was enforced in the definition of the Hilbert space HN- The reason is that the ground state energy on HN is the same as the ground state energy one would get if restricted to the fully symmetric subspace (i.e., the bosonic subspace). This is a fairly simple consequence of the facts that the Hamiltonian is fully symmetric in all variables and that the expected energy of any trial state does not increase if we replace it by its absolute value. It is a simple consequence of the classical Sobolev inequality (see below) that the right side of (7) is finite. Dyson proved in [11] that EN < —CN7/5, but not with the correct constant. His method and conjecture was inspired by the Bogolubov approximation, which had been previously used by Foldy [19] to calculate the energy asymptotics for the high density onecomponent Bose plasma (bosonic jellium). The Bogolubov approximation is usually applied to bosonic systems, it is therefore important that we can think of our system as such.

118

JAN PHILIP SOLOVEJ

In their original work on stability of matter [12] Dyson and Lenard obtained as a corollary a lower bound on the Bose energy EN > —C'N5/3 (see also Brydges and Federbush [3]). The correct exponent 7/5 was however first proved by Conlon, Lieb and Yau in [7], where one finds the lower bound E^ > —C"N7/5, but again not with the correct constant. Dyson's conjecture was finally proved in two papers establishing respectively an upper and a lower bound. An asymptotic lower bound of the form (7) was proved by Lieb and Solovej [30]. The corresponding asymptotic upper bound will appear in Solovej [34]. The rigorous calculation of Foldy's high density asymptotics for the energy of the onecomponent plasma can be found in Lieb and Solovej [29] (lower bound) and Solovej [34] (upper bound). The main goal here is to review the techniques used to prove Dyson's conjecture and to sketch the main steps in the proof. Many of the techniques used were developed to study stability of matter and I will therefore use the opportunity to briefly review these applications as well. The main strategy in proving the lower bounds in Theorems 3.1 and 3.2 is to estimate the Hamiltonian below by a simplified Hamiltonian for which one can get a fairly explicit lower bound. For Theorem 3.1 the simplified Hamiltonian is a non-interacting mean field type Hamiltonian. For Theorem 3.2 the simplified Hamiltonian is a non-particle conserving Hamiltonian, which in the language of second quantization is quadratic in creation and annihilation operators (a quadratic Hamiltonian). We first discuss how to treat the simplified Hamiltonians.

4. One-particle operators and quadratic Hamiltonians The most basic estimate on the ground state energy of a one-particle Hamiltonian, given in the next theorem, is a simple consequence of the Sobolev inequality / |VV>|2 > C(J |V'|6)1^3) C > 0 for functions i/) on K3. Theorem 4.1 (Sobolev estimate). Let V be a locally integrable function on R 3 , such that the Schrodinger operator —A + V may be defined as a quadratic form on C°° functions with compact support. Then

-\A + V>-CsJ\V(x)t/2dx where Cs > 0 and \t\- = max{—1,0}. This theorem of course immediately implies that the many-body operator N

1

sr--Ai+v(Xi) i=l

z

has the lower bound -CsN J \V(x)\_ dx. On the fermionic subspace we, however, have the much stronger Lieb-Thirring inequality [31].

The energy of charged matter

119

Theorem 4.2 (Lieb-Thirring inequality). On the fermionic space /\ integer) we have the operator inequality

H\ (M a positive

M

J T - 1 A , + V(Xi) > -Cvrq J

\V(x)fJ2dx,

where CLT > 0 V is a locally integrable function on R 3 . For the relativistic operator T R e l and the magnetic Pauli operator T P a u h one has similar inequalities on the space f\M H\{q = 2 for T P a u l i ). For T R e l (with m = 1) Daubechies [9] proved M

J2T™

+ V(xi)

> -Cuqc~3

/ \V{x)\idx

- Mc2.

(8)

For T P a u l i (with m = 1 and 2 = 1) the inequality AT ^-7;Pauli

+

K(a

* =1

,.)

/" 5 /2

3

> -Cft's J l ^ ( ^ ) | - ^ - ^ s c - /

2

ft \3/4 2 (^y |V x A | ^ j

f t \ X/4 (^y \V(x)\idxj

(9)

can be found in [26]. When studying bosonic systems one may, as explained above, use the Sobolev estimate to get a lower bound on the energy. This will in general not give the best dependence on the number of particles. E.g., for the charged gas this would lead to a lower bound —CN5^3 as in [3,12]. To get the sharp dependence —CN7/5 a more precise treatment of the interplay between the kinetic energy and the Coulomb potential is needed. This brings us to Bogolubov's method for calculating the energy of a Bose gas. In the Bogolubov approximation one assumes that most particles form a Bose condensate in a momentum zero state. The main contribution to the ground state energy will come from the correlation between two non-condensed particles with opposite momenta. This effect is most easily explained using the formalism of creation and annihilation operators. The following theorem gives a rigorous formulation of Bogolubov's method in the simple case used by Foldy in [19]. Theorem 4.3 (Bogolubov's method). Assume thatb± ± are four (unbounded) operators defined on a common dense domain on a Hilbert space, such that their adjoints are also defined on the same domain. Assume moreover that in the sense of quadratic forms on this domain we have the commutator identities [b*T,,z,XA

= [&r',,'A,,] = [h>,-,K,+] = [bT',+X,-]

=0,

forallz,z',T,T'

= ±,

and [6T)Z,b*iZ] < 1,

for allz,r

= ±.

Then for all t,g+,g- > 0 we have (again in the sense of quadratic forms) 1

XI r,2=±l

b

r,zK,z+

Yl

V9T9^zz'(b*+,zb+,z'+b*-,zb-,z'+K,zb-,z'

+b+,zb-,z')

z,z'=±l

> -(t + g++g.) + y/{t + g++g-)2 - {g+ +g-)2.

120

JAN PHILIP SOLOVEJ

The operator inequality in this theorem follows simply by completing squares (see also [30]). The importance of the lower bound is emphasized by the fact that it is sharp if the operators b±,± axe truly annihilation operators. In Bogolubov's method the operators are however not exactly annihilation operators. Rather one should think of b±tZ as the operator that annihilates a particle with charge z and momentum ± (here only the sign of the momentum is important) and creates a particle in the condensate (i.e., with momentum 0) with charge z. The value t represents the kinetic energy of the particles with momentum ± and g± represent the strength of the Coulomb interaction.

5. Reduction to a simplified Hamiltonian The reduction to a simplified Hamiltonian requires controlling the Coulomb interaction. It is often convenient to replace the Coulomb potential by the Yukawa potential

W

=* ^ ,

M>O.

Replacing the Coulomb potential YQ by a Yukawa potential Y^ with fj, > 0 amounts to introducing a long distance cutoff in the potential. It is easy to control this replacement since YQ — VM has positive Fourier transform (is of positive type). Hence N

J2

2

ZiZjiYo-Y^Xi-Xj)^-^^,

l
t=l

since (Y0 - VM)(0) = fi. In order to bound the many-body Hamiltonian H^ below by a one-body Hamiltonian one must estimate the two-body potential by a one-body potential. Theorem 5.1 (Onsager's correlation estimate). Given particles with charges at positions xi,..., XN, let

zi,...,Zff

Di = min{|:rj — Xj\ | ZiZj < 0 } ,

i.e. Di is the shortest distance from, the particle at Xi to a particle with the opposite charge. Then N

( 1

1

\

N ziZj Y^Xi - Xj) > - ]T>? ^12 (AM)2 + 2 (AM) + IJ Y^Di). l
Above Di depends only on Xi and on the positions of all the particles with the opposite charge of the particle at a^. This theorem allows us to separate the original Hamiltonian HN in a one-body Hamiltonian for all the negatively charged particles (with a one-body potential depending on the positions of all the positively charged particles) and a one-body Hamiltonian for all the positively charged particles (with a one-body potential depending on the positions of all the negatively charged particles). The proof of this theorem essentially goes back to Onsager [35] (for (i = 0), who addressed a stability question for classical charged matter. In [30] it is being used to introduce a short distance cutoff in the potential (i.e., to replace YQ hy YQ — Y^ for large /x). The one-body potential is controlled using the Sobolev estimate Theorem 4.1.

The energy of charged matter

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If both the negatively and positively charged particles are fermions, we may use Onsager's estimate together with the Lieb-Thirring inequality in Theorem 4.2 to prove stability of matter. This special case was treated by Dyson and Lenard in their first paper [12], where they also used a version of Onsager's correlation estimate, but did not have the Lieb-Thirring inequality at their disposal. In order to prove stability of matter as formulated in Theorem 3.1 one may use the following stronger correlation estimate due to Baxter [1]. Theorem 5.2 (Baxter's correlation estimate^.> Assume as before that all negatively charged particles have charge —1 Then

E T^-^- E (n-z^iziVDr1. l
l

3

>

i=l,Zi<0

The importance here is that the sum on the right is only over negatively charged particles. Thus the right side is a one-body potential for the negatively charged particles thinking of the positively charged particles as fixed. A sharper (although more complicated to state) estimate was given by Lieb and Yau in [32]. The estimate of Lieb and Yau is sharper in the sense that the coefficient 1 + 2z above may be changed to z, but additional bounded errors are needed. Remark 5.1 (The proof of stability of matter). A proof of stability of matter in the standard non-relativistic case may proceed as follows. We use Baxter's estimate to arrive at N

HN>

^2

(Ti-(l

+ 2max{zi})(D-1-R-1))-N(l

+

2msxx{zi})R-1

i=l,zt<0

> - C i V ( l + 2max{z i }) 2 , where we have used the Lieb-Thirring inequality Theorem 4.2 for the potential V = (l + 2ma,x{zi})(-D-1

+ i?" 1 )

and we chose R ~ (1 + 2max{zj}) _ 1 . The proof of stability of matter by Lieb and Thirring [31] used the No-binding Theorem of Thomas-Fermi theory (see Lieb and Simon [28]) instead of Baxter's estimate. If the fermions are described by the relativistic kinetic energy T R e l or the magnetic Pauli operator T P a u h the proof of stability is not quite as simple. Note in particular that a potential that has a Coulomb singularity is not integrable to the fourth power and one can therefore not directly use the inequalities (8) or (9). The reader is referred to [32] and [26] for the detailed proofs of Stability in these cases. 5.1. Many-body localization techniques In many aspects of many-body theory the method of localizing has been a very useful technique. Dyson and Lenard used it heavily in their papers [12,13]. We here discuss the two problems connected with localizing a charged system, i.e., the localization of the interaction and the localization of the kinetic energy.

122

JAN PHILIP SOLOVEJ

The simplest way to approach the localization of the kinetic energy is to do a Neumann localization. In fact, let — A ^ u be the Neumann Laplacian for a unit cube centered at y GM3. Then in terms of quadratic forms defined on smooth compactly supported functions we may write f ,,,-, -A= / -A^dy, (10) where —A is the Laplacian on all of R 3 . Of course, there is a rescaled version to cubes of other sizes. The structure of this identity, i.e., that we have written the original operator as an integral over localized operators is characteristic for the way we shall write localizations in this section. The idea is then that for each value of the integration parameter one has a localized problem that one will estimate. Afterward the estimate is integrated. Another approach is to use a smooth localization. Let x be a C°° function with compact support. Let Xyix) = x(x ~ v)- Then we have the identity

( | x2) (-A) = j

-Xy&Xy dy-J VX2,

(11)

which is often referred to as the IMS localization formula (see [8]). Instead of writing the localization as an integral one often uses a sum instead, but then one must introduce a partition of unity. For the problem of the charged Bose gas it turns out that we cannot use either of these localization methods. In this case we cannot completely localize the kinetic energy operator. We must still have some kinetic energy to control the variation between localized regions. The solution is to write the kinetic energy as a sum of a a high energy part and a low energy part. We then localize only the high energy part and use the low energy part to control the large distance variation. The result can be found in [30]. We shall state a simplified version here. Assume that \ with 0 < x < 1 is smooth has support in a unit cube centered at the origin in R 3 and that i / l - x 2 is also smooth. Let as before Xy(x) = x(x ~ ]))• Let a*(y) be the creation operator for a particle in a normalized constant function in the unit cube centered at y e R 3 . Let Vy be the orthogonal projection operator projecting on functions orthogonal to constants in the cube. Then for any bounded set fid3 and any 0 < s < 1 we have the following many-body kinetic energy localization estimate on the symmetric tensor product space 0 s y m I/ 2 (R 3 ) (the fully symmetric subspace of the full tensor product space)

(1 + £ (x, s)) £

-A, > ^

^vvV 3 r

• ] C ( ^ « o ( i / + ej)ao(v + ej) + 1/2 - y/<%(y)aQ(y) + 1/2

dy

3=

3vol(fi),

(12)

where ei,e 2 ,e3 is the standard basis in R 3 and e(x,s) —» 0 as s —» 0. Note that this is an explicit many-body bound in the sense that the right side of the operator inequality is not a one-body operator. The above estimate generalizes immediately to the situation where a charge variable is present. In this case there should be two creation operators, one for charge

The energy of charged matter

123

+ 1 and one for charge —1. There are rescaled version of (12), which hold if we replace the unit cube by cubes of other sizes. The first sum on the right side of (12) is the localization of the high energy part of the kinetic energy. High energy means much larger than s - 2 . The second sum on the right side of (12) represents the kinetic energy due to the variations between localized regions. Using the Cauchy-Schwarz inequality one sees that for a state *€®£mL2(R3)wehave

*> \Ja*o(y + ejK(y + ej) + 1/2 - y/<%(y)a0{y) + 1/2 j * > (V(*. K(y + tjH(y + **) +1/2)*) -

/ v

(*, (oSfoKte) +1/2)*)) • (13)

This allows us to think of the last term on the right side of (12) as a discrete Laplacian acting on the function y -» ^(*,(a5(y)Oo(y) + l / 2 ) * ) .

(14)

For the Bose gas we shall conclude that most particles are in the constant function state (the zero momentum state). More precisely, this means that ($,a,Q(y)a0(y)'$) is almost the expected total number of particles in the unit cube centered at y. The expectation of the last term on the right side of (12) will therefore essentially give a contribution equal to the Laplacian of the square root of the density (assuming that we can approximate the discrete Laplacian by the continuous Laplacian). We finally come to the discussion of the localization of the interaction. For the Coulomb or Yukawa interaction this can be done using a method of Conlon, Lieb, and Yau [7]. Let as before x be a smooth function supported in the unit cube centered at the origin. There is an LJ > 0 depending on \ s u c n that for all \i > 0 we have

a

X2J

53

z z

i jYn(xi-Xj)>

53

Xyi^Y^+Uxi-x^Xyix^dy-Nw.

(15)

Note that the effect of the localization function Xy o n the right side is that for each fixed value of the integration parameter y only particles that live in the unit cube centered at y interact with each other. Again it is easy to see how this estimate changes under rescaling. A very elegant version of this estimate was given by Graf and Schencker [21].

6. T h e lower b o u n d in Dyson's conjecture We shall now describe the main steps in the proof of the lower bound in Theorem 3.2. The reader should look in [30] for details. In each step we shall ignore certain errors and we shall not explain in details how these errors are estimated. In the detailed proof the errors are, in fact, only estimated at the very end when the main contributions have been identified. It is first of all important to understand that there are two relevant length scales in the problem. A long scale LQ, which is the diameter of the Bose gas and a short scale £o which is the distance on which the Bogolubov pairs interact. It will turn out that LQ ~ N~1^5 and to ~ N-V5.

124

JAN PHILIP SOLOVEJ

Step 1: We first localize the whole system into a large cube of size L » LQ ~ N~1/5. On the other hand L should not be too large in order to allow us to control the volume error in (12). This first localization is done using the Conlon-Lieb-Yau estimate (15) and an IMS-localization (11) of the kinetic energy. Note that after rescaling the error in (15) will be 2 2 2 NUJ/L < UJN6/5. The IMS localization error (i.e., N / ( V x ) / / X ) will be NL~ < AT 7 / 5 . Step 2: We do a second localization into boxes of size £, where £Q <SL £ <§C L0. This localization is done using the many-body kinetic energy localization (12) together with the Conlon-Lieb-Yau estimate (15). This time the error in using (15) is Nui/£ <§: NU)/£Q ~ UJN7'5.

The result of this localization is that the total ground state energy is estimated below by a sum of two terms. One term is the discrete Laplacian term discussed in the previous section. The other term is the local energy, which will eventually lead to the effective Hamiltonian for which we may apply the Bogolubov approximation. Step 3: Before proceeding with the analysis of the local short scale energy we introduce long and short distance cutoffs in the interaction. This is done as explained in the beginning of Section 5. Step 4: This is the final step in the reduction to the effective Hamiltonian. The localized two body potential is of the form W(x,x') = Xy(x)V(x — x')xy(x'), where V is a cutoff Coulomb potential, i.e, V = Y^ — Yv for some appropriate \x
5Z

E

zz

'{ua®Uf3,Wu0®us)a*azapz,aSz,ayz.

z,z'=±l af3-
The fourth step in the proof is to show that one may ignore (for a lower bound) all terms in this sum which do not have precisely two of the parameters a, /3,7,5 equal to zero. Moreover one may also ignore terms for which a = j = 0 or (3 = 5 = 0. This step in the proof is rather technical and uses heavily that we have been able to introduce cutoffs in the potential. The resulting two body interaction may be written

577^3 /

V
^^^'(KPJ+P^

+ b-PJ-P,z' + hXp,zh-v,f + b+PJ-p,*') * '

where V denotes the Fourier transform of the potential V, u± denote the total number of positively and negatively charged particles in the cube respectively, and b;,z =

(£3vz)-1/2az(VyXyeinaoz,

where a*(/) creates a particle of charge z in the state / and as before Vy is the projection orthogonal to constants. It is easy to see that for fixed p the operators &±p>z satisfy the conditions in Theorem 4.3 on the Foldy-Bogolubov method. It is also easy to see that the localized kinetic energy resulting from (12) may be estimated

The energy of charged matter

125

below by

where t(p) = |£ 3 p 4 (p 2 + is-2)-1. It is now immediate from the Foldy-Bogolubov method in Theorem 4.3 that the local ground state energy is estimated below by 2(2TT) :

/ (t(P) + Vyv[p)) - ^(t(Py + 2t(p)vyv(p)dP

where vy = u+ + v~ (which depends on the location of the cube, although this had been suppressed for u^). If we now ignore the various cutoffs (which may be justified) and simply replace t(p) by ^£3p2 and V(p) by 4tr/p2 we see that the above integral gives —Jvy ^~ 3 / 4 = -J(vy/t3)5/4£3, with J defined in Theorem 3.2. Step 5: In order to control the errors encountered in reducing to the effective Hamiltonian as well as in treating the discrete Laplacian term we must conclude that most particles are in the condensate, as explained in the previous section. To do this, we shall localize the number of excited particles. To be more precise, we may think of the Hilbert space as being a direct sum over subspaces with a definite number of particles in the condensate. The Hamiltonian is not diagonal in this representation. We want to conclude however that restricting any given state to only a small finite number of the subspaces will not significantly change its expected energy. This is achieved using the following result from [29]. Theorem 6.1 (Localization of large matrices). Suppose that A is an NxN Hermitian matrix and let Ak, with k = 0,1,...,N — 1, denote the matrix consisting of the kth supraand infra-diagonal of A. Let ip € CN be a normalized vector and set dk = (ip,Akil') and A = (ip,Aip) = ]Lfc=o dk- (ty need not be an eigenvector of A.) Choose some positive integer M < N. Then, with M fixed, there is some n € [0, N — M] and some normalized vector <j> € C ^ with the property that <j)j = 0 unless n + 1 < j
M-l

Af-l

2

(0,^)
(16)

k=M

where C > 0 is a universal constant. (Note that the first sum starts with k = 1.) Step 6: The final step is to combine the two parts of the energy described in Step. 2. The local energy was (approximately) bounded below by —J(vy/(.3)5^4£3 which when integrated over y and normalized by the volume of the cube gives

-jf

(-y/^f'dy. JR3

The other part of the energy is essentially the kinetic energy of the function in (14) (where we had actually ignored the charged variable). Using the result of Step 5 that most particles are in the condensate we may write this kinetic energy as (approximately)

v

U>( ^

/
2

dy.

126

JAN PHILIP SOLOVEJ

If we use that the total number of particles is N we have the condition that /M3 uy/£3 dy = N. Thus if we define (y) = N~4/5^l

i/5y/£3 we see by a straightforward scaling argument

that J 4>(x)2dx = 1 and that the ground state energy is approximately bounded below by N7/5 (\

f

(V(x))2 dx - J f

Mx))5'2

dy

which is of course bounded below by minimizing over all <j> as in (7).

7. The upper bound in Dyson's conjecture To prove an upper bound on E^ of the form given in Dyson's conjecture Theorem 3.2 we shall construct a trial function using as an input a minimizer <j> for the variational problem on the right side of (7). That minimizers exist can be seen using spherical decreasing rearrangements. Define <j>o(x) — N3/w{Nl/nx). Let 4>a, a = 1 , . . . be an orthonormal family of real functions all orthogonal to fa. We choose these functions below. We follow Dyson [11] and first choose a trial function which does not have a specified particle number, i.e., a state in the space © # = 0 ^ ^ , 0~to = C) more precisely in the bosonic subspace, i.e., the bosonic Fock space. We shall evaluate the expected value of the operator Y^N=O HN in our state. As our trial many-body wave function we now choose

*=X[{l-\i)1/AexJ-\l

+ \0aZ+ + \0al_-

£

£

^ « ' < z < « < ) |0),

(17)

where a*a z is the creation of a particle of charge z = ± 1 in the state (f>a, |0) is the vacuum state, and the coefficients Ao, Ai,... will be chosen below satisfying 0 < Aa < 1 for a ^ 0. It is straightforward to check that \P is a normalized function. Dyson used a very similar trial state in [11], but in his case the exponent was a purely quadratic expression in creation operators, whereas the one used here is only quadratic in the creation operators a*az, with a ^ 0 and linear in a^±. As a consequence our state will be more sharply localized around the mean of the particle number. Consider the operator oo

a=l

,2 a

A straightforward calculation of the energy expectation in the state VP gives that ( *, £

HNV ) = X20 /"(V0o) 2 + Tr ( I T ) + 2A* Tr ( / c ( r - v T ( r + l ) ) ) ,

(18)

where K. is the operator with integral kernel K{x, y) = (t>0{x) \x - 2/| _1 <j>0{y). Moreover, the expected particle number in the state \£ is 2AQ + Tr(r). In order for * to be well defined by the formula (17) we must require this expectation to be finite.

The energy of charged matter

127

Instead of making explicit choices for the individual functions a and the coefficients Xa, a ^ 0 we may equivalently choose the operator T. In defining T we use the method of coherent states. Let x be a non-negative real and smooth function supported in the unit ball in R 3 , with J V = 1- Let as before TV"2/5 < I < AT"1/5 and define xt(x) = e~3/2x(x/e). We choose T(x,y) =

(2TT)- 3

/

JWxR3

f(u, \p\)V£0 \0U
where P f is the projection orthogonal to (j>o, ^ l P ( i ) = exp(ipx) xe(x - u), and

1/

^ + IftrAfoodQ'

_A

We note that T is a positive trace class operator, T<po = 0, and that all eigenfunctions of r may be chosen real. These are precisely the requirements needed in order for V to define the orthonormal family (f>a and the coefficients Aa for a ^ 0. We use the following version of the Berezin-Lieb inequality [2,23]. Assume that £(£) is an operator concave function on K + U {0} with £(0) > 0. Then if Y is a positive semi-definite operator we have Tr(YZ(T)) > (27r)- 3 |e(/Kbl)) {KvV&YV&B^) dudp.

(19)

We use this for the function £(t) = ^/t(t+l). HY = I then (19) holds for all concave function £ with £(0) > 0. Of course, if £ is the identity function then (19) is an identity. This reduces proving an upper bound on the energy expectation (18) to the calculations of explicit integrals. After estimating these integrals one arrives at the leading contribution (in TV) A

2

/ ^ )

+ fj

2

( 5 P 2 + 2A O 2 0 O ( U ) 2 ^) f(u, \p\) - ^2\l4>0(ufv,f(u,

\p\)(f(u, \p\) + 1) dpdu

= A2|(V^0)2-j/(2A2)5/^/2, where J is as in (7). If we choose A0 = \/N/2 we get after a simple rescaling that the energy above is TV7/5 times the right side of (7) (recall that was chosen as the minimizer). We also note that the expected number of particles is 2A2, + Tr(r) = N + 6»(7V3/5), as N —• oo. The only remaining problem is to show how a similar energy could be achieved with a wave function with a fixed number of particles N, i.e., how to show that we really have an upper bound on E^. We indicate this fairly simple argument here.

128

JAN P H I L I P SOLOVEJ

We construct a trial function \&' as above, b u t with an expected particle number N' chosen appropriately close t o but slightly smaller t h a n N. Using t h a t we have a good lower bound on the energy EN for all N we may, without changing the energy expectation significantly, replace \1>' by a normalized wave function ^ t h a t only has particle numbers less t h a n N. Since the function N H-> E^ is a decreasing function we see t h a t the energy expectation in the state \I> is, in fact, an upper bound t o E^.

References 1. J. R. Baxter, "Inequalities for potentials of particle systems", Illinois J. Math. 24, 645-652 (1980). 2. F. A. Berezin, Izv. Akad. Nauk, ser. mat, 36 (1972). English translation: USSR Izv. 6 (1972). and F. A. Berezin, "General concept of quantization", Coram. Math. Phys. 40, 153-174 (1975). 3. David Brydges, Paul Federbush, "A note on energy bounds for boson matter", Jour. Math. Phys. 17, 2133-2134 (1976). 4. Luca Bugliaro, Jiirg Frohlich, Gian Michele Graf, "Stability of quantum electrodynamics with nonrelativistic matter", Phys. Rev. Lett. 77, 3494-3497 (1996). 5. Chandrasekhar, Subramanyan, Phil. Mag. 11, 592 (1931). 6. Joseph G. Conlon, "The ground state energy of a classical gas", Comm. Math. Phys. 94, 439458 (1984). 7. Joseph G. Conlon, Elliott H. Lieb, Horng-Tzer Yau, "The AT7/,S law for charged bosons", Comm. Math. Phys. 116, 417-448 (1988). 8. H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, "Schrodinger operators with application to quantum mechanics and global geometry", in Texts and Monographs in Physics, SpringerVerlag, Berlin, 1987. 9. Ingrid Daubechies, "An uncertainty principle for fermions with generalized kinetic energy", Comm. Math. Phys. 90, 511-520 (1983). 10. Ingrid Daubechies, Elliott H. Lieb, "One-electron relativistic molecules with Coulomb interaction", Comm. Math. Phys. 90, 497-510 (1983). 11. Freeman J. Dyson, "Ground state energy of a finite system of charged particles", Jour. Math. Phys. 8, 1538-1545 (1967). 12. Freeman J. Dyson, Andrew Lenard, "Stability of matter. I", Jour. Math. Phys. 8, 423-434 (1967). 13. Freeman J. Dyson, Andrew Lenard, "Stability of matter. II", Jour. Math. Phys. 9, 698-711 (1968). 14. Paul Federbush, "A new approach to the stability of matter problem. I, II" Jour. Math. Phys. 16, 347-351 (1975); ibid. 16, 706-709 (1975). 15. Charles L. Fefferman, "Stability of matter with magnetic fields", CRM Proc. Lecture Notes 12, 119-133 (1997). 16. Charles Fefferman, Jiirg Frohlich, Gian Michele Graf, "Stability of ultraviolet-cutoff quantum electrodynamics with non-relativistic matter", Comm. Math. Phys. 190, 309-330 (1997). 17. Charles Fefferman, Rafael de la Llave "Relativistic stability of matter. I", Rev. Mat. Iberoamericana 2, 119-213 (1986). 18. Michael Fisher, David Ruelle, "The stability of many-particle systems", Jour. Math. Phys. 7, 260-270 (1966). 19. Leslie L. Foldy, "Charged boson gas", Phys. Rev. 124, 649-651 (1961); errata, ibid. 125, 2208 (1962). 20. Jiirg Frohlich, Elliott H. Lieb, Michael Loss, "Stability of Coulomb systems with magnetic fields. I. The one-electron atom", Comm. Math. Phys. 104, 251-270 (1986). 21. Gian Michele Graf, Daniel Schenker, "On the molecular limit of Coulomb gases", Comm. Math. Phys. 174, 215-227 (1995).

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22. Ira W. Herbst, "Spectral theory of the operator (p 2 + m 2 ) 1 / 2 - Ze2/r", Comm. Math. Phys. 53, 285-294 (1977). 23. Elliott H. Lieb, "The classical limit of quantum spin systems", Comm. Math. Phys. 3 1 , 327-340 (1973). 24. Elliott H. Lieb, Michael Loss, "Stability of Coulomb systems with magnetic fields. II. The many-electron atom and the one-electron molecule", Comm. Math. Phys. 104, 271-282 (1986). 25. Elliott H. Lieb, Michael Loss, "Stability of a model of relativistic quantum electrodynamics", Comm. Math. Phys. 228, 561-588 (2002). 26. Elliott H. Lieb, Michael Loss, Jan Philip Solovej, "Stability of matter in magnetic fields", Phys. Rev. Lett. 75, 985-989 (1995). 27. Elliott H. Lieb, Heinz Siedentop, Jan Philip Solovej, "Stability and instability of relativistic electrons in classical electromagnetic fields", Jour. Stat. Phys. 89, 37-59 (1997). 28. Elliott H. Lieb, Barry Simon, "The Thomas-Fermi theory of atoms, molecules and solids", Adv. in Math. 23, 22-116 (1977). 29. Elliott H. Lieb, Jan Philip Solovej, "Ground state energy of the one-component charged Bose gas", Comm. Math. Phys. 217, 127-163 (2001); erratum, ibid. 225, 219-221 (2002). 30. Elliott H. Lieb, Jan Philip Solovej, "Ground state energy of the two-component charged Bose gas", preprint, 2003. 31. Elliott H. Lieb, Walter E. Thirring, "Bound for the kinetic energy of fermions which proves the stability of matter", Phys. Rev. Lett. 35, 687-689 (1975). 32. Elliott H. Lieb, Horng-Tzer Yau, "The stability and instability of relativistic matter" Comm. Math. Phys. 118, 177-213 (1988). 33. Michael Loss, Horng-Tzer Yau, "Stability of Coulomb systems with magnetic fields. III. Zero energy bound states of the Pauli operator", Comm. Math. Phys. 104, 283-290 (1986). 34. Jan Philip Solovej, in preparation. 35. Lars Onsager, "Electrostatic interaction of molecules", Jour. Phys. Chem. 4 3 , 189-196 (1939). 36. R. A. Weder, "Spectral analysis of pseudodifferential operators", J. Functional Analysis 20, 319-337 (1975).

Entropy production and convergence to equilibrium for the Boltzmann equation CEDRIC VILLANI (ENS

Lyon)

In this short paper, I shall address recent developments in the study of convergence to equilibrium for the Boltzmann equation, linked with the study of Boltzmann's H Theorem and entropy production. The presentation closely follows my lecture at the ICMP 2003 Conference in Lisbon. For pedagogy, I will use an informal style, at the cost of losing rigor and precision; but details and rigorous discussion can be found in the research papers quoted within the text.

1. A "simple" motivating problem In an empty box (modelled as a smooth open connected subset il of R 3 ) throw N ~ 1020 "independent" small particles (say billiard balls of radius r ~ l/\/N), according to some nice probability density fo(x,v)dxdv in phase space. Here "phase space" means the coordinate space of positions and velocities. Of course it is impossible to have really independent particles, since they have to exclude each other, but let us forget about this issue. The density of particles, or empirical measure, is given by the formula

fl(dxdv) :=

1

N

jjj^6^^)i=l

If N is very large, and the particles are (close to) being independent, some versions of the law of large number imply that at time 0, Jlo — fo(x,v) dvdx, in a sense which will not be made precise here. Now, at positive times the system evolves according to Newton's equations, with interaction between the particles, so the balls acquire new positions and velocities. Can one predict a good approximation of the density of particles at large times?? Here would be a physicist's guess. Let (ft)t>o be the solution of Boltzmann's equation starting from the initial datum /o. One expects (i) flt ~ ft (with very high probability), for given t, if TV is large enough; (ii) ft ~ Gaussian when t is large enough; B U T statement (i) cannot hold true when the time t is very, very large — because of Poincare's recurrence theorem, for instance. In fact, the Boltzmann approximation loses its validity under very, very long periods of time. Here "large enough", "very high", "very very large" should be quantified in terms of /o, interaction, shape of the box, number N of particles. Our seemingly simple problem actually appears to be tremendously complicated! To this date, neither statement (i) nor statement (ii) has received mathematical justification in a quantitative way. Statement (i) is related to Lanford's 1973 famous theorem (a

130

Entropy production and convergence to equilibrium for the Boltzmann equation

131

good account of this theorem can be found in Cercignani, Illner and Pulvirenti [3]). This talk is all about statement (ii). So the central question which will occupy this paper is the following: Can one establish quantitative rates of convergence to equilibrium for solutions of the Boltzmann equation? Before entering into the bulk of the subject, I should stress that the presentation which follows is quite informal and tries to conceal the extreme technicality of the subject. More precise information can be obtained in recent papers by the author, some of them in collaboration with Desvillettes [5,6,16] (all of them available from http://www.umpa.ens-lyon. f r / ~ c v i l l a n i ) . These references themselves contain pointers to various parts of the mathematical literature.

2. The Boltzmann equation The Boltzmann equation models the evolution of a dilute "chaotic" gas made of many particles interacting by binary, elastic, localized, instantaneous collisions. A lot of references and information about the mathematical theory of the Boltzmann equation can be obtained in the author's long survey paper [15]. The unknown in Boltzmann's equation is the time-dependent distribution function of the gas, in phase space: ft(x,v), where t = time; x = position 6 fl c R n (say n = 3); v = velocity. The quantity ft(x,v) can be thought of as the density of the gas around position x and velocity v, at time t. The Boltzmann equation itself reads (BE)

^ +

v-Vxf = Q(f,f). transport collisions

The collision operator Q may seem ugly to the non-familiar reader:

Q(f,f)=

[

f

\f(v')f(v'.)-f{v)f{y.)]B{v-v.,o)dcTdv*.

Here we used the notation ,

u + u* 2

\v — v,\ 2 '

, *

v + v* 2

\v-vJ 2

The reader can think of (v',i>*) as the velocities of two particles before they collided; after the collision, their velocities have become (v, v*). Moreover, B is the Boltzmann collision kernel, e.g. B = \v — v*\ (hard sphere kernel in dimension 3). The Boltzmann equation is not complete without b o u n d a r y conditions (for x £ dQ). Here are some simple examples of boundary conditions: bounce-back

specular reflection

f(x,v) = f(x,-v)

f(x,v) =

f(x,Rxv)

periodic

132

CEDRIC VILLANI

3. From kinetic to hydrodynamic description Most equations in fluid mechanics are written in terms of hydrodynamic models rather than kinetic models. From a kinetic description one can always define a hydrodynamic description, with the following recipes: p(t,x

:=

f dv

u(t,x

:= -

F(t,x :=

fvdv

(mean velocity),

f (v —u) <S> (v — u)dv

T(t, x := — t r P np

E(t,x

(density),

:= Jf\±dv

(tensor pressure field),

(temperature), = pl^f

+

^pT

(energy).

Besides their importance in physics, these fields will play a key role in the sequel.

4. Mathematical status of the Boltzmann equation The mathematical theory of the Boltzmann equation is still far from completion, and so far only scattered pieces of theory exist. Below are listed some of the most significant pieces; many details and references can be found in the author's discussion [15] of the Cauchy problem for the Boltzmann equation. — Existence of weak ("renormalized") solutions in great generality (for short-range interactions this is a celebrated theorem by DiPerna k, Lions (1989), which can be found in Cercignani, Illner and Pulvirenti [3]; for long-range interactions this is a recent result by Alexandre and the author [1]). All the rest is under restrictive assumptions: — A good theory of solutions close to equilibrium: this theory was started by Ukai (1974) and then continued by many authors since then. Among a great numbers of contributions, we highlight the recent works by Guo [8-10]. — A rather good theory for small solutions in the whole space (Kaniel-Shinbrot 1978, Illner, Toscani, . . . ) . — An excellent theory for spatially homogeneous solutions, i.e., no x variable (Carleman 1933, Arkeryd 1972, . . . ) . — It is still an outstanding problem to construct a decent theory for classical solutions in the large. Why is it so difficult to deal with the full Boltzmann equation? Here below are listed some of the main reasons: — the collision operator Q is quadratic; — Q is complicated, and fine analysis with it is quite a challenge;

Entropy production and convergence to equilibrium for the Boltzmann equation

133

— Q acts only on the velocity-dependence, not on the position, which introduces a fundamental degeneracy; — v • V x and Q get along awfully: virtually any trick which works well for one, is a disaster for the other. To give an idea of the present state of attempts towards a theory of smooth solutions in the large, here below is an example of conditional regularity result, worked out by the author: Under reasonable assumptions on B (say "hard potentials without cut-off", for instance) and periodic boundary conditions, I F one has the following a priori estimates: (i) the density and energy fields bounded from above, (ii) the density is bounded below (no vacuum!): p > 5 > 0, (iii) the pressure tensor is bounded below, in the sense of matrices: P > XIn (A > 0), T H E N one can construct very nice solutions: (1) all Sobolev norms (with derivatives in both x and v) of / are bounded; (2) all u-moments (J f\v\k dvdx) are bounded. N B : Condition (iii) above is rather natural, in the sense that / integrable =>• P > 0 />a.e.

5. B o l t z m a n n ' s H T h e o r e m : Let us define the H functional as the opposite of Boltzmann's entropy: H(f):=

f

f log f dvdx =

-S(f).

JClxRn

Let (ft)t>o be a nice solution of the Boltzmann equation. Then, (i) dH/dt < 0. More precisely, there exists a functional D>0,

jtH(ft) = -JD(ft(x,-))dx

acting on L 1 (R"), s.t.

<0.

entropy production (ii) Assume B > 0 a.e. Then D(f) = 0 iff f is a

Maxwellian:

Statement (ii) can be reformulated in terms of solutions of the Boltzmann equation: (ii1) The entropy production vanishes at time t iff ft is a local Maxwellian dynamical state": ft(x,v)

= MpuT(v),

p = p(x),

u = u(x),

= "hydro-

T = T(x).

To the H Theorem one can add the following complement: (iii) Assume appropriate boundary conditions and appropriate geometric conditions on £1; then, up to normalization of conservation laws, the B.E. admits a unique solution which is hydrodynamical (locally Maxwellian) at all times, (iv) Corollary: This is the only stationary state.

134

CEDRIC VILLANI

Main example: Assume n = 2 or 3, specular reflection, 0, connected and not axisymmetric, | 0 | = 1, total mass 1, total energy n/2. Then the only hydrodynamical solution of the Boltzmann equation is the global Maxwellian e ft(x,v) = M(v) = (2TT)"/ 2

6. Convergence to equilibrium Let / be a nice solution of B.E., with nice bounds, uniform in time. It is an easy task to prove convergence to a stationary state (for this you just have to combine the H Theorem with a bit of functional analysis). What is much, much more tricky is to find constructive bounds on the speed of convergence. Wrong strategy: Indeed,

Start by linearizing close to equilibrium and perform a spectral study.

— The "natural" estimates of Sobolev and moment bounds are not strong enough to allow the "natural" linearization of the Boltzmann equation in L2{M~l). — How long do we have to wait until the solution is sufficiently close to equilibrium, that it makes sense to linearize? This problem is not of technical nature but intrinsic to linearization. In fact linearization can only be understood as a complement to other "fully nonlinear" techniques.

7. Short history The problem of speed of convergence was pioneered by Kac [11] (1954), who tried to treat it in the spatially homogeneous case as the limit of a many-particle problem. On this occasion he introduced — "Kac's problem" about spectral gap in large dimension; on this problem there has been a number of recent contributions by Diaconis & Saloff-Coste, Janvresse, Maslin, Carlen & Carvalho & Loss, and the other; in particular, it is profitable to consult Carlen, Carvalho & Loss [2]; — the concept of "propagation of chaos"; — one of the first continuous mean-field limits; . . . but did not get anywhere about rates of convergence. Then McKean [12] (1966) introduced information theory into the game, for an oversimplified one-dimensional model called Kac's caricature. In the 1990's, Desvillettes, Carlen & Carvalho, Toscani & Villani obtained strong results in the spatially homogeneous case. These works are reviewed in a recent work [16] by the author, which will be discussed later on in this paper. The result to be presented in the sequel is about explicit rates of convergence in the spatially inhomogeneous case, for smooth solutions; most of it was worked out by the author in collaboration with Desvillettes.

Entropy production and convergence to equilibrium for the Boltzmann equation

135

8. Main difficulties In dealing with the problem of convergence to equilibrium one encounters difficulties which are reminiscent of the already mentioned difficulties for the Cauchy problem: — The complexity of Q, again. — The fact that the entropy production vanishes for hydrodynamical states, which can be seen as a reflection of the degeneracy of the collision operator. In particular, —• local Maxwellians may be a nuisance here! This is very different from the problem of hydrodynamical limit,

| £ + t , . V x / = |^Q(/,/),

Kn^O.

In that limit, on the contrary, one hopes (and in some cases proves) that the solution stays very close to be a local Maxwellian at each time. — One has to understand, and translate into equations, the crucial role of the transport operator v • Vx and the boundary conditions, which help selecting the equilibrium but do not produce any entropy! Remark: There are some common points between the way the problem was just formulated, and the hypoelliptic regularity problematic. The analogy can actually be pushed rather far.

9. M a i n result (Desvillettes & Villani, 2002) Here is our main result, formulated in a rather informal way; a precise statement can be found in the original research paper [6]. Let f be a solution of the Boltzmann equation such that (i) all Sobolev norms and moments of f are bounded, uniformly in t; (ii) ft(x,v) > K0 exp(-A0\v\i°). Then, after suitable normalizations, \\ft — M||£°o = 0(t~°°),

with explicit constants.

Remarks: — It was shown recently by Mouhot [13] that (i) => (ii) under realistic assumptions, at least for periodic boundary conditions. — In the spatially homogeneous case, say for "hard potentials with cut-off", a complete regularity theory can be worked out, and then such a theorem applies to prove convergence to equilibrium like 0(£~°°), see Mouhot and Villani [14]; but even if we take the regularity bounds for granted, going from spatially homogeneous to spatially inhomogeneous is still a monster headache; — The theorem as it stands applies almost verbatim to Guo's solutions, but does not need / to be close to equilibrium, and does not rely on linearization. — The usual strategy to attack the problem of convergence to equilibrium is to try to show that / looks like a hydrodynamic state in large time, then identify this state. Our proof goes somehow the opposite way, as we shall see.

136

CEDRIC VILLANI

10. Strategy of proof The functional used to measure the distance to equilibrium is the Kullback information of / with respect to M: H(f | M) = H(f) - H(M). This simplified form is due to the conservation of energy, actually the more general definition of the Kullback information is #(/!$) = / / l o g - , which is nonnegative as soon as / / = / g. Then our proof goes as follows: (1) Use a "sharp" quantitative H Theorem. (2) Establish a quantitative version of "instability of the hydrodynamic behavior". (3) Couple both features by — "additivity of the entropy" (total entropy = hydrodynamic entropy + kinetic entropy), — interpolation and geometrical inequalities. (4) Get a system of differential inequalities involving closeness to global and to local Maxwellians. Let us examine these various points in more detail one after another.

11. Quantitative H t h e o r e m Let / = f(v) be a distribution in velocity space. Let p, u, T be the associated density, mean velocity, temperature. This defines a Maxwellian distribution MpuT — Mpur{v). We know: £>(/) = 0 iff / = MfpuT. Our goal is to establish an inequality roughly looking like D{f) > "distance (/, MpuT)". Here D is the functional of entropy production, whose explicit form is

D(f) = \j(f(v')M)

- f(v)f{v,j)

log j^jf^Bdadvdv*

.

Presently, a good notion of "distance" is given by the Kullback information

H(f\MfpuT)=

I J

fVygf-dv. M

puT

Strictly speaking, this is not a distance (although often called Kullback-Leibler distance), but it behaves in some respects like the square of a distance. Cercignani's conjecture (around 1980): Under suitable positivity assumptions on the collision kernel B, holds the functional inequality D(f) > K H(f | MfpuT), where K depends on p and T — maybe on other a priori estimates on f as well.

Entropy production and convergence to equilibrium for the Boltzmann equation

137

To study this conjecture, it is sufficient to assume p = 1, u = 0, T = 1. In any case, it is false most of the time (Bobylev, Wennberg, Cercignani). The first positive results approaching it were established by Carlen & Carvalho (1992), with subsequent considerable improvement in Toscani & Villani (1999). A recent result by the author [16] establishes a sharp variant of Cercignani's conjecture which can be used in the spatially inhomogeneous problem under smoothness/moments/ lower bounds a priori estimates. The paper can be consulted as well for a review of the abovementioned previous results by Bobylev, Cercignani, Carlen, Carvalho, Toscani, Wennberg and the author. Theorem

(Villani, 2002):

(i) Let B — 1 -\-\v — u*|2 [nonphysical assumption!]; then Cercignani's conjecture holds true: D(f) > K H(f \ M) with

T*(f) := max eigenvalue of P. (ii) For any reasonable B (ex: \v — v*\), if f lies in all Sobolev spaces, has all moments finite, and satisfies a lower bound f > Koexp(—Ao\v\9°), then D(f)>K£(f)H(f\M)1+s.

Ve>0,

Here are some very sketchy elements about the proof of this theorem. To go from (i) to (ii), the key is to establish the tricky non-concentration estimate {f'fl-fU)\ogf-^do-dvdv*=0(5n-*H{f\M)1-*),

/ J\v—v.\<8

J J*

where 5 and e are arbitrarily small, and the constants depend on Sobolev norms of / as well as a (stretched exponential) lower bound. The fact that the exponent of 8 in the r.h.s. is close to n is not so important, but it is important that the exponent does not approach 0 as e —* 0, and that the exponent of H(f | M) be arbitrarily close to 1. This is what ensures that in the end the loss of exponent in the final inequality is arbitrarily small. Next, here is a vague sketch of proof for (i). The argument involves the relative Fisher information, well-known in information theory and statistics:

I(f\M) = Jf\Vvlog{f/M)\2dv, and the Ornstein-Uhlenbeck regularization semigroup (let's call it (St)t>o) generated by the PDE dtf = Avf + V„ • (/«). Note that {d/dt)H{Stf \ M) = -I(SJ | M). The key identity, partially algebraic, is a commutator identity: let £(F,G) := (F — 0)\og(F/G). Then

dt t=o

[St,£] = -J,

J(F,G)

= \V\ogF - V l o g G | 2 ( F + G).

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At the end of a quite intricate chain of inequalities, one can derive the following representation formula for a lower bound on D: /•+00

D{f)>K Jo

r

e" 4 "' / Jm2n

J(StF,StG)dvdv*dt,

where F(u,u») = f{v)f(v*) is a tensor product and G(v,v*) is the average w.r.t. a of f(v')f(v'*): ^ o n l y depends o n » | t ) » and \v\2 + |v*|2. This conflict of symmetries is the key: it implies, after another chain of inequalities, /

J{StF,StG)dvdv*

> KI(Stf

| M).

To conclude, one uses the inequality e-4ntI(g\M)>I(S2ntg\M) (a particular case of the Blachman-Stam inequality from information theory). The final result is

= • D(f) >K J

I(S{2n+1)tf

\M)dt=

^ - j - y H(f | M).

The complete proof is quite tricky and relies on the precise form of the collision operator; but once it is done, we can forget almost everything about the precise form of the Boltzmann collision operator, and only recall that -jtH(f\M)>KeH(f\MfpuT)^. In the spatially homogeneous case, this would be the end of the game: -jtH(f

| M) > K£H(f

| M)1+£

==*• By Gronwall, H(f \ M) = 0(t~i) and we are done. But in the spatially inhomogeneous case, p, u, T depend on x, so the entropy production does not control H(/ | M)!!! In fact, if / happens to be hydrodynamical at some time, then the instantaneous entropy production is zero this shows that we cannot aim at anything in Gronwall style.

12. Instability of hydrodynamic behavior Here is the main idea which we worked out with Desvillettes: If f approaches a local Maxwellian, not global, then f will depart "transversally" from the space M. of all local Maxwellian distributions. This vague statement will be quantified only on the average, and holds true only for generic local Maxwellian. In fact: — VT 7^ 0, or dev(u) ^ 0 = » departure from M; — symmetrized gradients of u =4> departure from the subspace of M with uniform temperature;

Entropy production and convergence to equilibrium for the Boltzmann equation

139

— gradients of p =$• departure from the subspace of M with uniform temperature and zero velocity field. In the above, we introduced the deviatoric part of the velocity field u, denoted by dev(u). It is defined as (Vit + ( V u ) r ) div(ti) 2 n i.e., the traceless part of the symmetric Reynolds tensor. It is well-known in hydrodynamics and plays an important role here too. Here is a schematic picture of the dynamics summarizing the above statements. The flow is supposed to represent in a fuzzy way the Boltzmann flow; the surface drawn here stands for the infinite-dimensional manifold of locally Maxwellian distributions; this manifold is unstable in some sense, except along a certain sub-manifold (no gradients of temperature, no deviatoric part); this sub-manifold itself is unstable, except for a sub-sub-manifold (not represented here; no gradients of temperature, no gradients of velocity); this sub-sub-manifold is in turn unstable, except for the global equilibrium M.

/

T=const dev(u)=0

The question is how to turn this picture into quantitative bounds. We used the following recipe (adapted from Desvillettes &; Villani [4]): compute a lower bound for d2 Let us answer some of the most immediate questions about that strategy: W h y the second derivative? 1) to estimate the speed of departure from M; since it is likely to vanish at second order in time if it ever vanishes, a lower bound on the second derivative will prevent it to vanish too much; H(f\MfpuT)

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2) by applying the equation twice, we shall have the transport operator v • V x enter the equations twice, and this will have the same effect as a A^; this is the way we recover "ellipticity". (Cf. the Chapman-Enskog procedure to go from the Boltzmann equation to the compressible Navier-Stokes system: the appearance of the viscosity in that process can be traced back to the repeated appearance of the transport operator.) W h y a square norm? 1) for smoothness: the square norm is easier to handle, than, say, the norm itself; 2) because H(f\g) anyway behaves somewhat like a square norm. In fact it is wellknown that ,, . ,,2 H{f | g) > KZJhL. W h y not look at (d2/dt2)H(f | M j u T ) ? Because it is very difficult, if not impossible, to control a Kullback information from above by something (almost) quadratic without using stronger estimates, like L°°(M~1 dvdx). Now that the goal is more clear, we can implement the strategy: after monster computations, we find ^

11/ ~ MfpuT\\h

> Kjj^*T\2

+ I tev(u)\2)dx - Ce(f) ||/ - MfpuT\\^H(f

|M)1^,

where Ce(f) depends on smoothness/moments of/ at high enough order (interpolation . . . ) , and a lower bound. Now there is no gradient of p in the r.h.s.! This reflects the existence of quasi-equilibria (the sub-manifold referred to above), for which the entropy production vanishes at high order in time (typically, order 4 instead of 2), but p is not uniform. To remedy this, we establish two "similar" inequalities with MpuT replaced by M

lu(T) —* presence of / | V s y m u | 2 d a ; ; M'p0l —• presence of / \Vp\2 dx.

We don't write down these inequalities explicitly here, the reader will find them all in Desvillettes & Villani [6].

13. Putting both features together It remains to "glue" together our quantitative H Theorem on one hand, and the instability of hydrodynamical regime on the other. This is not trivial, because these results are expressed in terms of different functionals. 1) First tool: Additivity of the entropy. H(f\M) where

We use formulas of the style

= H(f\MpfuT)

+

H(p,u,T)

Entropy production and convergence to equilibrium for the Boltzmann equation

H(p,u,T)= I

plogj^dx

= J(p]ogp-p + l) + J p^ 4>(T)=T

141

+^ ^

+

^T)),-^^)],

-logT-1.

There are two other "similar" formulas involving Mfu,T), 2) Second tool: "Geometrical" inequalities.

We use two types of inequalities:

j |VT| 2 dx > K{Q) \\T - (T)p\\

Poincare inequality: Korn inequality:

Mp01.

/ | V sym w| 2 dx > /C(fi) / | Vu| 2 dx.

—> For specular reflection (which results in the tangency condition: u tangent to the boundary), the Korn constant /C(0) is positive iff fl not axisymmetric! 3) Third tool: Relating entropies and L2 norms. H(f | MfpuT) > K£(f)(\\f

For this we use interpolation:

- MfpuT\\2L2)1+e,

y"(plo g/ 9-p + l ) < C ( / ) | | p - l | | 2 2 ,

etc.

etc.

At the end of the day, we obtain a system of 4 "nonlinear" differential inequalities (first and second-order in t) coupled by identities and inequalities. To deal with this system, we need a replacement for Gronwall's lemma. What will play this role is the following lemma: Lemma:

Let h(t) > 0 satisfy Vt € (ti, t2),

h"(t) + Ah(t)l~E

> a > 0,

for some e < 0.1. Then, a2n^iy

either t% —1\ is small:

t^ — t\ < 50 1 ! A21-12-') 1 r a ~ (\ 1 \ inf I —, —-^ ). KJ dt > 2 or h is large on the average: t2-hJ/ h(t) ~ 100 \A 'A J tl It is not completely the end: to apply this lemma, we need the differential inequalities to be valid on "not too short" time intervals, and therefore first rule out rapid oscillations of hydrodynamic quantities.

14. D a m p i n g of hydrodynamic oscillations This damping is the last step in our proof, and enables us to rule out the possibility of rapid oscillations in the hydrodynamical quantities in long time. Because M is stationary and / is smooth, one can establish inequalities of the style

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CEDRIC VILLANI

/

plogpdx


etc.

(we obtain three such new differential inequalities).

15. Conclusion of t h e proof After putting together all the pieces of the puzzle, we prove: for e < 0.01, the whole system of differential inequalities implies H(ft | M) =

0{t-1'™*).

This concludes the proof of the theorem. Let us now make a few remarks.

16. R e m a r k 1: Quasi-equilibria If p, u, T are such that VT = 0,

dev(u) = 0 ,

u • n = 0,

then the entropy production around the local Maxwellian MpuT vanishes up to order (almost) 4 in t. Do there exist nontrivial u's? This question amounts to ask whether the conformal group C(fi) is nontrivial. The answer to this question is not immediate, and depends on the dimension and the geometry of the domain: — n = 2, Q simply connected: C(fi) is always 3-dimensional (!); — n = 3: C(Q) is nontrivial iff fi = conformal image of an axisymmetric domain (this lemma was explained to us by Ghys; it is related to (but does not rely on) some famous results in the differential geometry, like the Liouville and Obata-Ferrand theorems). Remark:

Grad "proves": for fi non-axisymmetric, \dev(u)\2dx>K

\Vu\2dx

( u - n = 0).

This is obviously false in view of the above, but can probably be saved under more conditions on the domain. A good estimate of this kind would lead to better estimates in dimension 3 than in dimension 2 . . . [This criticism does not aim at diminishing the merit of Grad, whose intuition in the problem of convergence to equilibrium is actually very impressive.]

17. R e m a r k 2: Role of t h e geometry of fJ How does the shape of the domain affect the speed of convergence to equilibrium? This of course is a very natural and physically relevant question. In our proof, the shape of O only affects the values of Poincare and Korn constants! It is therefore of great interest to have explicit estimates on these constants.

Entropy production and convergence to equilibrium for the Boltzmann equation

143

For specular boundary conditions (u • n = 0), the positivity of the Korn constant K.(Cl), f \Vsymu\2 dx > /C(fi) f |Vu| 2 dx, quantifies departure of CI from axisymmetry. If ft is convex, /C(fi) is bounded below in terms of G(Q) — Grad's number (here for n = 3): G(n) := inf2 inf j||V sym M||?,2 (n) ; V • u ~ 0, curl(w) = a, u • n = o\ . Estimating this number turns out to be related to a Monge-Kantorovich minimization problem! (Desvillettes & Villani [5]) G(Sl) > K inf

W2(Cn,CT''af,

where C = Lebesgue measure; "sym; a" means "symmetrized around axis (g,a)"; W2 is the 2-Wasserstein distance,

W^{n, v)2 = infr#^=t/ / \x — ^(a;)! 2 d^(x).

18. Remark 3: Are there oscillations in the kinetic entropy?? The proof suggests that H(f | M* T), which measures how close / is from being hydrodynamic, may show strong time-oscillations. On some integrable baby models (Gaussian diffusion semigroups . . . ) , this is true. But does this tendency persist for the Boltzmann equation?? I do think that this may be the case, and to back this theory I invite the reader to take a look at the following numerical simulations performed by Filbet on a simplified geometry (periodic boundary conditions, 1-dimensional in position, 2-dimensional in velocity) with an accurate spectral code [7] (the length L is the size of the periodic box, the Knudsen number is of order 0.25): Local Relative Entropy Global Relative Entropy

0.1

Local Relative Entropy Global Relative Entropy

0.01 0.001 0.0001 1e-05 1e-06 1e-07 1e-08

0.2

0.4

0.6

0.8

L=l Figure 1. Time-evolution of the "local" (kinetic) and "global" entropies.

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C E D R I C VILLANI

In these experiments, the initial d a t u m was chosen to be in local equilibrium. Since this is a logarithmic plot, the convergence seems to be in fact exponential, as could be expected, and it is faster for a small box. W h a t is much more surprising is the fact t h a t , in t h e first picture, the dynamics behaves almost in a spatially homogeneous way after some time, contrary to the possibly natural guess t h a t the hydrodynamical regime would be attained first. In t h e second picture, t h e oscillations in t h e entropy production are quite well-marked, and definitely suggest t h a t the solution oscillates between states where it is quite close t o hydrodynamic, and states where it is not at all. This kind of behavior in any way is a clear indication t h a t things in the spatially inhomogeneous context can be much, much more subtle t h a n in the spatially homogeneous one. No such weird behavior is ever observed in numerical simulations of the spatially homogeneous Boltzmann equation.

References 1. R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55, 30-70 (2002). 2. E. Carlen, M. C. Carvalho, M. Loss, "Determination of the spectral gap for Kac's master equation and related stochastic evolutions", to appear in Acta Math. 3. C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, New York, 1994. 4. L. Desvillettes, C. Villani, Comm. Pure Appl. Math. 54, 1-42 (2001). 5. L. Desvillettes, C. Villani, ESAIM Control Optim. Calc. Var. 8, 603-619 (2002). 6. L. Desvillettes, C. Villani, "On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation", preprint, available from h t t p : //www. umpa. ens-lyon. fr/"cvillani. 7. F. Filbet, G. Russo, J. Comput. Phys. 186, 457-480 (2003). 8. Y. Guo, Comm. Math. Phys. 231, 391-434 (2002). 9. Y. Guo, Arch. Ration. Mech. Anal. 169, 305-353 (2003). 10. Y. Guo, Invent. Math. 153, 539-630 (2003). 11. M. Kac, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. Ill, 1956, pp. 171-197. 12. H. P. McKean, Jr., Arch. Rational Mech. Anal. 21, 343-367 (1966). 13. C. Mouhot, "Quantitative lower bounds for the full Boltzmann equation", preprint (2004). 14. C. Mouhot, C. Villani. "Regularity theory for the spatially homogeneous Boltzmann equation with cutoff", preprint, available from h t t p : / / w w w . u m p a . e n s - l y o n . f r / ~ c v i l l a n i . 15. C. Villani, in Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 71-305. 16. C. Villani, Comm. Math. Phys. 234, 455-490 (2003).

Aspects of free probability DAN VOICULESCU (U. California at Berkeley) Free probability theory provides a probabilistic framework for quantities with the highest degree of noncommutativity. This brings out the ties among von Neumann algebras of free groups, the large TV limit of random multimatrix models, the operators of the Boltzmann Fock space and combinatorics of noncrossing partitions. Many concepts of classical probability theory have free probability counterparts. In particular, the free analogues of entropy and of Fisher's information will be one of the main focuses of the talk. The analysis of the free variables involved, relies on free difference quotient derivations, which give rise to bialgebras in the class with derivation comultiplication, which appears to be selfdual. Duality for such bialgebras underlies the free entropy and analytic aspects of the free Markov property.

1. Introduction Free probability theory is a probabilistic framework for quantities with the highest degree of noncommutativity. In particular, this means that the same laws of randomness occur in different situations with maximum noncommutativity. Here, after a brief look at the theory as a whole, we will go into somewhat more detail about the free analogue of entropy and about certain free analysis questions related to the free difference quotient derivations. The subject originates in the study of operator algebras and we shall discuss links to random matrices and to duality for the class of bialgebras with derivation comultiplication. Besides a few references quoted in the text, at the end of each of sections 2, 3 and 4 there are brief bibliographical notes to connect with the list of references at the end.

2. Free probability 2.1. A game, which is rather often played, is to modify an axiom of a basic theory and to examine the consequences. In the case at hand the basic theory is noncommutative probability (a.k.a. quantum probability) theory. The modification occurs in the definition of independence, the independent quantities, instead of commuting, will be highly noncommuting in general. 2.2. The framework for noncommutative probability theory will be similar to that for quantum mechanical quantities. A noncommutative probability space will be an algebra A, 1 G A, of bounded operators on a Hilbert space H. endowed with an expectation functional

C, if (a) = « , 0 for some ( e H , ||£|| = 1. Elements a £ A, will be called noncommutative random variables.

145

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DAN VOICULESCU

2.3. A family of subalgebras (Ai)i£i, 1 € A%, in (A, ((oi - C, fJ.a(P) = 0

Aspects of free probability

147

let £(h)£ = h<£>£, h €.H, £ € TH be the left creation operators. Then if (7ii)iei are pairwise orthogonal subspaces in H, the family of sets of operators

ui = {e(h),f(h)\ he?U},

iel,

will be free w.r.t. the vacuum expectation T* e M, T(ST) = T(TS) and T(T*T) = 0 iff T = 0. In case such M is a factor, i.e., Centre(M) = C l and M is infinite-dimensional, M is called a factor of type Hi and the trace-state turns out to be unique. 2.11. If G is a group and, A(g)e/l = egh is the left regular representation on £2(G), then the von Neumann algebra L(G) generated by X(G) has a natural trace r(-) = (-ee,ee). In case G has infinite conjugacy classes (abbreviated i.c.c.) L{G) is a factor of type Hi. Suppose G = Gi * G2 is a free product of subgroups. Then the von Neumann subalgebras generated by A(Gi), A(G2) are free in (L(G),T) (and are isomorphic to L{Gi),L{G2)). 2.12. The large N limit of n-tuples of N x N random matrices also produces freeness and von Neumann algebras with trace states. Let Tjtff = T?N be such a n-tuple which is also hermitian. Then the GNS-construction of operator algebra theory can be used to construct a von Neumann algebra with faithful trace-state (M,T), generated by a n-tuple Tj = T*, 1 < j < n so that r(Th ... Tip) = lim A T ^ T r ^ . . . Tip,N) ((j an ultranlter and assuming the limits exist and yield bounded operators). 2.13. Theorem ([33, Cor. 2.14]). Let TjtN = T*N, 1 < j < n, n = n x + . . . + np be n-tuples of random matrices with classical distributions /ijv € Prob(A / ljy)". Let (Ui,... ,Up-i) e J7(A r ) p_1 act on (A\,..., An) by conjugating Aj byUq ifni + ... + n Q _i < j < ni + . . . + nq (if ni + . . . + n p _i < j < n, then Aj stays the same) and assume /xjy is invariant. If additionally || supp£t./v||oo < R then in the large N limit von Neumann algebra the sets Qs = {Tj I ni -I

h n s _i < j < ni H

(- n s } ,

1 < s
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DAN VOICULESCU

i.i.d. random variables in the classical theory. Moreover the large iV-limit von Neumann algebra, i.e., the von Neumann algebra generated by Si,...,Sn is isomorphic to L(F(n)) where F(n) is a free group on n generators. 2.15. The fact that L(F(n)) is asymptotically generated by a n-tuple of Gaussian random matrices has been useful both ways. It has added to the understanding of the large N limit of these random matrices, while at the same time it has been the source of much progress on the L(F(n))'s. For instance if P = P* = P2 G L(F(oo)), P ^ 0 is a projection then L(F(oo)) and PL(F(oo))P are isomorphic (the case T(P) G Q was done by the author in [25] and then completed by Radulescu also for T(P) G R\Q [14]). This scaling result, which concerns von Neumann's fundamental group of L(.F(oo)) uses in an essential way the symmetry and scaling properties of Gaussian random matrices and their interplay with asymptotic freeness. 2.16. The Hi-factors (M, r) of von Neumann are a wonderful world of geometries of linear subspaces, corresponding to the selfadjoint projections P = P* = P2 £ M with dimensions T{P) G [0,1]. The most basic and smallest such M is the hyperfinite Ili-factor, which by a deep theorem of Connes can be defined as the Ili-factor L(G) obtained from any countable amenable i.c.c. group. Free probability theory and, in particular, the results obtained via the asymptotic random matrix realization has made the extended class of free group factors (i.e., factors PL(F(n))P for some projection P G L{F{n))) the second important class of Ill-factors. For instance, some of the deepest results in subfactor theory are about subfactor inclusions of hyperfinite Hi-factors N C M. On the other hand not all inclusion data (the standard invariant) have hyperfinite realization, but it has been shown (combined work of Shlyakhtenko, Ueda and Popa) that there are always realizations as subfactors of a free group factor. 2.17. Note also that like the hyperfinite class which has an extension to type III there is an emerging class of "free type III" factors (Shlyakhtenko, Radulescu). 2.18. At this time it seems that such "free" (or "free group") von Neumann algebras may arise from mathematical models in physics either via creation and annihilation operators on the full Fock space or from some associated subfactor inclusion without hyperfinite realization or, last but not least, from random multimatrix models. The last possibility is demonstrated for instance by the large N 2D Yang-Mills QCD (work of I. M. Singer, M. Douglas, R. Gopakumar and D. Gross). It seems natural to formulate the following 2.19. Guess: the large N limit of random multimatrix models with distributions daN = cN exp(-JV Tr P(Ai

,...,An))d\,

P a polynomial, give rise to von Neumann algebras in the class of free group factors. 2.20. Bibliographical notes. Expositions of free probability and more references are in [38] and [35]. For the combinatorial approach see [20]. The first paper on free probability theory is [24]. Applications of free probability and especially of the random matrix model [26] to von Neumann algebras are in [25], [14], [7], [15], [17], [13], [23]. About connections with some models in physics see [6], [9], [19].

Aspects of free probability

149

3. Free entropy 3.1. Shannon's differential entropy of a n-tuple of numerical random variables ( / i , . . . , fn) with joint distribution p(t\,..., tn) dt\... dtn is given by H(fi,...,fn)

= - /

p(ti,...,tn)\ogp(ti,...,tn)dt1...dtn.

By free entropy we mean a quantity x(-^i> • • • ,Xn) where Xj = X* G (M,T) (a von Neumann algebra with faithful trace state) which has similar properties to H under the free/classical parallelism, i.e., under free independence instead of classical independence, semicircle law instead of Gauss law, etc. There are two approaches: one based on microstates, the other on the free analogue of Fisher's information. 3.2. The Boltzmann formula S = klogW provided the idea for the first approach to free entropy. In view of the asymptotic freeness results for random matrices it was natural to look for matricial microstates. Let r # ( X i , . . . , Xn; m, k, e) be the set of n-tuples (Ai,..., An) G (M%a)n, || A,-1| < R, 1 < j < n, so that \k-1Trk(Ah...Aip)-T(Xil...Xip)\<e for all 1 < ij < n, 1 < j < p, 1 < p < m. Then x(-X"i> • • •, Xn) is defined by taking sup inf inf R>0

e>0 m£N

of limsup (fc _ 2 logvoir(. • •) + — logfc) fc-^oo ^

where vol is the Euclidean volume on structure.

2

(Msha)n

/

w.r.t. the Hilbert-Schmidt Hilbert space

3.3. If n = 1, Tji(X;...), R > ||X|| is approximately a tube around the unitary orbit {UAU* | U G U(k)} of a microstate A. This rough idea underlies the (rigorous) proof of the formula X(X) = J J log \s ~ t\ Ms) dfx{t) + ^ + ~ log 2TT where // = Hx is the distribution of X. 3.4. If n > 1 and Xi,...,Xn are free, then the asymptotic freeness result for random matrices implies that T(X\,..., Xn;...) is close in measure as k —> oo to a product r j 1 < < n r ( X J ; . . . ) . This "explains" the fact that, assuming x{Xj) > —oo (1 < j < n) we have

x(xu...,xn) iff X\,...,

= x(x1) + --- + x(xn)

Xn are free.

3.5. There is also a change of variable formula X(F1(X1,...,

Xn),...,

Fn(Xu ...,Xn))=

log \det\(J(F))

+ X(*i,

...,Xn)

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DAN VOICULESCU

where F\,...,Fn are noncommutative power series, |det| the Kadison-Fugles positive determinant in Mn ®M® Mop, J(F) <E Mn M Mop a Jacobian of partial free difference quotients, etc. Free difference quotients are explained later in this article, for other details and conditions concerning this formula we refer the reader to our original paper. 3.6. Other properties of free entropy which are similar to the classical ones are semicontinuity, subadditivity and a semicircular bound (analogue of the Gaussian bound). 3.7. The use of microstates poses some yet unresolved technical problems. Some problems arise from the limsup in the definition of Xi which we would like to be actually a lim as k —• oo. This difficulty affects our ability to prove the analogue of the property of x ln 3.4 when the singletons {X\},..., {Xn} are replaced by groups of variables. The best we can do at present is to establish this property for a modification Xw °f X where the lim sup is replaced by the limit after some ultrafilter u>. 3.8. There are also surprises, properties not predicted by the classical theory, like degenerate convexity. If X\,..., Xn, n > 1 generate M and r = Or' + (1 — 6)T" where T', T" are tracestates, T' ^ T", 0 < 6 < 1 then x(Xi,...,Xn)

= -oo.

(The condition is equivalent to M not being a factor and n > 1.) 3.9. Free entropy has had important applications to the solution of problems on von Neumann algebras. For instance, the question whether every separable Hi factor M has a Cartan subalgebra (i.e., a maximal abelian von Neumann subalgebra A C M, so that its normalizer {u e U{M) \ uAu*} generates M), was answered in the negative by showing that L(F(n)) is a counterexample. The question means, roughly, whether all separable Hi factors arise from measurable ergodic equivalence relations, i.e., originate in ergodic theory. The idea of the proof we gave is that on one hand L(F(n)) is generated by a semicircular system Si,...,Sn for which x(5i>• • • >Sn) = nx(Si) > ~°°! while on the other hand we showed that the existence of a Cartan subalgebra implies that x(^i) • • • >^n) = — °° f° r any generator T\,... ,Tn. Several other results in this direction were obtained by L. Ge, M. Stefan, D. Shlyakhtenko, K. Jung and K. Dykema. 3.10. An important problem for free entropy is the variational problem of maximizing X(T1,...,Tn)-T(P(T1,...,Tn))

where P is a given selfadjoint noncommutative polynomial. It is the natural generalization of the n = 1 case, which is equivalent to the classical problem of maximizing / /

log \s - t\ dfx(s) dn(t) - J P(t) dn(t).

3.11. The second approach to free entropy is based on a free analogue of the Fisher information. Classical Fisher information can be defined as the derivative of entropy under an independent Gaussian increment:

lH(f

+ e1/2g)\e=0=FiSher(f)

Aspects of free probability

151

where / , g are independent and g is (0,1) Gaussian. A function can be recovered (up to a constant) from its derivative, similarly the notion of entropy from the Fisher information. If / has distribution p(t) dt where p is smooth with compact support, then „'\2

Fisher(/) = f

SOI*

L2(p)

where Jj is the differential operator of derivations acting on test-functions in L2(R,pdt)

p

and

\dtj

is the statisticians score function. The Fisher information naturally extends to the free nvariable context. Besides the usual dictionary entries Gauss/semicircle, independent/free, etc., there is an additional entry for the derivative. 3.12. The free difference quotient replaces in free probability (more generally in "free analysis") the usual derivative. Let 1 G B c M be a selfadjoint subalgebra and let X = X* G M. Assume there is no nontrivial algebraic relation between B and X, so that the algebra generated by them identifies with B(X), the one-variable polynomials with noncommuting (except scalars) coefficients in B. We define the derivation dx..B : B{X) ^ B{X) ® B(X) by dx-.BboXbiX,..

.,bn = ]>J

b0X,...,bj-1®bjX,...,bn.

l<j
In case Xj = Xj € M, 1 < j < n, satisfy no nontrivial algebraic relation, they generate a unital algebra isomorphic to C ( X i , . . . , X n ) , and putting Bj = C{Xi,..., XJ-I,XJ+I, ..., Xn) we have C ( X i , . . . ,Xn) = Bj(Xj). We define the j - t h partial free difference quotient derivation dj:C(X1,...,Xn)^C(X1,...,Xn)®2 tobedxy.Br Note that in case B = C, C(X) identifies with C[X] (the commutative polynomials) and under the isomorphism C[X]®2~C[X,F] the derivation dx-.c corresponds to the difference quotient v

;

X-Y

This explains our terminology. 3.13. The conjugate variable of X w.r.t. (a.k.a. free score) is defined by J(X

: B) = d*x,Bl <8> 1

152

DAN VOICULESCU

(if the adjoint exists). Here dx-.B is viewed as an unbounded operator defined on B(X) C L2(M,T)

taking values in L2(M,T)

® L2(M,T)

and L2(M,T)

is the noncommutative L2-

space for the scalar product (mi, 1712) = Tfm^mi). Similarly we define Jk=d'kl®l

=

J{Xk:Bk)

when given Xj = Xj G M, 1 < j < n. The existence problem for conjugate variables is overcome by semicircular perturbations. If S G M is (0,1)-semicircular and free with B(X) then \\J(X + eS:B)\\ < 2 e " 1 (i.e., J(X + eS : B) is actually in M). The free Fisher information of a n-tuple X\,..., in M is **(X1,...,Xn)= ]T r(J2).

Xn

l
The corresponding free entropy is defined by x%Xu...,Xn)

J™(-^t-$%X1+t1/2S1,...,Xn+t1/2)n}dt

= ^log27re +

where Si,..., Sn is a semicircular system free with {Xi,..., Xn}. x* is sometimes called the nonmicro states free entropy or micro states-free free entropy. Several classical results in this context carry over to the free setting (Cramer-Rao inequality, Stam inequality, informationlog-Sobolev inequality, etc.). 3.14. The conjugate variables are also related to the free entropy defined in the matricial microstates approach. If Pk — P* £ C ( X i , . . . ,Xn) then —X(Xi+ePi,...,Xn

+ ePn) 6=0

provided Ji,...,Jn

= £ r(JkPk), l
exist.

3.15. The theory of x* is incomplete due to some unresolved technical problems like the continuity problem for $*(Xi +t^2Si,..., Xn + t1/2Sn) (only right semicontinuity has been proved). 3.16. An important question is unification, i.e., proving x = X* (maybe under certain conditions). For n = 1 this is easily verified. For n > 1 only the inequality x < X* has recently been established by P. Biane-M. Capitaine-A. Guionnet [2]. Though it is the easier of the two inequalities comparing x a n d X* t n e proof is quite difficult. The approach relies on the connection to large deviations for a n-tuple of Gaussian random matrices. Here x is related to the quantity to be estimated while x* to the rate function. This brings to the proof some powerful probabilistic techniques from large deviations and stochastic analysis. 3.17. We have also studied a notion of mutual free information for a pair A, B of von Neumann subalgebras in (M,T). The approach is similar to the free Fisher information approach to x* • I n * m s c a s e dx-.B is replaced by a new derivation dA..B : A V B -» (i4 V B)®2

Aspects of free probability dA-.Ba — a ® 1 - 1 a,

153

OA-.BB = 0

where A V B is the algebra generated by >1 U 5 and its is assumed there is no nontrivial algebraic relation between A and B. 3.18. Bibliographical notes. The basic references for most of the material about entropy are the author's papers [29-33]. A survey of free entropy is in [36], where the unification problem and variational problem are briefly discussed. Applications to Neumann algebras are in [30], [8], [21], [18]. Some large deviations papers connected to entropy are [5], [2]. Further work about free entropy is in [3] and [12].

free also von free

4. Duality for t h e coalgebra of dx-.B 4 . 1 . The free analysis questions around free difference quotient derivations are perhaps best understood within the bialgebra duality framework to which we turn now. We have coassociativity relations: {dx-.B ® i d ) O dx-.B = ( i d ® 9 x : s ) O dx-.B

and (di id) o dj = (id ®dj) o di. Actually due to reduction tricks like the identification in case n = p2 B(X1,...,X1?)~(B®MP)(X) the (d\,..., dn) and dx-.B situations are related and we will consider only dx-.B here. The coassociativity makes (B(X),8X-.B) a bialgebra the comultiplication of which is a derivation. (In the case B = C, this structure on C[X] for the classical difference quotient played a key role in G.-C. Rota's revival of umbral calculus.) 4.2. For the B(X) C (M,T) situation occurring in the nonmicrostates approach to free entropy, it is appropriate to let dx-.B a c t on a larger ring (like for the usual derivation, which we don't want to restrict to polynomials). As a consequence of using some closure of dx-.B: w e w m have inverses and, in particular, resolvents and their generalizations at hand. 4.3. A generalized difference quotient ring (GDQ ring) (A, /x, d) is a unital algebra (^4, /i) with a coassociative comultiplication d (did)od = (idd)od which is a derivation d o n = (id ®/i) o (d ® id) +

(M ® id)

o (id ®<9).

This structure has the remarkable feature of being selfdual. Let A' denote the dual of the vector space A. If (A,fi,d) is a GDQ ring then (A',d',fj,*) is a (topological) GDQ ring ("topological" refers to using a topology on A' and a topological tensor product). The one-line proof is that dualizing the relation for d o \i we have: tf o 8' = (9* ® id) o (id gi/x*) + (id «><9*) o (/i* ® id).

154

DAN VOICULESCU

4.4. The corepresentations (grouplike elements) of (A, fi, d), i.e., matrices a €E M.n{A) so that (d id.Mn)a — a ®Mn a-> are important in understanding the dual. If there is X € A so that dX = \®1 then elements of the form where j3j € A4„(kerd), are corepresentations. In the (M,T) context where we have many invertible elements, the above formula provides a good supply of corepresentations. Actually, we will use only those with /33 = pi = / , i.e., generalized matricial resolvents

a=

((3-X®In)-\

Roughly, the idea for a duality transform is to consider a map A'

9 if -* 0 ( y > i d ^ d i m a

a

) (a) e 0

Mdlm a a

with a running over some set of corepresentations. We will do better by having the range of the map in a natural topological GDQ ring and the transform a morphism. 4.5. Let B be a C*-algebra (i.e., a selfadjoint algebra of operators on Hilbert space, closed in the norm-topology) and assume 1 € B. A fully matricial B-set is fi = (Cln)n>i where fi„ C Mn(B) are such that fim+„ n (Mm(B) © Mn(B)) = 0 m © Q.n and (AdS®IdB)(ft„)=fi„ where S € GL(n,C) and (Ad S)(T) = STS"1. CI is open if each Cln is open. Similarly, we define a scalar fully matricial analytic function on fl, to be / = (/n) n >i where / „ : 0 „ —* .M„(C) is analytic, /m+n|fim©fi„

=

fm © fn

and / „ o (Ad S ® id B ) = Ad S o fn. The set of such / will be denoted by A(fl). Fact ([37]). A (f2) is a topological GDQ ring. In case Q. = Cl* A(Q,) has a natural involution and a "dual positivity" relation. Here Cl* = (f^)„>i. For the definition of the comultiplication and dual positivity we refer to [37]. In case B = C, and fln = {Te Mn(C) | a(T) C Qi} then the comultiplication on the f\-component

of / is just the difference-quotient

h{z\)- hi**) z\ - Z2

on analytic functions. 4.6. Returning to B(X) C M, we can ensure that dx-.B is closeable in a suitable topology by a free semicircular perturbation X —» X + eS, in which case dx-.B will a l s o be defined on the

155

Aspects of free probability

ring A obtained by adjoining to B(X) the matrix-coefficients of all matricial B-resolvents (/? - X In)'1 where 0 e pn(X; B) = {(3e Mn(B)

\0-X®In

invertible}.

Then p(X;B) = (pn(X;B))n>i is an open fully matricial B-set we shall call the fully matricial B-resolvent of X. The duality transform A'Bip->((tp®id.Mn)(»

-X®

In)~l)n>i

e

A(p(X;B))

is then a GT>Q-ring morphism intertwining involutions and dual positivities. In case B = C, this boils down to the n = 1 component and to the map t)'xdp{t)

Meas (o-(X)) 9 p -> f (z -

where the right-hand side is C(p)(z) the Cauchy transform of p defined on C\a(X). dual multiplication, when defined, is such that C(p1#p2)(z)

=

The

C(p1)(z)C(p2)(z)

and the comultiplication

c(dp)(Zl,Z2) =

c

^ Z\ -

c

^ .

Z2

4.7. The dual multiplication underlies the definition of conjugate variables. Indeed, we have T(*J(X : B)) = (T ® r ) o dX:B(») = T#T. 4.8. The B-resolvents are quite natural in free probability. Indeed the "conditional probabilities" amount to replacing C by an algebra B, i.e.,

B, B which is given by the orthogonal projection of L2{M,T) onto L'2(B,r). 4.9. The coalgebra structure and corepresentations for dx-.B are at the heart of some important analytic subordination results in free probability. Here is the simplest example. Let X = X*, Y = Y* be free in (M,T). Then there is an analytic function u : {Imz > 0} - » { I m z > 0} so that T((X

+

Y-ZI)-1)=T((X-U(Z)T1).

ls a The key to this result is the fact that the conditional expectation EVN(X+Y)\VN(X) GDQ-ring homomorphism of the bialgebra of dx-.c to the bialgebra of dx+Y-.c and hence maps corepresentations to corepresentations. (The subordination result is essentially due to the author, was followed by the realization of P. Biane that it was the consequence of

156

DAN VOICULESCU

an operator-valued subordination result and was t h e n taken u p again by us, generalized t o B-resolvents and explained via the bialgebra structure.) 4 . 1 0 . T h e use of matricial B-resolvents has been essential in some remarkable random matrix results obtained via free probability methods. One example is the recent result of U. Haagerup and S. Thorbjoernsen ([10]) t h a t for any noncommutative polynomial and the Gaussian i.i.d. n-tuple of hermitian random matrices limjv->oo \\P(TitN,...,Tn,N)\\ = | | P ( 5 i , . . . ,-Xn)|| almost surely. T h e other example is the work of D. Shlyakhtenko [16] on Gaussian random band matrices. 4 . 1 1 . B i b l i o g r a p h i c a l n o t e s . T h e main references for this section are [35], [37]. Concerning the combinatorics and the classical difference quotient bialgebra see [11]. For conditional free probability, freeness with amalgamation, etc., see [24], [27]. T h e analytic subordination work is in [28], [1] and [35].

Acknowledgments T h e author was supported in part by N S F Grant DMS-0079945. He t h a n k s Mrs. Faye Yeager (U.C. Berkeley) for typing the manuscript. P a r t of the work on these notes was done while the author held an International Blaise Pascal Research Chair, from the State and lie de France Region, managed by the Foundation de l'Ecole Normale Superieure and visited the Institut de Mathematiques de Jussieu. T h e author's participation in the International Congress of Mathematical Physics was supported in part by funds from the Clay Mathematics Institute as a CMI Emissary.

References 1. P. Biane, "Processes with free increments", Math. Z no. 1, 143-174 (1998). 2. P. Biane, M. Capitaine, A. Guionnet, "Large deviation bounds for matrix Brownian motion", Invent. Math. 152, 433-459 (2003). 3. P. Biane, R. Speicher, "Free diffusions, free entropy and free Fisher information", Ann. Inst. H. Poincare Probab. Statist. 37, 581-606 (2001). 4. E. Brezin, C. Itzykson, G. Parisi, J. B. Zuber, "Planar diagrams", Coram. Math. Phys. 59, 35-51 (1978). 5. T. Cabanal-Duvillard, A. Guionnet, "Large deviations upper bounds for the laws of matrixvalued processes and noncommutative entropies", Ann. Probab. 29, 1205-1261 (2001). 6. M. R. Douglas, "Stochastic master fields", Phys. Lett. B 344, 117-126 (1995). 7. K. J. Dykema, "Free products of hyperfinite von Neumann algebras and free dimension", Duke Math. J. 69, 97-119 (1993). 8. L. Ge, "Applications of free entropy to finite von Neumann algebras II", Ann. of Math. 147, 143-157 (1998). 9. R. Gopakumar, D. J. Gross, "Mastering the master field", Nucl. Phys. B 451, 379-415 (1995). 10. U. Haagerup, S. Thorbjoernsen, "A new application of random matrices: Ext(C£(F2)) is not a group", preprint (2002). 11. A. S. Joni, G.-C. Rota, "Coalgebras and bialgebras in combinatorics", in Studies in Applied Mathematics 6 1 , Elsevier North Holland, 1979, pp. 93-139. 12. A. Nica, D. Shlyakhtenko, R. Speicher, "Some minimization problems for the free analogue of the Fisher information", Adv. Math. 121, 282-347 (1999). 13. S. Popa, D. Shlyakhtenko, "Universal properties of L(Foo)", in Subfactor Theory, preprint (2002).

Aspects of free probability

157

14. F. Radulescu, "A one-parameter group of automorphisms of L(Foo) ® B(H) scaling the trace", C. R. Acad. Sci. Paris t. 314, Serie I, 1027-1032 (1992). 15. F. Radulescu, "Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group of noninteger index", Invent. Math. 115, 347-389 (1994). 16. D. Shlyakhtenko, "Random Gaussian band matrices and freeness with amalgamation", International Math. Res. Notices 20, 1013-1025 (1996). 17. D. Shlyakhtenko, "Free quasi-free states", Pacific J. Math. 177, 329-368 (1997). 18. D. Shlyakhtenko, "On prime factors of type III", Proc. Nat. Acad. Sci. 97, 12439-12441 (2000). 19. I. M. Singer, "On the master field in two dimensions", Functional Analysis on the Eve of the 21st Century, Essays in Honor of the 80th Birthday of I. M. Gelfand, Progress in Math. 131, 263-283 (1995). 20. R. Speicher, "Combinatorial theory of the free product with amalgamation and operator-valued free probability theory", Mem. Amer. Math. Soc. 627 (1998). 21. M. B. Stefan, "The indecomposability of free group factors over nonprime-subfactors and abelian subalgebras", preprint. 22. G. t'Hooft, "A two-dimensional model for mesons", Nucl. Phys. B75, 461-470 (1974). 23. Y. Ueda, D. Shlyakhtenko, "Irreducible subfactors of index A > 4", preprint (2000). 24. D. Voiculescu, "Symmetries of some reduced free product C-algebras", in Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Math 1132, Springer, 1985, pp. 556-588. 25. D. Voiculescu, "Circular and semicircular systems and free product factors", in Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, Progr. Math. 92, Birkhauser, Boston, MA, 1990, pp. 45-60. 26. D. Voiculescu, "Limit laws for random matrices and free products", Invent. Math. 104, 201-220 (1991). 27. D. Voiculescu, "Operations on certain noncommuting operator-valued random variables", Asterisque no. 232, 243-275 (1995). 28. D. Voiculescu, "The analogues of entropy and of Fisher's information measure in free probability theory", Comm. Math. Phys. 155, 71-92 (1993). 29. D. Voiculescu, "The analogues of entropy and of Fisher's information measure in free probability theory II", Invent. Math. 118, 411-440 (1994). 30. D. Voiculescu, "The analogues of entropy and of Fisher's information measure in free probability theory III: The absence of Cartan subalgebras", Geom. Fund. Anal. 6, 172-199 (1996). 31. D. Voiculescu, "The analogues of entropy and of Fisher's information measure in free probability theory V: Noncommutative Hilbert transforms", Invent. Math. 132, 182-227 (1998). 32. D. Voiculescu, "The analogues of entropy and of Fisher's information measure in free probability theory VI: Liberation and mutual free information", Adv. Math. 146, 101-166 (1999). 33. D. Voiculescu, "A strengthened asymptotic freeness result for random matrices with applications to free entropy", International Math. Res. Notices No. 1, 41-63 (1998). 34. D. Voiculescu, "Lectures on free probability theory", Lectures on probability theory and statistics, Ecole d'ete des Probability's de Saint-Flour XXVIII, Lecture Notes in Math. 1738, Springer, 1998, pp. 280-349. 35. D. Voiculescu, "The coalgebra of the free difference quotient in free probability theory", International Math. Res. Notices 2, 79-106 (2000). 36. D. Voiculescu, "Free entropy", Bull. London Math. Soc. 34, 257-278 (2002). 37. D. Voiculescu, "Free analysis questions I: Duality transform for the coalgebra of dx-.a", to appear in International Math Research Notices (2004). 38. D. Voiculescu, K. J. Dykema, A. Nica, Free random variables, CRM Monograph Series 1, Amer. Math. S o c , Providence, RI, 1992. 39. E. P. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions", Ann. Math. 62, 54 (1955).

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Invited session talks

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Condensed matter physics Session organized by J. YNGVASON (Wien) and Y. AvRON (Haifa)

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Adiabatic transport, Kubo formula and Anderson localization in some lattice and continuum models A.

ELGART

(Courant

Institute)

The different explanations of the Quantum Hall Effect rely on the validity of the linear response theory for a system that has infinite extent. We will present recent results on the adiabatic charge transport in this context for two dimensional lattice (joint work with M. Aizenman and J. Schenker) and continuum (joint work with B. Schlein) models of a non-interacting electron gas. It is proved that if the Fermi energy falls in the localization regime then the Hall transport is correctly described by the linear response Kubo formula. The localization condition is set forth by the fractional moment method, which is by now extended also to continuum models (joint work with M. Aizenman, S. Naboko, J. Schenker and G. Stoltz). In the present talk, besides localization criteria, we will discuss some ideas — Nenciu's asymptotic expansion, generalized space-momentum inequalities, and finite speed of propagation estimates — which enter the proof.

1. Introduction Since the discovery of the Quantum Hall Effect (QHE) by von Klitzing in 1980 [16] a great variety of different problems associated with the behavior of a two-dimensional electron gas in the presence of a magnetic field has been studied both experimentally and theoretically. In particular, the remarkable phenomenology associated with the QHE was generated by a number of theoretical physicists, notably by Laughlin [17], Thouless et al. [20], and Halperin [14]. Von Klitzing realized that a two dimensional electron gas at very low temperatures and strong magnetic field displays a quantization of the Hall conductance, that is the conductance measured in the direction transversal to the applied current. Specifically, the graph of the Hall conductance as a function of the magnetic field is a staircase function, where at the plateaux the value of the conductance is an integer multiple of e2/h, with astonishing accuracy. The robustness of the quantization is especially intriguing because on the microscopic level the samples have a lot of impurities, and are characterized by different concentration of electrons and by different geometries. The topological nature of the quantization, which explains such insensitivity, was first suggested by Thouless [20], and was understood latter in terms of topological invariants: Chern numbers [6] and Fredholm indices [9]. Two pictures were introduced for a description of QHE: "Edge currents picture" and "Bulk currents picture". The edge current picture [14] suggests that the Hall current flows in the narrow regions along the sample boundaries, so that the Hall voltage drops entirely in these regions. On the other hand, the description in terms of bulk currents [4,6] suggests that the Hall voltage drops gradually across the sample, and one can think about a system with an infinite extent. The natural proposal raised by Halperin suggests that in reality one should expect an intermix of these two pictures, and that the edge and the bulk conductance should coincide. Such a link between the edge and the bulk conductance was recently

163

164

A. ELGART

rigorously established by Kellendonk, Richter, and Schulz-Baldes [15], using methods of non commutative geometry, and shortly thereafter by Elbau and Graf [11], with functional analytic techniques, provided that the bulk conductivity is equal to the index of the pair of projections associated with the model and that there is a spectral gap at the Fermi energy. Here we present a derivation of the bulk conductivity for QHE models from first principles. As far as we are aware of, the analogous theory of the edge quantization has yet to be constructed. As was mentioned above, for the bulk models we can assume that the system has an infinite extent to begin with, and is characterized by two parameters, u> and A (we will use the atomic units e = m = h = 1 throughout the paper). The parameter w describes the frequency, or the rate at which the system is driven by the external field, while A describes its strength. There are two natural quantities that can be studied here: the Hall conductance ae and the Hall conductivity ay. In the first case the parameter A is equal to the total voltage drop AV across the sample and ae is a coefficient of proportionality in the (expected) Ohm's law: / = aeAV, where J is a Hall current. In the second case A is associated with the strength E of the (linear) electric field, and oy relates the current density j with E: j = ayE. In (and only in) two dimensional systems two quantities are expected to coincide: ae = ery.a Our results deal exclusively with the Hall conductance in 2D, with underlying Hilbert space being £ 2 (Z 2 ) or £ 2 (R 2 ). If one hopes for the linear relation / = aeAV to hold, and expects a total of N electrons to be transported during one cycle, a natural constraint on the two parameters w and A = AV arises: The total charge transported in the system is an integral over the time of the current which should behave as a*-1 • AV. Accordingly, we are forced to keep the latter relation fixed in the weak field regime (AV —> 0), where the linear relation between the current and the field is naturally anticipated13. Such a coupling is encoded in the adiabatic setup [8], where the Hamiltonian Ho of the system in equilibrium is perturbed by an external field of the form eg(et)Ai, with t describing a (physical) time, g G C^(0,1), e being an adiabatic parameter (eventually we will be interested in the limit e —> 0), and A standing for some (fixed) perturbation. In order to describe the electric field applied in the x\ direction, we choose Ai in the form of a multiplication operator depending on the variable x\ only (we will use the jargon "switch function in xi direction") in the coordinate basis, with a profile satisfying Ai(xi) = 0 for very negative values of the coordinate xi, and Ai(^i) = 1 for very positive values of x\. The corresponding voltage drop across the sample is AV = eg(et) at time t. This time dependent electric field will generate the Hall current, flowing in the x-i direction, which we then measure by detecting electrons crossing some given line perpendicular to x%. The spatial location of the line is essentially irrelevant in the limit e —> 0, since one expects that the current will be incompressible. In particular, one can sample the currents along different lines, and then compute some weighted average. This observation suggests the choice of the current operator (i.e., the observable whose quantum expectation gives the value of the current) in the form I := [H(t), A2], where A2 is another switch function, but in x? direction. The above picture, being very accessible from the mathematical viewpoint, is well tied by a b

For a cubic sample of size L in d dimensions aeL2~d ~ cry. T h e theory becomes simpler if the linear response limit, AV —* 0 is undertaken before the DC frequency limit, ui —» 0. However, such a setup does not describe the charge transport due to the above argument. For recent developments in this direction, see [10].

Adiabatic transport, Kubo formula and Anderson localization ...

165

the Faraday's law with the experiment of von Klitzing, where the voltage drop is induced by applied currents. The resulting formula for the conductance
2. Results Let us first glue together what was discussed in the introduction. We start with an equilibrium system on 1? or R 2 , described by some Hamiltonian Ho, acting on the corresponding Hilbert space (?{I?) or £2(R2). We then perturb it by time dependent external field, so that the full Hamiltonian is H := Ho + eg(et)A\, and we are interested in computing the number of electrons, which will cross the sample in the transversal direction X2 as a result of the perturbation (we will use the terminology "excess charge" for this quantity, and use the notation (Qe))- We assume that initially (at any time t < 0) the system is in an equilibrium state, described by the density matrix p(Ho). Then the instantaneous state pe(t) is given by the solution of the Heisenberg equation: idtPt{t) c

= [H{t),Pt{t)}\

Pe(0) = p(ffo).

(1)

Of course the natural definition of the localized regime in this context will be the constancy of ae in the corresponding range of the (Fermi) energies. Accordingly, the description of localized regime, given by (A2,A4') should be called a sufficient condition for localization. However, it is believed that in the context of random Schrodinger operators there is a jump discontinuity in the behavior of the transport exponents, so that for two dimensional systems either (Al) is satisfied or the system is characterized by at least diffusive behavior (see [13] for some rigorous results in this direction). This fact, together with the formal computation for conductance [1], indicate that in 2D (Al) can be also viewed as the necessary condition for localization, and therefore we omit the adjective "sufficient" from the description of (A2).

166

A. ELGART

In order to compute (Qe) we first consider the quantum mechanical expectation of the excess current (Ie)(t), given by (Q(t) := ti(Pe(t) - p(H0))I(t)

= tv(p£(t) - P(H0))[H(t),

A 2 ],

(2)

which should be viewed as a difference between the total current (coming from the pe part), and the equilibrium current (associated with p(H0) part). We then integrate (Ie) over the time in order to get (Qe). If the Ohm's law Ie(t) = aeAV{t)

= eg(et)

holds, then /•OO

/»1

{Qe) =&e

eg(et) dt = ae / g(s) ds. Jo Jo The results, established in two consequent works with Aizenman and Schenker (for the lattice case), and with Schlein (for continuum), state that under the assumptions (Al)-(A3) below Theorem 2.1. The following relation holds: lim(Qe)=K ->°

£

[ g(s)ds, Jo

(3)

with K:=2ntTp(H0)[[Aup(H0)},[A2,p(Ho)}},

(4)

which can be therefore identified with the conductance ae. Remark 2 . 1 . The formula for K, introduced in the theorem, is known as the Kubo formula for conductance. This object has a number of important properties: (1) the stability under deformations of the switch functions Ait2, that can be viewed as robustness with respect to the changes in the voltage distribution inside the sample and the incompressibility of the Hall current; (2) as a function of the Fermi energy, Ep, K remains constant inside the localization regime, described by the assumption (A2) — which explains the staircase form of
Adiabatic transport, Kubo formula and Anderson localization ...

167

the underlying Hamiltonian in order to control the trace norms, associated with quantum mechanical expectations. On the other hand, the Landau type models have been and remain the main prototype for the studies on QHE, therefore driving our motivation to prove the Kubo formula for them. We believe however that the theorem can be proven in a more general setup of Schrodinger operators, under the localization condition (A4') stated below, using methods derived for the lattice case and available results for the continuum. 2.1. Assumptions For the lattice case we set forth the following conditions: (Al) Locality of Hamiltonian: A self adjoint operator H on f 2 (Z 2 ) is said to be short range if there are fi > 0 and A < oo such that (x\H\y)\

<,Ae~^-

for all x ^ y.

(5)

(A2) Localization at the Fermi energy Ep: Let Hu be a random self-adjoint operator on £ 2 (Z 2 ). We say that Hu satisfies the fractional moment condition at E € R if for some 0 < a < 1, /x > 0, and A < oo, E

sup \ee[-l,l]

1


H.-E-ie

(6)

For the continuum we introduce the family of Landau Hamiltonians (A3) H0 acting on £ 2 (R 2 ): / B \2 / B \2 Ho = [Pl - —X2J + [P2 + —Xt) + V , where the potential V is smooth and relatively weak: || V||„ i00 < Cn for some integer n large enough (n = 5 will do), and ||V||oo < B/2 (which means that there is a spectral gap between the different bands associated with the Landau levels, and we choose Ep to fall into one of these gaps). What we really expect to be good assumptions in the continuum case are (A3') Ho is a Schrodinger operator on £2(R2) with sufficiently smooth potential — analog of (Al). (A4') Localization at Fermi energy Ep: E

sup Xz Ho-E-ie W-i,i]

Xw


where Xz is the indicator function of the cube of size one centered at z — analog of (A2). This condition corresponds to the natural generalization of the AizenmanMolchanov criteria for continuum Schrodinger operator, developed in a joint work with Aizenman, Naboko, Schenker, and Stolz [3].

168

A. ELGART

3. Basic tools Let us first highlight some physically and mathematically interesting features of the problem, and then describe natural tools required to overcome the corresponding difficulties. (1) Importance of the localized regime. The justification of the Kubo formula becomes a relatively easy task in the case when the Fermi energy EF falls into a gap (use of Nenciu expansion and propagation bounds below solve the problem). However, from the physical viewpoint, it is important to solve the problem in the absence of a gap. Indeed the widths of the Hall plateaux correspond to the band of localized states, since in the gap the filling factor remains constant (the integrated density of states does not change). The absence of the gap in the spectrum suggests that (naively) there is no intrinsic time scale associated with such a system. (2) Interesting topology. As was already stressed above, the Kubo formula is associated with certain Fredholm index, which already suggests that the effect comes from non trace class perturbation of Hamiltonian HQ. In fact it is the product of the density p£(t) — P{HQ) with the current operator I(t) which makes equation (2) trace class. (3) Non trivial time dependence. The total variation in the Hamiltonian H(t) is of order e, but we are interested in the cumulative effect during the time e _ 1 , therefore the simple perturbation theory will not work. The tools that help to tackle these difficulties are as follows: (1) We will argue below that the time scale is nevertheless present here, and comes from the density of the states in the vicinity of the Fermi energy. (2) Finite speed of propagation, equation (8) below, will be crucial in studying the trace class properties of the corresponding operators. Obtaining such a bound is a simple task on the lattice under (Al), but requires a substantial work for the continuum case, basically due to the time dependence of H(t). (3) The Nenciu expansion below (in the presence of the gap) and the adiabatic evolution introduced by Avron, Seiler, and Yaffe [8] are very handy tools for working with (slowly) time dependent problems. (4) In addition to these three items, one has to develop the various trace class estimates for the continuum case, where we have to tackle also the infrared and ultraviolet problems typical for Schrodinger operators. 3.1. Finite speed of propagation Let us start with an explanation of why such bounds are important. Morally, the density pt{t) — p0 that appears in equation (2) behaves like -if

Ue(t,s)[Aupo]Ue(s,t)g(es)ds, Jo

where Ue is the unitary propagator, satisfying: ieU€(s,t) = H(s)Ue(s,t),

Ue{t,t) = \.

(7)

Adiabatic transport, Kubo formula and Anderson localization

169

If the kernel of po is sufficiently localized (as it happens when Ep lies in a gap or (almost surely) in the localized regime (A2,A4')), then [p0, Ai] looks like a bump in the x\ direction. On the other hand [H0, A2] looks like a bump in the i 2 direction, which suggests that ||[Ai, / o 0 ][jy 0 ,A 2 ]||! < 00. If we substitute equation (7) into equation (2), a similar expression will appear, but with Uc lying between the two commutators. Finite speed of propagation below guarantees that also with Ut we get a trace class operator. We will use the following notation: (xi) = (I + 2; 2 ) 1 / 2 . Lemma 3.1. (Finite speed of propagation in continuum) Fix n,m £ N/2, and assume that V £ ^2(n+m),oo(IR2)- Then there is a constant D = D(n,m) such that, for i = 1,2, \\(xi)n(H(s)+irUc(s,t)(H(t)

+ i)-m-n(xi)-n\\


(8)

for all s , t £ l . 3.2. Nenciu expansion We want to solve the following Heisenberg equation iepe(s) =

\H(s),pe(s)},

where pc(0) is a spectral projection of the operator H(0). Naturally, one can look for asymptotic series pe(s) ~ B0(s) + eBxis) + e2B2(s) + ••• . Substitution into the Heisenberg equation leads to a sequence of differential equations iBj(s) = [H(s),Bj+1(s)]

j=0,...

(9)

However, these equations do not have a unique solution. The idea of Nenciu was to use an extra constraint, namely the fact that p 2 (s) = pe(s), so that: 3

Bj(s) = £

Bm(s)Bj_m(s)

j = 0,...

(10)

m=0

It turns out that the system of hierarchical relations 9, 10 has a unique explicit solution (Nenciu [18]). 3.3. Adiabatic evolution and effective gap Without the spectral gap, Nenciu expansion does not work, since for every power of e in the expansion we have to pay a price in singularity of Bj. However it is clear that pe(t) is not singular at all, so a different method is needed in the absence of a spectral gap. With introduction of a fictitious adiabatic evolution, suggested by Avron, Seiler and Yaffe [8], the problem essentially reduces to the control of the following object: Po Pe(l) (1 ~ Po) • Let us use the notation P A for the spectral projection onto the energy interval A around Ep • Then the following result holds:

170

A. ELGART

L e m m a 3.2. (1) po(l - PA)pe(l)

(1 - po) = 0(e f c A- f c ) for any k e N.

(2) If Ep falls in the localized regime then P&pe = PAQ + 0(ekA~k), Q is bounded and has (a.s.)

exponentially

localized

where the operator

kernel.

This result suggests t h a t one can create an "effective gap" of width A , as far as A > e. T h e price one has t o pay for t h e creation of this gap (i.e., t h e corresponding error) is dictated by the density of t h e states in t h e A-interval around the Fermi energy, namely by trxo-Pe- T h e interesting open question here is whether this upper bound on t h e error (clearly, t r Xo-Pe —* 0 as e —> 0) is also sharp. T h e belief of t h e author is t h a t t h e answer t o this question is yes, and it comes from the close similarity t o another time dependent problem. It t u r n s out t h a t an a t o m subjected t o t h e quantized radiation field and slowly varied external field behaves very differently with and without spectral gap between t h e ground state and a.c. spectrum [5]. In t h e presence of t h e gap t h e error of t h e adiabatic evolution is 0(ek) for any k, b u t in its absence t h e error is of some fixed power of e, which depends entirely on t h e density of t h e states in t h e vicinity of the ground state.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

M. Aizenman, G. M. Graf, J. Phys. A: Math. Gen. 3 1 , 6783 (1998). M. Aizenman, A. Elgart, J. H. Schenker, preprint. M. Aizenman, A. Elgart, S. Naboko, J. H. Schenker, G. Stolz, preprint. H. Aoki, T. Ando, Solid State Commun. 38, 1079 (1981). J. E. Avron, E. Elgart, Comm. Math. Phys. 159, 399 (1999). J. E. Avron, R. Seiler, Phys. Rev. Lett. 54, 259 (1985). J. E. Avron, R. Seiler, B. Simon, Comm. Math. Phys. 159, 399 (1994). J. E. Avron, R. Seiler, L. G. Yaffe, Comm. Math. Phys. 110, 33 (1987). J. Bellissard, A. van Elst, H. Schulz-Baldes, J. Math. Phys. 35, 5373 (1994). J. M. Bouclet, F. Germinet, A. Klein, J. H. Schenker, in preparation. P. Elbau, G. M. Graf, Comm. Math. Phys. 229, (2002). A. Elgart, B. Schlein, to appear in Comm. on Pure and Applied Math.. F. Germinet, A. Klein, preprint. B. I. Halperin, Phys. Rev. B 25, 2185 (1982). J. Kellendonk, T. Richter, H. Schulz-Baldes, Rev. Math. Phys. 14, 87 (2002). K. v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980). R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). G. Nenciu, Comm. Math. Phys. 152, 479 (1993). R. E. Prange, S. M. Girvin (eds.), The Quantum Hall Effect. Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1987. 20. D. J. Thouless, M. Kohomoto, M. P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).

Transport in adiabatic quantum pumps G. M. G R A F (ETH Zurich), A. ELGART (Courant

L.

SADUN

(U. Texas), K.

SCHNEE

Institute),

(Schrodinger Institute,

Vienna)

A quantum pump is an externally driven device coupled to reservoirs at equilibrium with one another. We consider transport phenomena when the electrons are independent and the driving is slow compared to the dwell time of particles in the pump. The charge transport associated with a given change of pump parameters is characterized in terms of 5-matrices pertaining to time-independent junctions. In fact, several transport properties (charge, dissipation and, at positive temperature, noise and entropy production) may be expressed in terms of the matrix of energy shift which, like Wigner's time delay to which it is dual, is determined by the S-matrix. We discuss transport at a semiclassical level, including geometric aspects of transport such as the question of charge quantization. On the analytical side we present an adiabatic theorem on transport for open gapless systems.

1. Introduction An adiabatic quantum pump is a time-dependent scatterer connected to several leads. The idealized setting is as follows: a pump, whose internal configuration varies slowly in time in a prescribed manner, is connected to n channels, along each of which independent electrons of charge e = 1 can enter or leave the pump. We assume that electrons do not have spin nor ^

^

1 2

Figure 1. The pump proper with n channels.

degrees of freedom transverse to the channel and may be thought of as a (non-relativistic) particle moving on a half line. The incoming electron distribution, at zero temperature, is a Fermi sea with Fermi energy /z common to all channels. As a rule, this does not apply to the distribution of the outgoing electrons, as their energies may have been shifted while scattering at the pump. Because of this imbalance a net current is flowing in the channels. The expected charge transport is expressed by the formula [9,11,12] dQ.

=

±{{dS)S')n

•

(1)

Here S = {Sij) is the n x n scattering matrix at energy [i computed as if the pump were frozen into its instantaneous configuration. A change of the configuration is accompanied

171

172

G. M. GRAF, A. ELGART, L. SADUN, K. SCHNEE

by a change S —> S + dS of the scattering matrix and by a net charge dQj leaving the pump through channel j . We recall that the definition of the S matrix involves the comparison of the dynamics with that generated by a reference Hamiltonian, e.g. a configuration with channels disconnected from the pump. Equation (1) is of a form well-known in thermodynamics for describing exchanges between a system and a reservoir, like dW = —pdV for the reversible work exchanged through a piston. In fact, dQj depends on the change of the system, but not on its rate, provided it is slow. Moreover dQj is expressed through S which, like p and V, is determined statically and from outside the system. As it is appropriate for a pump, denoting by F(/i) the projection onto the states of energy below /J, the Krein spectral shift is formally defined as £(//) = Tr(P(/i) — PQ(M))» where the 0 stands for the reference configuration, and counts the excess of electrons in the Fermi sea. Thus, by the Friedel sum rule [15] or Birman-Krein formula [22] d e t 5 = e 2 , n ^ M \ we have n

V

dQj = ^- tr((dS)5*) = -^-dlogdetS = - d £ .

r~~f

27T

Z7T

Here tr(-) denotes the trace of a n x n matrix, in contrast to Tr(-) applying to operators on the (single particle) Hilbert space. As a result, the Kirkhoff law only holds when integrated over a cycle: / £ ) " = i dQj = 0. Derivations of equation (1) and of related transport properties have been given in various settings and at different levels of rigor [3,7,9,11,12,16].

2. Semiclassical description and quantization of charge transport A semiclassical description of the effect of the pump on the electrons may be given pretending that they have (i) classical positions and momenta as long as they freely move in a channel and (ii) that these phase space coordinates are changed according to quantum mechanical scattering when they traverse the pump. As for (i), canonical phase space coordinates of a particle are conveniently chosen as its time i 6 R of passage at some fiducial point, e.g. at the connection x — 0 of the channel [0, oo) with the pump, and its energy E G [0, oo). The aspect (ii) can be described in terms of the frozen scattering matrix S(E) of the pump in effect at the time it is being traversed by the electron. The scattering matrix so acquires a parametric time dependence, S(E,t). Indeed, the shift induced by the scattering on the coordinate t is the well-known Wigner time delay [14,21] T

(E,t)

= -i-^(E,t)S*(E,t).

(2)

More precisely its diagonal element Tjj(E, t) has the meaning of the average time delay of a particle exiting channel j . In fact, consider an incoming wave packet J dke~i(-kx+u(-h^/dk). The part of it scattered into lead j is Jdkel(-kx^u^k^Sji(uj(k),t)ip(k) and is associated with x = v(ko)(t — (arg Sji)') + c (with ' = d/dE), which implies a time delay of (arg Sji)'. Averaging this with

Transport in adiabatic quantum pumps

173

the probability \Sji\2 for the particle to have come from channel i, we find for the average delay n

n

£|%|2(arg%)' = I m ^ % ^ =7^, i=l

i=l

Similarly, the scattering process affects the coordinate E by the energy shift [4] AC

£(E,t) = i~(E,t)S*(E,t).

(3)

While this claim appears plausible in view of the duality between equations (2, 3) one may find it puzzling because each matrix S(E, t) originates from a frozen Hamiltonian and hence from an energy-conserving dynamics. Of course, the dynamics of the electron in an operating pump is generated by a time-dependent Hamiltonian H(t) and the appropriate (dynamical) scattering operator Sd is that relating it to the reference Hamiltonian HQ. Moving the time origin from 0 to r results in H (t) —> H(t + r) and in a unitarily equivalent scattering operator Sd{r), Sd{T) = e-iHorSdeiH°T. (4) Written as an equation of motion, dSdjdr = — i[Ho, Sd(r)}, the equation can be reorganized as [17] H0 = Sd(t)H0S*d(t)

+^S*d(t),

assuming unitarity of Sd. Conjugation by the scattering matrix takes outgoing observables to incoming observables. The last term on the r.h.s. is thus to be identified with the energy shift [17]. The relation to its frozen counterpart (3) emerges in the adiabatic limit and can be clarified by means of coherent states [5] or pseudodifferential operators [6]. The net charge leaving the pump through channel j in the time interval [0, T] is the difference between the outgoing and incoming contributions:

Qj = ^J

dE' J dt'p{E)- — j

dEj

dtp(E),

(5)

where p is the occupation of incoming states, e.g. p(E) = Q(p, — E) in the zero temperature situation considered in section 1. In the first integral E is given through the map

$ : (E',f) ~ (E,t) =

(E'-£ij(E',t!),t'-Tjj(E',t)),

which describes the effect of the pump on the energy and the time of passage of an electron in terms of the outgoing data (E',t'). The denominator 2TT is the size of the phase space cell of a quantum state. By linearizing in the small energy shift £jj we obtain Qj{t)

=

1 ~2^J

r°°dE

P'(E)£n(E,t),

(6)

(' = d/dt), which reduces to 1 Qi{t) =—SjMt), 2TT

i.e., to equation (1), at zero temperature.

(7)

174

G. M. GRAF, A. ELGART, L. SADUN, K. SCHNEE

An alternate expression for the transported charge may be derived from equation (5) by using $ as a change of variables in the first integral. Its Jacobian is 1 — Q,jj(E',t'), where Cljj is the divergence of the displacement (£JJ,TJJ). Remarkably, ttjj is a diagonal entry of the commutator of time delay and energy shift Q = i[T,£\.

(8)

The result is the following balance equation fT

2-KQJ^

Jo

/»oo

dtp(0)£j:j(0,t)

-

Jo

dEp(E)Tj:i{E,t)

/*oo

t=T t=0

+

pT

dE dtp(E)Q.j:i(E,t). Jo Jo

(9)

The first term on the r.h.s. describes the release and trapping from bound states. It can in fact be seen [6] that £jj(0,t) consists of delta functions peaked at times t when an eigenvalue is branching off or merging with the continuum at E = 0. The middle term describes the depletion of the outgoing flow as a result of a time delay increasing over time, since effectively no charge is exiting during a time Tjj\\z£. The last term describes electrons that are being reshuffled between leads without delays, i.e., consistently with Kirkhoff's law, since Y^j=i tyj = trfi = 0 by equation (8). In addition to (7), but without explanation, we mention further transport properties (pointwise in time) [4,6] which can be expressed in terms of the energy shift: (i) Dissipation is the part of the energy flow Ej into lead j except for the part, p.Qj, which can be recovered from the lead (or reservoir) by reclaiming the transported charge. The expression at zero temperature is Ej(t) - (iQjit) = i j ; ( f 2 ) W (/i,t) > 0. (ii) The entropy and noise currents are given as the difference between the outgoing and incoming entropy (or noise) currents. The expressions are sJ(t,^) = ^-k((S%j-(£jj)2)>0, <we

k = {2 e n t r ° P y ; 16 noise,

(10)

the results being valid for small temperatures / 3 _ 1 provided however that /3 remains small w.r.t. to the adiabatic time scale.

3. Quantization of charge transport Early work [20] on pumps showed that a system which is both space and time periodic with filled bands of electrons exhibits quantized transport at zero temperature and in the adiabatic regime for reasons related to Chern numbers. In our case, where the pump is spatially compact, equation (9) allows for a comparison with that situation. Assuming that the system has no bound states, the charge transported during a period T is Qj = w~ I &jj(E,t)dE lit Jc

A dt,

(11)

where the integration is now over a finite cylinder (E, t) 6 C — (M mod T) x [0, /x]. Actually, since £(0, t) = 0 by the assumption just made, the bottom of the cylinder may be pinched

Transport in adiabatic quantum pumps

175

to a point and the cylinder turned to a disk. Consider now the complex line bundle with base C and fiber C C n spanned by the state feeding channel j \i)(E,t)) = S*(E,t)\j),

|j) = (*«)(i=i,...n).

The connection Pjd, Pj(E,t) = \tp(E,t)){ip(E,t)\, Ejjdt — TjjdE and curvature

(12)

has connection 1-form i[(dS)S*]jj —

-i(dS A dS*)j:j = dEjj Adt + dEA dTj:i = Slj:jdE A dt, which exhibits the integrand of equation (11) as a Chern character. However the analogy with [20] ends here and does not provide Chern numbers as a rule, since the manifold C has a boundary and, besides, equation (12) provides a global section, trivializing the bundle. There is an exception to this rule, however. The bundle is the pull-back of the canonical line bundle L over P C " - 1 via 7r o ip, where ip : C —> C" is given by equation (12) and 7r : C™ H-> PC™ -1 is the Hopf map. Hence, Qj = (27r) _1 L 0 1 I U C ) w, where (27r)_1u; is the first Chern character of L. The exception thus is: If d~K{%j){C)) = 0 , i.e., if \ip((i,t)) is constant in t up to a phase along the period, then Qj is an integer. This last conclusion can be reached more quickly from equation (7) since, by Sjj(/j,,t) = —i(il>\ip), the charge Qj =
4. An adiabatic theorem The basic equation (1) may be given a more firm footing on the basis of the Schrodinger equation. We sketch here (see [7] for full details) the mathematical framework. The singleparticle Hilbert space is given as H = H0®L2(R+,Cn),

(13)

where states in L2(R+,Cn) = ®" = 1 L 2 (R + ), resp. in Ho, describe an electron in one of the leads j = 1, n, resp. in the pump proper, cf. figure 1. Let Ho : Ji —> H denote the projection onto H.Q. We consider Hamiltonians H on H satisfying Hi/> = -d2ijj/dx2

for V € C^°(M + ,C n ),

||(if + i)~ m n 0 ||i < C for some fixed C> 0 and m £ N, <7pp(ff) n (0, oo) = 0. The first assumption states that particles in the channels are free. The second one, where || • ||i denotes the trace class norm over H, essentially requires that the pump proper contains

176

G. M. GRAF, A. ELGART, L. SADUN, K. SCHNEE

finitely many states below any fixed energy. The third one forbids embedded eigenvalues. These assumptions guarantee that the pump proper is finite, but place no restriction on its constitution except that it should leak at all positive energies. Since the pump configuration is supposed to change slowly in time t, we consider the evolution of the electrons in an adiabatic limit, where s = et is kept fixed as e > 0 tends to 0. In terms of the rescaled time coordinate s, called epoch, the propagator Ue(s,s') on H satisfies the non-autonomous Schrodinger equation idtUe(s, s') = £-xH{s)Ue{s,

s'),

UE(s', s') = 1,

where the Hamiltonians H(s) are of the above kind with H(s) — H(s') bounded and smooth in s. It is convenient to start the system from an equilibrium state. This is done by positing its 1-particle density matrix to be of the form p{H{so)) at some initial epoch so- Here p is a function of bounded variation with suppiip C (0, oo). In particular, an admissible choice is the Fermi sea, where p(E) = 9(p, — E). The time evolution then acts as p(H(s0)) ^ Pe(s) = U£(s, so) p(H(s0)) Ue(s0, s).

(14)

It should be noted that P£(s) is not an equilibrium state for H(s). In a single channel R + 9 x, the operator detecting a particle past some fiducial point x = a is the (smooth) characteristic function F(x > a). For technical reasons to be described below we found it preferable to use \A\ instead of x, where A is (px + xp)/2 (or a related operator) on the subspace L 2 (R+) of a chosen channel j and 0 on the remaining subspaces in equation (13). The current operator then consistently is Ij(a) = i[H(s),F(\A\ > a)], and Tr(Pe(s)Ij(a)) is its expectation value. The parameter a may be vaguely interpreted as the position of the ammeter. The fundamental result equation (1) may thus be given the following reformulation [7,19]. T h e o r e m 4 . 1 . Under the above assumptions we have for s > SQ

lim l i m e - ^ P e O O J ^ o ) ) = - — / o—too £-»o

2ir J0

dp(E)( — S*) \ as

,

(15)

/ jj

where S(E, s) is scattering n x n matrix (fiber) of the scattering operator for the pair (H(s),Ho) with Ho being any fixed reference Hamiltonian. The double limit is uniform in s £ I, I being a compact interval, whence it carries over to the transfered charge f Jo

dt'Ti{Pe{et')Ij{a))

= e-1 f ds'Tr(F £ (s')^(a))Jo

We shall conclude this section with several remarks about the theorem and its derivation. They are mostly heuristic, but may serve as a guide through the complete proof. The limit a —» oo is done in order to obtain a scattering description. Though a (positive energy) particle may require a large time (not epoch) interval to complete scattering, its length remains bounded in e small. Therefore the adiabatic limit e —> 0 is to be performed first. Equation (15), like equation (6), reduces to equation (1) at zero temperature. The above result is a statement about the adiabatic behavior of certain open, gapless systems, as explained shortly. Typically, adiabatic theorems are concerned with the evolution

Transport in adiabatic quantum pumps

177

Ue(s,so)PUe(s,so)*, where P is the spectral projection of H(so) onto a separated part of its spectrum [8,18] or on an embedded eigenvalue [2,10], and possibly with its deviation from the corresponding projection of H(s). The same is true here, see equation (14), except that P = p(H(so)) corresponds to a gapless part of the continuous spectrum. In fact, the theorem is implied by the following statement, which probes that deviation: Split the current 7,-(o) = i[H,F(\A\ > a)} = i{H,F(A < - a ) ] + i[H,F{A > a)] = ir(a) + I+(a) into incoming and outgoing parts. Then, in the double limit of equation (15), e-1Tr(Pe(s)I-(a)) 1

e- Tr(Pe{s)I+(a))

= e-1Tv(p(H(s))lr(a))

+ o(l),

1

= e" Tr (p(H(s))I+(a))

+ Tr([S^(s),

p(H(s))}I+(a))

+ o(l),

where oo

/

dtteiHMtH{s)e-iH^)t.

(17)

-oo

According to these estimates the state (14) deviates from equilibrium only in its outgoing part and may be approximated on that part by p(H(s))+s[S^(s),p(H(s))}

+ ..-.

(18)

The asymmetry w.r.t. the incoming part, p(H(s)), is of course due to the assumption s > s0, i.e., to the fact that the current measurement occurs after the preparation of the state. It should moreover be noted that in the sum of the r.h.s. of equations (16) the equilibrium contributions cancel, Tr(p(H(s))(I~(a) + I+(a))) = 0. The theorem also represents an instance where linear response theory can be justified rigorously, cf. [13], without there being a dissipation mechanism. Indeed, by the chain rule for wave operators, we may pretend that the reference Hamiltonian used in equation (15) is H(s) itself and thus prove the result with (dS/ds)S* replaced by ds>S(s',s)s<=s, where S(s',s) is the scattering operator for the pair of autonomous Hamiltonians (H(s'),H(s)). However, as explained in connection with equation (4), the relevant object is the dynamical scattering operator, now relating the ^-dependent Hamiltonian H(s+et) — H(s) +sH(s)t + ••• to H(s). In linear response approximation it equals Sd = 1 + eS^(s) + ••-, and equation (18) comes from Sdp(H(s))S^ and can be rewritten by means of [S^(s),p(H(s))}

= -ids,S(s',s)s<=3p'(H(s)),

(19)

with oo

/

iH t dte

^ H(s)e-iHMt.

(20)

•oo

Here, 1 + (s' — s)ds>S(s',s)a>=s + ••• is the Born approximation for S(s',s). Formally, equation (19) follows for ^(A) = e1^*, (t e U), by a change of variables in equation (17) and hence for general functions p. The importance of equation (19) is twofold. First, it reduces matters to frozen scattering data and their derivatives, as required by the theorem. Second, it makes it clear why only the variation dp(E) matters and, in particular, why for the Fermi sea p(A) = 8(n — A) only states at the Fermi energy p, contribute to the current. As a final remark we mention that the integrals (17, 20), once multiplied with if {a), can be shown to converge using propagation estimates.

178

G. M. G R A F , A. ELGART, L. SADUN, K. SCHNEE

Acknowledgments We t h a n k J. Avron, without whom t h e results presented here would not have been possible. This work is supported in p a r t by the Texas Advanced Research P r o g r a m and N S F Grant PHY-9971149; t h e E U grant H P R N - C T - 2 0 0 2 - 0 0 2 7 7 .

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

A. Andreev, A. Kamenev, Phys. Rev. Lett. 85, 1294 (2000). J. E. Avron, A. Elgart, Coram. Math. Phys. 203, 445 (1999). J. E. Avron, A. Elgart, G. M. Graf, L. Sadun, Phys. Rev. B 62, R10618 (2000). J. E. Avron, A. Elgart, G. M. Graf, L. Sadun, Phys. Rev. Lett. 87, 236601 (2001). J. E. Avron, A. Elgart, G. M. Graf, L. Sadun, J. Math. Phys. 4 3 , 3415 (2002). J. E. Avron, A. Elgart, G. M. Graf, L. Sadun, arXiv:math-ph/0209029. J. E. Avron, A. Elgart, G.M. Graf, L. Sadun, K. Schnee, arXiv:math-ph/0209029. J. E. Avron, R. Seiler, L. G. Yaffe, Comm. Math. Phys. 110, 33 (1987). P. W. Brouwer, Phys. Rev. B 58, 10135 (1998). F. Bornemann, Lecture Notes in Mathematics, vol. 1687, Springer, 1998. M. Biittiker, A. Pretre, H. Thomas, Phys. Rev. Lett. 70, 4114 (1993). M. Biittiker, H. Thomas, A. Pretre, Z. Phys. B 94, 133 (1994). A. Elgart, these proceedings. L. Eisenbud, Dissertation, Princeton University, 1948, unpublished. L. D. Landau, E. M. Lifshitz, Statistical Mechanics, Pergamon Press, 1978. L. S. Levitov, H. Lee, B. Lesovik, J. Math. Phys. 37, 4845 (1996). P. A. Martin, M. Sassoli de Bianchi, J. Phys. A 28, 2403 (1995). G. Nenciu, Comm. Math. Phys. 152, 479 (1993). K. Schnee, Dissertation, ETH-Zurich, 2002, unpublished. D. Thouless, Phys. Rev. B 27, 6083 (1983). E. P. Wigner, Phys. Rev. 98, 145 (1955). D. R. Yafaev, Mathematical Scattering Theory, AMS, 1992.

One-dimensional behavior of dilute Bose gases in traps ROBERT SEIRINGER, ELLIOTT

H. LIEB (Princeton), JAKOB

YNGVASON

(U. Wien)

We report on a rigorous analysis of the ground state of a dilute Bose gas in an external confining potential in the limit where the confinement becomes very strong in two dimensions and as a result the system behaves essentially one-dimensional. We show that the system is well described by a one-dimensional model of Bosons with delta function interaction that was solved long ago by Lieb and Liniger. We explicate 5 parameter regions in which various types of behavior of the system occur. This is relevant for the interpretation of recent experiments on ultra-cold atomic gases in strongly elongated traps.

1. Introduction Recently it has become possible to do experiments in highly elongated traps on ultra-cold Bose gases that are effectively one-dimensional [1-4]. These show peculiar features predicted by a model of a one-dimensional Bose gas with delta function two-body interaction analyzed long ago [5,6], like quasi-fermionic behavior [7], the absence of Bose-Einstein condensation (BEC) in a dilute limit [8-10], and an excitation spectrum different from that predicted by Bogoliubov's theory [6,11,12]. The theoretical work on the one-dimensional behavior of the system in elongated traps has so far been based either on variational calculations, starting from a 3D pseudo-potential, or on numerical Quantum Monte Carlo studies. In contrast, in [13] a rigorous study of the behavior of the system, starting from the basic Schrodinger equation, was undertaken, and we give a summary of this work here. The complicated nature of the problem can be seen from the fact that for an interaction potential with a hard core, as we consider here, the true ground state wave function does not approximately factorize in the longitudinal and transverse variables (otherwise the energy would be infinite) and the effective one-dimensional interaction potential can not be obtained by simply integrating out the transverse variables of the three-dimensional interaction potential. Nevertheless we are able to demonstrate rigorously that the one-dimensional behavior really follows from the fundamental Schrodinger equation. It is also important to delineate, as we do here, precisely what can be seen in the different parameter regions. The proofs of our assertions are too long to be given here and can be found in [13]. We emphasize that everything can be rigorously derived from first principles.

2. Setting and main results We shall now describe the setting more precisely. It is convenient to write the Hamiltonian in the following way. We choose units such that H = 2m = 1, where m denotes the particle mass. The Hamiltonian is than given by JV

HN,L>r>a = '£(-Aj+Vrx(Xf)

+ VL(zj))+

j=l

Yl l
179

«a(|xi-x,-|)

(1)

180

ROBERT SEIRINGER, ELLIOTT H. LIEB, JAKOB YNGVASON

with x = (x, y, z) = (x-1, z) and with ^ ( x ^ ^ K ^ x

1

/ ^ ,

VL{z) = ±V{z/L),

U o (|x|)

= lt;(|x|/a).

(2)

Here, r,L,a are variable scaling parameters while V-1, V and v are fixed. The interaction potential v is supposed to be nonnegative, of finite range and have scattering length 1; the scaled potential va then has scattering length a. (See, e.g., [14] or [15] for the definition of the scattering length.) The external trap potentials V and V1- confine the particles in the longitudinal and the transversal directions, respectively, and are assumed to be locally bounded and tend to oo as |z| and Ix-1! tend to oo. To simplify the discussion we find it also convenient to assume that V is homogeneous of some order s > 0, namely V(z) = \z\s, but weaker assumptions, e.g. asymptotic homogeneity [16], would in fact suffice. The case of a simple box with hard walls is realized by taking s = oo, while the usual harmonic approximation is s = 2. It is understood that the lengths associated with the ground states of —d2/dz2 + V(z) and —Ax + y- L (x J -) are both of the order 1 so that L and r measure, respectively, the longitudinal and the transverse extensions of the trap. We shall always be concerned with the ground state of the Hamiltonian (1) and with large particle number, N 3> 1, which is appropriate for the consideration of actual experiments. The other parameters of the problem, o, r and L, will satisfy a
= N J l * o ( x , x 2 , . . . , X i V )| 2 d 3 x 2 • • -cftcjv •

(3)

On the average, this 3D density will always be low in the parameter range considered here (in the sense that the mean distance between particles is large compared to the scattering length a). However, the effective ID density can be either high or low. Note that, in contrast to 3D gases, high density in ID corresponds to weak interactions and vice versa [5]. In parallel with the 3D Hamiltonian (1) we consider the Hamiltonian for n Bosons in ID with delta function interaction and coupling constant g > 0, i.e., H™=ir-dydz]+g j=l

Y,

Sizi-zj).

(4)

l
We consider this Hamiltonian for the Zj in an interval of length £, and are interested in the thermodynamic limit t —> oo, n —> oo with p = nji fixed. The ground state energy per particle in this limit is independent of boundary conditions and can, according to [5], be written as elD(p) = P2e(g/p), (5) with a function e(t) determined by a certain integral equation. Its asymptotic form is e(t) K \t for t < 1 and e(t) -> TT 2 /3 for t -> oo. Thus elD(p) » \gp for g/p « 1

(6)

4D(P) « (?r 2 /3)p 2 for g/p » 1.

(7)

and

One-dimensional behavior of dilute Bose gases in traps

181

This latter energy is the same as for non-interacting fermions in ID, which can be understood from the fact that (4) with g = oo is equivalent to a Hamiltonian describing free fermions. Taking /oeJD(p) as a local energy density for an inhomogeneous ID system we can form the energy functional oo

/

( | V v ^ ) | 2 + VL(z)p(z) + p{zfe{glp{z)))

dz.

(8)

-oo

Its ground state energy is obtained by minimizing over all normalized densities, i.e.,

E1D(N,L,g)=hdU\p] : p > 0, J" p(z)dz = N\ .

(9)

Using convexity of the map p — i > p3e(g/p), it is standard to show that there exists a unique minimizer of (8) (see, e.g., [15]). It will be denoted by p}p£„g- We also define the mean ID density of this minimizer to be 1

r°°

P=xJ_oo(p1N,L,g(z))

i

dz.

(10)

Note that, in general, this mean density depends on g besides N and L. The order of magnitude of p in the various parameter regions will be described in the next section. Our main result relates the 3D ground state energy of (1), ECiM(N,L,r,a), to the ID 1D density functional energy E (N,L,g) in the large TV limit provided r/L and a/r are sufficiently small. For this to be true the parameter g has to be chose appropriately, of course. To give the precise statement, let e x and b(x±), respectively, denote the ground state energy and the normalized ground state wave function of — A x + V ± (x- L ). The corresponding quantities for -A1 + V ^ x 1 ) are e x / r 2 and ^ ( x - 1 ) = ( l / r ) 6 ( x ± / r ) . In the case that the trap is a cylinder with hard walls 6 is a Bessel function; for a quadratic V1- it is a Gaussian. In any case, 6 is a bounded function, and in particular b £ L 4 (R 2 ). We can therefore define 9 by

l6(x±)l4rf2x± = 8™ / I M x ^ l 4 ^ .

9-^rJ

(ii)

Our main result is: Theorem 2.1. Let N —> oo and simultaneously r/L —> 0 and a/r —> 0 in a way such that r2p • min{p, g} —> 0. Then l m

EQM(N,L,r,a)-Ne±/r2 E">{N,Ltg)

= 1

-

(12)

An analogous result holds for the quantum mechanical ground state density. Define the ID QM density by averaging over the transverse variables, i.e.,

Let L := N/p denote the extension of the system in ^-direction, and define the rescaled density p by PlN,L,g{z) = jP(z/L).

(14)

182

ROBERT SEIRINGER, ELLIOTT H. LIEB, JAKOB YNGVASON

Note that, although p depends on N, L and g, ||p||i = ||p]|2 = 1, which shows in particular that L is the relevant length scale in z-direction. We have T h e o r e m 2.2. In the same limit as considered in Theorem 2.1,

in weak Ll sense. Note that because of (6) and (7) the condition r2p • min{p~, g} —> 0 is the same as eJ D (p) « 1/r2,

(16)

i.e., the average energy per particle associated with the longitudinal motion should be much smaller than the energy gap between the ground and first excited state of the confining Hamiltonian in the transverse directions. This is sufficient for ID behavior, it is not necessary to have a ~ r, as might have been supposed. This is an intrinsically quantum-mechanical phenomenon.

3. Discussion We will now give a discussion of the various parameter regions that are included in the limit considered in Theorems 2.1 and 2.2 above. We begin by describing the division of the space of parameters into two basic regions. This decomposition will eventually be refined into five regions, but for the moment let us concentrate on the basic dichotomy. In earlier work [15,16] we proved that the 3D Gross-Pitaevskii formula for the energy is correct to leading order in situations in which iV is large but a is small compared to the mean particle distance. This energy has two parts: The energy necessary to confine the particles in the trap, plus the internal energy of interaction, which is N4:irap3D. This formula was proved to be correct for a fixed confining potential in the limit N —> oo with a3p3D —> 0. However, this limit does not hold uniformly if r/L gets small as N gets large. In other words, new physics can come into play as r/L —> 0 and it turns out that this depends on the ratio of a/r2 to the ID density, or, in other words, on g/p. There are two basic regimes to consider in highly elongated traps, i.e., when r < L . They are — the ID limit of the 3D Gross-Pitaevskii regime; — the 'true' ID regime. The former is characterized by g/p oo the particles are effectively impenetrable; this is usually referred to as the Girardeau-Tonks region.) These two situations correspond to high ID density (weak interaction) and low ID density (strong interaction), respectively. Physically, the main difference is that in the strong interaction regime the motion of the particles in the longitudinal direction is highly correlated, while in the weak interaction regime it is not. Mathematically, this distinction also shows up in our proofs. The first region is correctly described by both the 3D and ID theories because the two give the same predictions there. That's why we call the second region the 'true' ID regime.

One-dimensional behavior of dilute Bose gases in traps

183

In both regions the internal energy of the gas is small compared to the energy of confinement. However, this in itself does not imply a specifically ID behavior. (If a is sufficiently small it is satisfied in a trap of any shape.) ID behavior, when it occurs, manifests itself by the fact that the transverse motion of the atoms is uncorrelated while the longitudinal motion is correlated (very roughly speaking) in the same way as pearls on a necklace. Thus, the true criterion for ID behavior is that g/p is of order unity or larger and not merely the condition that the energy of confinement dominates the internal energy. We shall now briefly describe the finer division of these two regimes into five regions altogether. Three of them (Regions 1-3) belong to the weak interaction regime and two (Regions 4-5) to the strong interaction regime. They are characterized by the behavior of g/p as N —> oo. In each of these regions the general functional (8) can be replaced by a different, simpler functional, and the energy Elu(N,L,g) in the theorem by the ground state energy of that functional. Analogously, the density in Theorem 2.2 can be replaced by the minimizer of the functional corresponding to the region considered. The five regions are • Region 1, the Ideal Gas case: In the trivial case where the interaction is so weak that it effectively vanishes in the large TV limit and everything collapses to the ground state of —d2/dz2 + V(z) with ground state energy e", the energy E1D in (12) can be replaced by Ne^/L2. This is the case if g/p
This case is

oo

/

(I V^p(z)\2

+ VL{z)p{z) + \gp{zf)

dz ,

(17)

-oo

corresponding to the high density approximation (6) of the interaction energy in (8). Its ground state energy is EGP(N,L,g) = NL~2EGP(1,1, NgL), by scaling. • Region 3, the I D TF case: N~2 < g/p < 1, with p ~ (N/L)(NgL)~1/(-s+1), where s is the degree of homogeneity of the longitudinal confining potential V. This region is described by a Thomas-Fermi type functional oo

/

{VL{z)p{z) + \gp{z)2)dz.

(18)

-oo

It is a limiting case of Region 2 in the sense that NgL » 1, but a/r is sufficiently small so that g/p -C 1, i.e., the high density approximation in (6) is still valid. The explanation of the factor (NgL)1^3+1') is as follows: The linear extension L of the minimizing density of (17) is for large values of NgL determined by VL(L) ~ g(N/L), which gives L ~ (NgL)1/(-s+1^L. In addition condition (16) requires gp < r~ 2 , which means 1/( s+1) that Na/L(NgL) < 1. The minimum energy of (18) has the scaling property ETF(N, L, g) = NL-2{NgLy/(s+^ETF(l, 1,1). • Region 4, the LL case: g/p ~ 1, with p ~ (N/L)N~2^s+2\ functional

described by an energy

oo

/

(VL(z)p(z) + p(z)3e(g/p(z)))dz. -oo

(19)

184

ROBERT SEIRINGER, ELLIOTT H. LIEB, JAKOB YNGVASON

This region corresponds to the case g/p ~ 1, so that neither the high density (6) nor the low density approximation (7) is valid and the full LL energy (5) has to be used. The extension L of the system is now determined by V L ( £ ) ~ (N/L)2 which leads to L ~ LN2^s+2\ Condition (16) means in this region that Nr/L ~ NsKs+2)r/L -> 0. Since Nr/L ~ (p/g)(a/r), this condition is automatically fulfilled if g/p is bounded away from zero and a/r -> 0. The ground state energy of (19), £ LL (iV, L,g), is equal to Nj2Ehh(l, 1,5/7), where we introduced the density parameter 7 := (7V/L)AT-2/(s+2).

(20)

• Region 5, the GT case: g/p » 1, with p ~ (N/L)N~2^s+2\ described by a functional with energy density ~ p 3 , corresponding to the Girardeau-Tonks limit of the the LL energy density. It corresponds to impenetrable particles, i.e, the limiting case g/p —> 00 and hence formula (7) for the energy density. As in Region 4, the mean density is here p ~ 7. The energy functional is oo

/ with minimum energy EGT(N,L)

(VL(z)p(z) + (ir2/3)p(z)3)dz,

(21)

•00

= A^7 2 £^ GT (1,1).

As already mentioned above, Regions 1-3 can be reached as limiting cases of a 3D GrossPitaevskii theory. In this sense, the behavior in these regions contains remnants of the 3D theory, which also shows up in the the fact that BEC prevails in Regions 1 and 2. This is what we discuss in the next section.

4. Bose-Einstein condensation In this final section we remark about BEC in the ground state. We are able to prove its occurrence in Regions 1 and 2. It probably also occurs in part of Region 3, but we cannot prove this and it remains an open problem. BEC means that the one-body density matrix 7 Q M (x, x'), which is obtained from the ground state wave function * 0 by QM 7

( x , x') = N J * 0 ( x , x 2 , . . . , XJV ) * 0 ( x ' , x 2 , . . . , x w )*
(22)

factorizes approximately, for large N, as Nip(x)ip(x') for some L2-normalized function ip. This, in fact, is 100% condensation and this is what we prove occurs in Regions 1 and 2. The condensate wave function ip is the square-root of the minimizer of the ID GP functional (17) times the transverse function 6 r (x ± ). This is stated in the following Theorem. Its proof is similar to the work in [17]. Theorem 4 . 1 . In the limit N —> 00, r/L —» 0 with NgL fixed, lim ^ V

M

( ( r x \ L*), ( r x ' \ Lz')) = 6(x x ) 6(x' X ) $GP{z) 4>GP(z>)

(23)

in trace norm. Here 4>GP is the square root of the minimizer of the GP functional (17) with N — 1, L = 1 and interaction parameter NgL.

One-dimensional behavior of dilute Bose gases in traps

185

B E C is not expected in Regions 4 and 5. In fact for a homogeneous gas of I D impenetrable bosons Lenard showed [8] t h a t t h e largest eigenvalue of 7 ^ M grows only as Nll2 and, according t o [18,19], this holds also for impenetrable bosons in a quadratic t r a p potential.

Acknowledgments E. H. L. was partially supported by the U.S. National Science Foundation grant No. PHY 01 39984. R. S. was supported by the Austrian Science Fund. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

K. Bongs et al., Phys. Rev. A 6 3 , 031602 (2001). A. Gorlitz et al., Phys. Rev. Lett. 87, 130402 (2001). M. Greiner et al., Phys. Rev. Lett. 87, 160405 (2001). F. Schreck et al., Phys. Rev. Lett. 87, 080403 (2001). E. H. Lieb, W. Liniger, Phys. Rev. 130, 1605 (1963). E. H. Lieb, Phys. Rev. 130, 1616 (1963). M. D. Girardeau, J. Math. Phys. 1, 516 (1960). A. Lenard, J. Math. Phys. 5, 930 (1964). L. Pitaevskii, S. Stringari, J. Low Temp. Phys. 85, 377 (1991). M. D. Girardeau, E. M. Wright, J. M. Triscari, Phys. Rev. A 6 3 , 033601 (2001). A. D. Jackson, G. M. Kavoulakis, Phys. Rev. Lett. 89, 070403 (2002). S. Komineas, N. Papanicolaou, Phys. Rev. Lett. 89, 070402 (2002). E. H. Lieb, R. Seiringer, J. Yngvason, arXiv:cond-mat/0304071 and arXiv:math-ph/0305025 . E. H. Lieb, J. Yngvason, Phys. Rev. Lett. 80, 2504 (1998). E. H. Lieb, R. Seiringer, J. Yngvason, Phys. Rev. A 6 1 , 043602 (2000). E. H. Lieb, R. Seiringer, J. Yngvason, Comm. Math. Phys. 224, 17 (2001). E. H. Lieb, R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002). T. Papenbrock, Phys. Rev. A 67, 041601(R) (2003). P. J. Forrester, N. E. Frankel, T. M. Garoni, N. S. Witte, arXiv:cond-mat/0211126.

Semiclassical dynamics of an electron moving in a slowly perturbed periodic potential STEFAN T E U F E L , GIANLUCA PANATI

(Tech. U. Miinchen)

The semiclassical dynamics of a quantum particle in a slowly perturbed periodic potential are discussed. Based on recent results [16] we explain how the well known semiclassical model of solid state physics is related to the Schrodinger equation. We also present the less known first order corrections to the semiclassical model and discuss their relation to the quantization of the Hall conductivity.

1. From q u a n t u m dynamics t o t h e refined semiclassical model A basic problem of solid state physics is to understand the motion of electrons in the periodic potential which is generated by the ionic cores. While this problem is quantum mechanical, many electronic properties of solids can be understood already in the semiclassical approximation [1,11,20]. It is common lore in solid state physics that for suitable wave packets, which are spread over many lattice spacings, the main effect of a periodic potential Vr on the electron dynamics corresponds to changing the dispersion relation from the free kinetic energy Efvee(k) — \ k2 to the modified kinetic energy En(k) given by the n t h Bloch function. Otherwise the electron responds to slowly varying external potentials A, as in the case of a vanishing periodic potential. Thus the semiclassical equations of motion are r = V£„(K),

k = -V(r) +rx

B(r),

(1)

where K — k — A(r) is the kinetic momentum and B = curlA is the magnetic field. (We choose units in which the Planck constant h, the speed c of light, and the mass m of the electron are equal to one, and absorb the charge e into the potentials.) The corresponding equations of motion for the canonical variables (r, k) are generated by the Hamiltonian Hsc(r,k) = En{k-A(r))+<j>(r),

(2)

where r is the position and k the quasi-momentum of the electron. Note that there is a semiclassical evolution for each Bloch band separately. In a recent work [16], we use adiabatic perturbation theory in order to understand on a mathematical level how these semiclassical equations emerge from the underlying Schrodinger equation i ds i/>(y, * ) = ( £ ( - iV y - A(sy))2 + Vr(y) + (ey)) iP(y, s)

(3)

in the limit e —> 0 at leading order. In addition, the order e corrections to (1) are established, see equation (7). In (3) the potential Vr : Kd —> R is periodic with respect to some regular lattice T generated through the basis {71,... ,7
r = lx e Rd •. x = Yfj=iaj a

186

for s o m e

« e zd|

Semiclassical dynamics of an electron moving in a slowly perturbed periodic potential

187

and Vr( • + 7 ) = Vr(-) f° r all 7 S T. The lattice spacing defines the microscopic spatial scale. The external potentials A(ey) and (ey), with A : Rd -> R d and <> / : R d -> R, are slowly varying on the scale of the lattice, as expressed through the dimensionless scale parameter £, e are of order e and therefore have to act over a time of order e _ 1 to produce finite changes, which is taken as the definition of the macroscopic time scale. Hence, one is interested in solutions of (3) for macroscopic times. The macroscopic space-time scale (x, t) is defined through x = ey and t = es. With this change of variables equation (3) reads iedt i>E(x,t) = ( j ( - ieV x - A{x))2 + Vr(x/e) + cf>(x)^j i>e{x,t)

(4)

with initial conditions tpe(x) = e~d/2ip(x/e). If Vr = 0, then the limit e —> 0 in equation (4) is the usual semiclassical limit with e replacing H. The problem of deriving (1) from the Schrodinger equation (3) in the limit e —* 0 has been attacked along several routes. In the physics literature (1) is usually accounted for by constructing suitable semiclassical wave packets [11,13,20]. The few mathematical approaches to the time-dependent problem (4) extend techniques from semiclassical analysis, as the Gaussian beam construction [5,8], or Wigner measures [2,7]. Separation of time-scales plays a fundamental role in the understanding of the dynamics of many physical systems. Bloch electron are not an exception. We have shown [9,16] that (4) can be fruitfully studied using adiabatic perturbation theory [15,18]. The results obtained in this way constitute not only a derivation of the semiclassical model (1) with general electric and magnetic fields, but they allow to compute systematically higher order corrections in the small parameter e. It turns out that the electron acquires a fc-dependent electric moment An(k) and magnetic moment Mn(k). If the n t h Bloch band En(k) is nondegenerate and isolated, with Bloch eigenfunctions i/jn(k,y), the electric dipole moment is given by the Berry connection An{k) = i(i>n{k)y^n{k)),

(5)

and the magnetic moment by the Rammal-Wilkinson term Mn(k)

= i (ViMfc), x(Hper(k)

- En(k))Vi>n(k))

•

(6)

For sake of a simpler exposition, we focus on the case d = 3 and use the vector product notation (a x b)i = ei:>lajbi for vectors a, b G R 3 . The inner product in L 2 (M 3 /T) is denoted by (•, •) and Hpel(k) is H of (3) with <j> = 0 = A for fixed Bloch momentum k, see (11). Note that En, An and Mn are T*-periodic functions of k, where T* is the lattice dual to T. Hence one can as well think of them as functions on the domain M* = R 3 /T*, the first Brillouin zone. As our main result [16] we find, cf. Theorem 2.1 in this note, that the semiclassical

188

STEFAN TEUFEL, GIANLUCA PANATI

equations of motion including first order corrections read r =

VK(E„(K)

k = -VP (#r)

-eB{r)

•

A1„(K))

- e k x £!„(«) ,

- e J3(r) • M „ ( K ) ) + r x B(r),

(7)

with fin(fc) = V x An(k) the curvature of the Berry connection. In terms of the eigenprojectors Pn(k) of Hpei(k) corresponding to the eigenvalues En{k) one finds that f U * ) = i Tr(P„(fc) [VP„(/c), xVPn(k)}),

(8)

and Mn(k)

= i Tr (Pn(fc) VP„(fc) x (H0(k) - En(k)) VP„(fc)) .

(9)

This shows immediately that iln(k) and M.n{k) do not depend on the choice of the phase of the Bloch function tpn(k), i.e., are invariant with respect to the Bloch gauge, and therefore have an intrinsic physical meaning. Notice, in parentheses, that the equations (7) are in Hamiltonian form with respect to a non standard, e-dependent symplectic form[16]. The equations of motion (7) were discovered only recently by Niu et al. [4,17] using coherent state solutions of equation (4). As the most striking application note that they provide a simple semiclassical explanation of the Quantum Hall Effect. To this end one adds in (3) a strong uniform magnetic field BQ, i.e., the vector potential AQ{X) = ^BQ X X. If its magnetic flux per unit cell is rational, then the Hamiltonian in (3) is still periodic at the expense of a larger unit cell and replacing the usual translations by magnetic translations. The semiclassical equations of motion (7) remain formally unaltered, only En(k) now refers to a magnetic Bloch band. Let us specialize (7) to two dimensions and take B(r) = 0,
jn= f JM'

dkr{k)= [

dk(WkEn(k)-£±nn(k))

JM*

= -£± f

dk£ln(k).

JM-

It is well known that fM, dfc £ln(k) is the Chern number of the magnetic Bloch bundle and as such an integer [19]. A more detailed exposition of the previous argument and further applications related to the semiclassical first order corrections are presented in Chapters 12 and 13 of ref. [3]. Among the latter are the anomalous Hall effect [10] and the thermodynamics of the Hofstadter model [6].

2. Transport of observables and Wigner functions: mathematical results We now come to the precise relationship between the semiclassical equations (7) and the Schrodinger dynamics (3), and we outline how this problem fits in the general framework of space-adiabatic theory. For the unperturbed periodic Hamiltonian HpeT = - A + VT

Semiclassical dynamics of an electron moving in a slowly perturbed periodic potential

189

the separation between slow and fast degrees of freedom is realized by the Zak transform (UMk,y)

:= Y, e- i(y+7) - fc V(2/ + 7), (*,y) £ R 2d ,

(10)

which defines a unitary operator U : L2(RJ) -» L2(Mfe*) ® L2(Td) * L2{M*k,L2(Td))

=-. H.

The transformed operator is fibered, i.e., UH^U-1

+ k)2 + Vr(y) = HpeT(k).

= l(-iS7y

(11)

Hper(k) is an operator-valued multiplication operator acting on the vector valued functions in L2(M£,L2(Ty)). Its pointwise spectrum defines the Bloch band structure through Hper(k)i>n(k)

=

En(k)i)n(k).

If Pn(k) denotes the eigenprojection corresponding to aBlochband En(k), then the projector Pn = JM, dkPn(k) commutes with Hper, thus defining a subspace which is invariant but not necessarily spectral. Let us now consider the perturbed Hamiltonian #m = §( - iVx - A{ex)f

+ Vr{x) + 4>{sx)

(12)

appearing in (3), where the index m reminds us of the microscopic spatial scale. Its Zak transform is easily computed to be Hi:=UH^U-1

= ^-iWy

+ k-A(ieVl))2

+ Vr(y) + ci>(ieVl),

where iV£ is the gradient on L2(M£, L2(Ty)) with suitable boundary conditions. In contrast to the unperturbed case, this operator is not fibered. However, instead of an operator-valued multiplication operator, one can make sense of it as an pseudodifferential operator with operator-valued symbol. Indeed, Hez is the Weyl quantization H(k, ieVj[) of the unboundedoperator valued symbol H(k, r) = \ ( - i V x + k- A(r)f

+ Vr(x) + <j>(r),

which acts on the Hilbert space Hi = L2{Td). It is crucial that, even in the perturbed case, to each isolated Bloch band En there corresponds an almost invariant subspace HnH. Only for states which start in this subspaces and thus, by construction, remain there up to small errors for long times, the semiclassical equations of motion (7) can have any significance. We now introduce some further notation and assumptions to prepare the statement of the main result. The flow of the dynamical system (7) is denoted by $£ : R2d —> M.2d or in canonical coordinates (r, k) = (r,K + A(r)) by $te(r,k) =

($lr(r,k-A(r)),&eK(r,k-A(r))+A(rj).

190

STEFAN TEUFEL, GIANLUCA PANATI

The existence of the smooth family of diffeomorphisms $* is not completely obvious from (7) alone, but follows from the Hamiltonian formulation of (7). For the formulation of the semiclassical limit it is more convenient to switch to the macroscopic Schrodinger equation (4). Its Hamiltonian is denoted by Hs = I ( - ieV* - A(x))2 + Vr(x/e) +
(13)

Under the following assumption on the potentials He is self-adjoint on H2' Assumption. Let Vr be infinitesimally bounded with respect to —A and assume that <j> € C£°(R d ,R) and Aj £ C^°(Rd,R) for any j £ { 1 , . . . ,d}. Here C£°(Rrf) denotes the space of smooth functions which are bounded together with all their derivatives. <0> , equipped with the metric dc

Finally we introduce the Frechet space C = C^ induced by the standard family of semi-norms \\a\\a = \\daa\\00,

a£N2d,

and the subspace of T*-periodic observables Cper = {aeC:

a(r, k + 7*) = a(r, k) V 7 * e T*} .

We abbreviate dc{a) := dc{a,0). Our main result [16] on the semiclassical limit of (4) is the following Egorov-type theorem. Theorem 2.1. Let En be an isolated, non-degenerate Bloch band. For each finite timeinterval J c K there is a constant C < 00, such that for all a € Cper with Weyl quantization a — a(x, —ieVx) one has (e i f f £ t/e~e-iHH/e

<

a°<£>o)n* B(L 2 (R d ))

eCdc{a),

(14)

and n^(eiff£'/eae-iffEt/£-ao^)^

<e2Cdc{a).

(15)

R e m a r k 2.1. The corresponding statement in [16] does not make explicit the dependence of the error on the observable a. However, the more precise version formulated here is a standard consequence of the Calderon-Vaillancourt theorem and the fact that composition with $ £ is a continuous map from C into itself. 0 Theorem 2.1 provides a semiclassical description of the evolution of observables. The most direct way to turn it into a description for the semiclassical evolution of states is via duality, i.e., via the Wigner function. Recall that according to the Calderon-Vaillancourt theorem there is a constant C < 00 depending only on the dimension d such that for a € C one has \(ip,aiP)L2{Rd)\< Cdc(a) HVII2. (16) Hence, the map C B a *-^> (ijj, aip) £ C is continuous and thus defines an element wf of the dual space C, the Wigner function of ip. Writing (V>, aip) =: (wf, a)C',c =• /

dgdp

a{q,p)wf(q,p)

(17)

Semiclassical dynamics of an electron moving in a slowly perturbed periodic potential and inserting into (17) the definition of the Weyl quantization for o e

191

S(R2d)

one arrives at the usual formula wf{q,v)

= ~ 3 JRd dte^riq

+ e£/2) 1>{q - e£/2)

(18)

for the Wigner function. Direct computation yields \\wt\\L2(R2d)=e-d(27rrd/2\\i;\\l2m. Therefore, wf £ L2(R2d) for all e > 0, which explains the notion of Wigner function. Although wf is obviously real-valued, it attains also negative values in general. Hence, it does not define a probability distribution on phase space. However, it correctly produces quantum mechanical expectations of semiclassical observables via (17). With this preparations we obtain the following corollary of Theorem 2.1, which says that the Wigner function of the solution of the Schrodinger equation (4) is approximately transported along the classical flow of (1) resp. (7). Corollary 2.1. Let En be an isolated, non-degenerate Bloch band. Then for each finite time-interval I C K there is a constant C < oo such that for t £ I, a £ Cper and for i>o e IlnL2(Rd) one has ( ( i ^ ' - i ^ o * , , ' ) , * ) ^

<sCdc(a)

||Vo|

<e2Cdc(a)||Vo| Here ipt = e~lH

t e

^ ipo

25

the solution of the Schrodinger equation (4).

As mentioned, the strategy of the proof of Theorem 2.1 is based on adiabatic methods, i.e., on separation between slow and fast degrees of freedom. In particular one realizes that Hz = UH'U-1

= ~ ( - iV„ + k - A(ieVJE))2 + Vr(y) +
is given by the Weyl quantization of an unbounded-operator valued symbol H{k, r) = ~ ( - i V x + k- A(r))2 + Vr(x) + 0(r), which acts on the Hilbert space H[ = L2(Td) for the fast degrees of freedom. This representation allows one to apply a suitable generalization of space-adiabatic perturbation theory [14,15] in order to construct the following objects. — Almost-invariant subspaces. To any isolated non-degenerate Bloch band En corresponds an orthogonal projection n ^ € B{H) which almost commutes with Hz, i.e., \\[Hi,Iln}\\=0(e°°). Hence, the subspace U.nH is approximately invariant under the time-evolution generated by i ? | .

192

STEFAN T E U F E L , GIANLUCA PANATI

— I n t e r t w i n i n g u n i t a r i e s a n d effective H a m i l t o n i a n . There exist a unitary operator U% : WnU -» L2(M*), such t h a t || (e- , H S« - f / r e - 1 ^ 4 K)K

|| = 0(e°°(l

+ \t\)),

where hn is the Weyl quantization of a semiclassical symbol. Its principal symbol hnfi = En(k — A(r)) + (r) is given through Peierls substitution. Theorem 2.1 follows from a semiclassical analysis of t h e effective Hamiltonian, including contributions from its subprincipal symbol. T h e gauge invariant form (7) is obtained by unitarily transforming the results back from L2(M*) to L 2 ( R 3 ) using (U^)^1 and U~l.

References 1. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Saunders, New York, 1976. 2. P. Bechouche, N. J. Mauser, F. Poupaud, "Semiclassical limit for the Schrodinger-Poisson equation in a crystal", Comm. Pure Appl. Math. 54, 851-890 (2001). 3. A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger. The geometric phase in quantum systems, Texts and Monographs in Physics, Springer, Heidelberg, 2003. 4. M. C. Chang, Q. Niu, "Berry phase, hyperorbits and the Hofstadter spectrum: Semiclassical dynamics and magnetic Bloch bands", Phys. Rev. B 53, 7010-7023 (1996). 5. M. Dimassi, J.-C. Guillot, J. Ralston, "Semiclassical asymptotics in magnetic Bloch bands", J. Phys. A 35, 7597-7605 (2002). 6. O. Gat, J. E. Avron, "Magnetic fingerprints of fractal spectra and the duality of Hofstadter models", New J. Phys. 5, 44.1-44.8 (2003). 7. P. Gerard, P. A. Markowich, N. J. Mauser, F. Poupaud, "Homogenization limits, Wigner transforms", Commun. Pure Appl. Math. 50, 323-380 (1997). 8. J. C. Guillot, J. Ralston, E. Trubowitz, "Semi-classical asymptotics in solid state physics", Comm. Math. Phys. 116, 401-415 (1988). 9. F. Hovermann, H. Spohn, S. Teufel, "Semiclassical limit for the Schrddinger equation with a short scale periodic potential", Comm. Math. Phys. 215, 609-629 (2001). 10. T. Jungwirth, Q. Niu, A. H. MacDonald, "Anomalous Hall effect in ferromagnetic semiconductors", Phys. Rev. Lett. 88, 207208 (2002). 11. W. Kohn, "Theory of Bloch electrons in a magnetic field: the effective Hamiltonian", Phys. Rev. 115, 1460-1478 (1959). 12. P. L. Lions, T. Paul, "Sur les mesures de Wigner", Revista Mathematica R>eroamericana 9, 553-618 (1993). 13. J. M. Luttinger, "The effect of a magnetic field on electrons in a periodic potential", Phys. Rev. 84, 814-817 (1951). 14. G. Nenciu, V. Sordoni, "Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory", Math. Phys. Preprint Archive mp_arc 01-36 (2001). 15. G. Panati, H. Spohn, S. Teufel, "Space-adiabatic perturbation theory", Adv. Theor. Math. Phys. 7 (2003). 16. G. Panati, H. Spohn, S. Teufel, "Effective dynamics for Bloch electrons: Peierls substitution and beyond", to appear in Comm. Math. Phys. (2003). 17. G. Sundaram, Q. Niu, "Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects", Phys. Rev. B 59, 14915-14925 (1999). 18. S. Teufel. Adiabatic perturbation theory in quantum dynamics, Springer Lecture Notes in Mathematics 1821, 2003. 19. D. J. Thouless, M. Kohomoto, M. P. Nightingale, M. den Nijs, "Quantized Hall conductance in a two-dimensional periodic potential", Phys. Rev. Lett. 49, 405-408 (1982). 20. J. Zak, "Dynamics of electrons in solids in external fields", Phys. Rev. 168, 686-695 (1968).

Dynamical systems Session organized by M. VlANA (Rio de Janeiro) and H.

ELIASSON

(Paris)

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Linear quasi-periodic systems — reducibility and almost reducibility L. H. ELIASSON (U. Paris 7) Reducibility is a central notion for linear quasi-periodic systems. We present a perturbative result about a slightly weaker property — almost reducibility — and we discuss this notion and its relation to reducibility.

1. Introduction We consider two types of quasi-periodic linear systems: time-continuous systems x'(t) = A(d + tw)x(t),

t£R,

(1)

with A : Td —» gl(RD), and time-discrete systems x(t + l) = A(9 + tu)x(t), d

D

*GZ,

d

d

(2)

d

with A : T -> Gl(R ). T is the d-dimensional torus K /Z . The mapping A is at least continuous, but we shall frequently assume that it is smooth and even analytic. w G M.d is the frequency vector. For the system (1) we can assume without restriction that the components of u> are rationally independent, i.e., (n,w)/0,

Vn€Zd\0.

This condition defines a dense Qs of full measure. Frequently we shall assume that w satisfies a Diophantine condition, i.e., for some 0 < K, 0 < r |(n,o;)| > - ^ - V n £ Z d \ 0 . (3) \n\T A vector w with rationally independent coefficients is said to be Liouville if it is not Diophantine. The set of Liouville vectors is a dense Qs (if d > 2), but almost all vectors are Diophantine. For the system (2) we shall assume that (w, 1) is rationally independent and, frequently, that (u, 1) satisfies a Diophantine condition. The flow of (1) is the mapping on T d x GL(RD) DC(K,T):

{0,X)»(0

+ tu,xe{t)X),

(4)

where x$(t) is the matrix solution of (1) defined by xg(0) — Id — the fundamental solution or monodromy matrix. x$(t) verifies the co-cycle property: xe{t + s) = xe+su,(t) xe{s),

\/t, s ^6 £ Td.

For the system (2) the flow has the same form (4) but in this case the fundamental solution xe(t) is defined by xg(l) = A{9) and the cocycle property.

195

196 Time-1-map.

L. H. ELIASSON

If xe{t), t € R, is the fundamental solution of (1) then xe(t),

t G Z,

is the fundamental solution of (2) with the new A(9)=xe(l). (This follows from the co-cycle property.) Clearly such an A(ff) is necessarily homotopic to Id, so we cannot obtain all time-discrete systems in this way. But by suspension we can obtain all time-discrete systems, even those that are not homotopic to Id. Indeed for a given A(9) of class Cr we can construct a matrix A(6,8d+i), also of class C r , such that the fundamental solution x$(t) of x'(t) = A((9,9d+1)

+ t(u,l))

x(t),

verifies x$(l) = A{9). (It is not known if this result holds also in the analytical category.)

2. Reductibility in t h e sense of Lyapunov Lyapunov introduced the notion of reducihility. He said that the system (1) is reducible if its fundamental solution can be represented as xe(t) = z(t)etB, for some B in gl(RD) and for some z(t) £ GL(ED) \z(t)\ and

(5) with

\z{t)~l\

bounded (uniformly in t). It is easy to see that (5) is equivalent to z'(t) = A(6 + tw) z(t) - z(t)B.

(6)

Proposition 2.1. If (1) is reducible for some 6, then it is reducible for all 9 and B{6) = B. Proof. If (1) is reducible for some 6 then clearly it is reducible for all 6+tu) with B(6+tu>) = B. Let first tn be a sequence such that 9 + tnu -> 9 in Td. By taking a subsequence we can assume that z(tn,9)

-» C.

Let now w'(t)=A(0

+ tLj)w(t)-w(t)B,

w(0) = c.

Since the solution depend continuously on the equation it follows that Z(t + tn)->w(t),

Z(t + tn)'1

-»

Hence, both \w(t)\ and |u;(t) _ 1 | are bounded uniformly in t.

Wit)-1. •

Linear quasi-periodic systems — reducibility and almost reducibility

197

The discrete system (2) is reducible if its fundamental solution can be represented as xe(t) = z(t)B\

(7)

for some B in Gl(RD) and for some z(t) as before. The equation (7) is equivalent to z(t + l) = A(6 + tLj)z{t)B-\

(8)

Proposition 2.2. (1) is reducible if, and only if, its time-1-map is reducible. Proof. We only need to show the implication to the left, so let's assume that the time-1map. Define now simply ze(t) = xe(t) e~tB. By assumption zg(t) is bounded on t G Z, hence on t € M.

•

Reducible systems which are not constant exist of course. For example, if A(8) € so(R n ) then all solutions of (1) are bounded and (1) is therefore reducible. 3. Obstructions t o reducibility Let v be a non-zero vector in Rd and consider X(9,v) = limsup - log\xe(t)v\.

(9)

t—»oo t

This number, which measures the exponential growth of the solution xg(t)v, can take only one out of at most D values

Xi(e) < • • • < xd(e).

(io)

These are the characteristic numbers of Lyapunov. By the theorem of Oseledec, the lim sups are limits for a.e. 8 £ Td and all v £ M D , and these limits are independent of 6 (for a.e. 6). These numbers are known as the Lyapunov exponents. Irregular systems.

The characteristic numbers of (1) always verify 1 /"*

V A J - ( 0 ) > lim - /

Tr(A(6 + SUJ) ds

(11)

— see [19]. The system is said to be regular if we have equality in (11), and it follows from the theorem of Oseledec that (1) is regular for almost every 6. It is clear that if (1) is reducible (for some, and hence for all, 6) then the system is regular for all 6. But there are almost-periodic linear systems that are non-regular [11,17]. Non-uniformly hyperbolic systems. (w+(8)

(1) is hyperbolic if the spaces

= {v£RD

: \(6,v) D

< 0}

1 W_(0) = { » £ l : X(9,v) > 0}

198

L. H. ELIASSON

span RD for a.e. 9, i.e., W+(6) + W-(e)=RD,

a.e. 0 <E T d .

We say that (1) is uniformly hyperbolic if there exists A+ < 0 < A_ such that \x8(t)v\ < cste.e a +M, tA

\x6{t)v\ < cste.e -|w|,

Vt > 0,Vu G W+{6), Vt < 0,Vw e W_(6>).

If it is not uniformly hyperbolic, we say that it is non-uniformly hyperbolic. Clearly a reducible system cannot be non-uniformly hyperbolic. There are many examples of non-uniformly hyperbolic systems. The best studied are the time-continuous Schrodinger equation Heu(t) = -u"{t) + V{9 + tu)u{t) = Eu(t),

t € R,

and the time-discrete Schrodinger equation Hgu(t) = -(u{t + 1) + u(t - 1)) + V{6 + tuj)u{t) = Eu(t),

t e Z.

It is known that these equations are uniformly hyperbolic if and only if E does not belong to the spectrum of Hg a [12], so if the Lyapunov exponent is positive for an E in the spectrum then the corresponding system is non-uniformly hyperbolic. For the discrete Schrodinger equation there are many analytic [3,10,21] as well as nonanalytic [4,9,20] examples of non-uniformly hyperbolic systems. Time-continuous systems are less well studied but also in this case there are analytic [9, 21] and non-analytic [1] examples. Non-uniformly hyperbolic systems which are not Schrodinger are constructed in [22]. Non-polynomial growth. A reducible system with zero Lyapunov exponents has solutions which are either bounded or have polynomial growth. Examples of systems with solutions with non-polynomial growth are given in [5].

4. Reductibility For quasi-periodic systems it is natural to consider a more special notion which goes back to Floquet. We say that (1) is reducible in the sense of Floquet, from now on simply reducible, if it is reducible in the the sense of Lyapunov with z(t) = Z(tu),h Z : (2T) d -> GL{RD) continuous. In this case (6) takes the formc d„Z(6) = A{6)Z{6) - Z(6)B. a

T h e spectrum of Hg is independent of 8. B y (2T) d we mean that Z is 2-periodic in each variable. c By du we means the directional derivative on T d in the direction u>.

b

(12)

Linear quasi-periodic systems — reducibility and almost reducibility

199

Another way to state this is to say that the change of variables = Z(9r1x)

(6,x)~(6,y

(13)

transforms the system (1) to the constant coefficient system y'(t) = By(t). In the time-discrete case equation (8) takes the form Z{e + u) =

A(6)Z(6)B-1

and the transformation (13) transforms (2) to the constant coefficient system y(t+l)

= By(t).

Reducibility is a stronger notion than reducibility in the sense of Lyapunov since it takes into account the "rotational" behaviour of the solutions. Periodic systems are always reducible, but quasi-periodic are not. Consider for example

and assume that d

I

t

a(0 + su) — (a) ds

is unbounded. Then (1) is not reducible. However, if the integral is bounded, which is always the case if w is Diophantine and a is sufficiently smooth, then the system is indeed reducible. Another example is (2) with A(0\ = ( y>

C0S

^

sin e

( )\

sin

V- (0) cos(0y"

This is an analytic system but not reducible unless the system is periodic. Ergodicity.

On a compact group like Td x SO(R3)

the Haar measure is invariant under the flow of (1). If the system is reducible then it has an invariant foliation and uncountably many ergodic measures. Examples of analytic systems with Diophantine frequency vector that are uniquely ergodic, and hence not reducible, are given in [6] — see also [18] for similar results for smooth systems with w Liouville. The examples [5,6] show that reducibility is not an open property not even in an analytic topology. In general the structure of the set of reducible system is badly understood. There are some relevant results in C°-topology — for SL (K2) [2] and for the Schrodinger equation [8] — and in C°°-topology — for T x SU{C2) [13] and for T x SL(R2) [14]. (a) is the mean value of the function a.

200

L. H. ELIASSON

5. A l m o s t r e d u c t i b i l i t y We say that (1) is almost reducible if, for all e > 0 there exist Z : (2T) d -> GL(RD)

continuous

and B G gl{RD) such that duZ(0) = A(0)Z(6) - Z(0)(B + A(0)),

B G gl(RD),

(14)

and such that | i | and I Z P H Z - 1 ! < £ .

(15)

This property implies that the system, for arbitrarily long time, behaves as a reducible system. For example, (14) and (15) imply \xe(t) - Z{6 + tu)etBZ{6)-l\

< e\t\e^tB\

\t\ < - . e

A consequence is that an almost reducible system, like a reducible one, never can be irregular nor non-uniformly hyperbolic. However, it may have asymptotic behaviour that is different from that of a reducible system. For example, an almost reducible systems may have nonpolynomial growth [5] or may be uniquely ergodic [6]. The reason is that such behaviour can occur close to constant coefficients and any system close to constant coefficients is almost reducible, as follows from the theorem described below [7]. Assumptions. Let A : Td -> gl(RD),

\A\r = sup \A(x)\ < oo, \Sfx\
and A0£gl(RD),

U£DC{K,T).

Theorem 5.1. There exists a constant £o = £o(r, K, r, \Ao\,D,d) \A-A0\r

< £0,

then for all e > 0 there exist JZ :(2T)d ^GL{RD),

\Z\p < oo

such that duZ(6) = A(6)Z(0) - Z(6)(B + A(6)), with

{

|Z|o, | Z - 1 | o bounded, \A\P,

\ZA\p<e.

such that, if

Linear quasi-periodic systems — reducibility and almost reducibility

201

Notice that the transformed system B -f- A(6) is defined only on (2T) d , so one cannot iterate this statement without loosing "periodicity". This is an essential complication in the proof. Notice also that the domain of analyticity of Z, measured by p, depends on e and may go to 0 as e —> 0. A result of similar flavour was proven in [15] for certain compact groups. It follows from this theorem that the set of almost reducible systems contains an open neighbourhood (in | | r -topology) of the constant systems. Question 5.1. Is the set of almost reducible systems an open set? contained in the closure of its interior?

Or perhaps, is it

Question 5.2. Is the set of almost reducible systems dense among all systems? Among all systems without exponential growth (all Lyapunov exponents being zero)? Among all systems with bounded solutions?

6. A r e m a r k a b o u t t h e proof and t h e second Melnikov condition 6.1. Floquet exponents Suppose that (1) is reducible with Z{6) and B verifying (12). The eigenvalues of B are the Floquet exponents. Proposition 6.1. The real parts of a(B) are unique — these are the Lyapunov exponents of (1). The imaginary parts of a(B) are determined modulo

{i(n,W):nGZd}. Proof. Let xe(t) = Z{9 + tLo)eBtZ(9)-1.

Then

lim -J- /

xe(t)e-^dt

has some finite and ^ 0 component only if Re(7) £ Re(a(B))

and

S(7)eS((7(S)) + | i ( n , W ) : n G Z d

• The following example shows that the imaginary parts are not unique in general. Example 6.1. If B = l

0 -(n,u)


for some n £ Z d , then 0

Zfc(*)=expi

(k,6) }

Q

verifies xe(t) = Zk(e + tuj)exp(t( for any k £

Q

_k

{n A

o)

0

*'u))\Zk{0)

-l

202

L. H. ELIASSON

6.2. Reduction of Floquet exponents. Let a = ( a i , . . . , ar>) be the eigenvalues of B and let a =

(ai,...,aD)

be D arbitrary complex numbers submitted to the conditions6

{

ak = &j

if ak = atj,

&k = &j

if ak = a j .

We define B(a) be "the same" matrix as B but with the eigenvalues a replaced by the eigenvalues a. It is well-defined by the first condition and real by the second condition. Take now for any eigenvalue ak a vector mk &\lid such that (mk=mi

if ak=

\mk

if

= -mi

an,

ak=a~i,

and define Y{6) = exp B i m (0) = exp[S - B(a - i(m, 0))]. Then Y : (2T) d -> Gl(RD),

Y(0) = 1,

and verifies duY{6) = BY{0) - Y{6)B(a -

i(m,u)).

Hence, (0,x)~(6,y

=

Z(er1x)

transforms the constant coefficient system x' = Bx with Floquet exponents a to a new constant coefficient system with Floquet exponents a = a — i(m,Lo). 6.3. The second Melnikov condition The eigenvalues of the operator ad B : X -> [B, X] are precisely all the differences of the eigenvalues of B. B is said to satisfy a "second Melnikov condition" if there exist K' > 0, T > 0, such that for all ak, a.\ we have \ak-ai-i{n,u)\>—-, e

z denotes the complex conjugate of z.

K'

VneZd\0.

(16)

Linear quasi-periodic systems — reducibility and almost reducibility

203

This condition is central in perturbation theory for reducibility and for lower-dimensional tori. Usually it is sufficient that, for each "scale" of the perturbation theory, this condition is fulfilled for n in some finite, but sufficiently large, domain in Zd. Another important point is that condition (16) is not required to hold for n — 0. If w satisfies the Diophantine condition (3), for each atk,ai there can be at most one vector n = rikj € Z d in the domain

I-I*"'-(£)* violating (16). Of course, there may be no such n at all in which case condition (16) is fulfilled in \n\ < N'. Let U = {(fc, I) : n,k,i exist}. Then one verifies that there exist vectors nik G | Z d such that nk,i = mlc-mi,

V(k,l)eU,

and ma = m/3

if a = /?,

ma = —771/3 if a = /9. Using these vectors we can construct Y(0) = exp B , m (0) which transforms the system x' — Bx to a new system y' — By with Floquet exponents a = a — i(m,Lj). By construction, these exponents will satisfy the Melnikov condition in some finite but large domain \n\ < N" PS N'. This operation will permit us to carry out the perturbation theory to any order of accuracy and, hence, obtain almost reducibility. The price to pay is that one has to use transformations which are close to an "exponential" of the type exp B , m (0).

(17)

The transformation Z in the Theorem is therefore not close to the identity but to a finite product of such "exponentials".

7. Reducibility and almost reducibility An infinite product of exponentials (17) will typically not converge. Therefore, if the (appropriate) Melnikov condition is violated at infinitely many "scales" of the perturbation theory we cannot expect reducibility. On the other hand, if it is violated only at finitely many "scales" we will get reducibility. This analysis lies behind the following conjecture.

204

L. H. ELIASSON

Conjecture 7.1. For u> £ DC(K,T) Ax{&) in \AX-A0\r

and for any generic one-parameter family of analytic <e0(r,K,T,\A0\,D,d),

the system x'{t) = Ax{9 + tw) x{t) is reducible for a.e. A. For reasonable one-parameter families of such systems, reducibility holds for a very large set of parameter values. For almost all parameter values reducibility has been proven for the Schrodinger equation [5] and for certain one-parameter families on certain compact groups [16]. On the other hand, not all systems close to constant coefficients are reducible. Conjecture 7.2. For w £ DC(K,T)

and for any generic analytic A{6) in

\A-Ao\r

<s0(r,K,T,\A0\,D,d)

the system x'(t) = A(9 + tuj)x(t) is non-reducible and, in case of a compact group, uniquely ergodic. This has been proven in the case Td x SO (R 3 ) [6]. Question 7.1. Are "almost all" almost reducible systems reducible? Question 7.2. Is the generic almost reducible system, non-reducible?

References 1. K. Bjerklov, "Positive Lyapunov exponents for a class of 1-D quasi-periodic Schrodinger equations — The continuum case", manuscript (2003). 2. J. Bochi, "Genericity of zero Lyapunov exponents", Ergod. Th. & Dynam. Sys. 22, 1667-1696 (2002). 3. J. Bourgain, M. Goldstein, "On non-perturbative localization with quasi-periodic potential", Ann. of Math. 154, 835-879 (2000). 4. L. H. Eliasson, "Discrete one-dimensional quasi-periodic Schrodinger operators with pure point spectrum", Acta Math. 179, 153-196 (1997). 5. L. H. Eliasson, "Floquet solutions for the one-dimensional quasi-periodic Schrodinger equation", Comm. Math. Phys. 146, 447-482 (1992) 6. L. H. Eliasson, "Ergodic skew systems on Td x SO(3,R)", Ergod. Th. & Dynam. Sys. 22, 1429-1449 (2002). 7. L. H. Eliasson, "Almost reducibility of linear quasi-periodic systems", Proceedings in Symposia in Pure Mathematics 69, 679-705 (2001). 8. R. Fabbri, R. Johnson, R. Pavani, "On the nature of the spectrum of the quasi-periodic Schrodinger operator", Nonlinear Analysis 3, 37-59 (2002). 9. J. Frohlich, T. Spencer, P. Wittver, "Localization for a class of one-dimensional quasi-periodic Schrodinger operators", Comm. Math. Phys. 132, 5-25 (1990). 10. M. Herman, "Une methode pour minorer les exposants de Lyapunov", Comment. Math. Helvetici 58, 453-502 (1983).

Linear quasi-periodic systems — reducibility and almost reducibility

205

11. R. A. Johnson, "The recurrent Hill's equation", J. Diff. Equations 46, 165-193 (1982). 12. R. A. Johnson, "Exponential dichotomy, rotation number, and linear differential equations with bounded coefficients", J. Diff. Equations 6 1 , 54-78 (1986). 13. R. Krikorian, "Global density of reducible quasi-periodic cocycles on T 1 x SU(C2)", Ann. of Math. 154, 269-326 (2001). 14. R. Krikorian, "Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic co-cycles on T x SL(R2)", manuscript (2003). 15. R. Krikorian, "Reductibilite des systemes produits-croises a valeurs dans des groupes compacts", Asterisque 259 (1999). 16. R. Krikorian, "Reductibilite presque partout des flots fibres quasi-periodiques a valeurs dans des groupes compacts", Ann. Sci. Ecole Norm. Sup. (4) 32, 187-240 (1995). 17. V. M. Millionscikov, "Proof of the existence of irregular systems of linear differential with almost periodic coefficients", Diff. Equations 4, 203-205 (1968). 18. M. Nerurkar, "On the construction of smooth ergodic skew-products", Ergod. Th. & Dynarn. Sys. 17, 311-326 (1988). 19. V. V. Nemytskii, V. V. Stepanov, Qualitative theory of differential equations, Princeton University Press, Princeton, New Jersey, 1960. 20. Ya. G. Sinai, "Anderson localization for the one-dimensional difference Schrodinger operator with a quasi-periodic potential", J. Stat. Phys. 46, 861-909 (1987). 21. E. Sorets, T. Spencer, "Positive Lyapunov exponents for Schrodinger operators with quasiperiodic potential", Comm. Math. Phys. 142, 543-566 (1991). 22. L. S. Young, "Lyapunov exponents for some quasi-periodic cocycles", Ergod. Th. & Dynam. Sys. 17, 483-504 (1997).

On discrete Schrodinger operators with stochastic potentials W. SCHLAG (Caltech)

1. Introduction Given an ergodic map T : Td -> Td, a potential Vx(n) = v(Tnx), analytic and nonconstant, define the Hamiltonian Hx:=-Az

where v : Td -> R is

+ XVx.

(1)

This is a version of the well-known Anderson's model. In the late 1950s, Phil Anderson predicted that random impurities could turn conductors into insulators. In mathematical terms, he predicted that random potentials should lead to pure point spectrum with rapidly decreasing eigenfunctions — at least for large disorders. Instead of random potentials, one considers here potentials given in terms of deterministic dynamics. The "randomness" is given by the parameter x £Td. Basic questions: Does (1) display pure point spectrum for large A, and does Anderson localization take place (exponentially decaying eigenfunctions)? One can also ask about dynamical localization. This refers to the property sup t ||(l + |ra|) A e tti? ^o||2 < °o for all A > 0, provided tpo decays very rapidly. Other topic of interest is the limiting distribution of the eigenvalues of Hx restricted to \—N, N] as N —> oo (this is known as the integrated density of states or IDS). Other very interesting questions concern the presence of a.c. spectrum for small A, as well as the structure of the spectrum (Cantor set). We cannot review the long and rich history of this subject here, but rather refer the reader to the monographs by Carmona and Lacroix [9] as well as Figotin and Pastur [11] for this purpose. Another resource which is closely related to this note is the forthcoming book by J. Bourgain [3]. Let us merely mention the fundamental results by Dinaburg, Sinai [10] (a.c. spectrum for small A), Fiirstenberg [14] (positivity of the Lyapunov exponent), Goldsheid, Molchanov, Pastur [15] (A.L. for the one-dimensional model), Frohlich, Spencer [12] ("multiscale analysis" in the random case), Aizenman, Molchanov [1] ("fractional moment method"), Frohlich, Spencer, Wittwer [13] and Sinai [18] (A.L. for the quasiperiodic model), as well as the many deep and important contributions by Avron, Bellissard, Campanino, Carmona, Delyon, Eliasson, Figotin, Gordon, Jitomirskaya, Kirsch, Klein, Kotani, Last, Martinelli, Oseledec, Pastur, Ruelle, Simon, Simon-Wolff, Souillard, Stolz, Thouless, Wegner. We now consider the eigenvalue equation (HxiP)n = -4>n+1 - V„_i + \v(Tnx)ipn Examples of maps T are: the shift Tx = x+u)(modZd),

206

= E1>n.

(2)

the doubling map Tx = 2a: (mod 1),

On discrete Schrodinger operators with stochastic potentials

207

and the skew-shift T(x, y) = (x + y,y + ui) (mod I?). It follows from the covariance relation HTx = UHXU* with U = left shift that Y,x := spec(H x ) is constant for a.e. x. The same also holds for the spectral parts E'ac x '

25". ££ c .

Recall the following basic objects in the study of (2). The monodromy matrices and Lyapunov exponent are defined as 1

Mn(x,\,E):=Y[ Ln(X,E):=L(\,E):=

\\v(T*x)-E

f n JTd

-1 0

log\\Mn(x,\,E)\\dx

lim Ln(\, E) = Urn - log ||Mn(a;, A, E)\\,

where the last relation holds for a.e. x by Kingman's subadditive ergodic theorem. Furthermore, by the Ishi-Pastur theorem E£ c C {E : L(\,E) = 0}, whereas Kotani's converse statement is meas [{E : L(X, E) = 0} \ S"c] = 0, a.e. x. Let Tx = T^x := x + w (mod 1) be the one-dimensional shift. We denote the Lyapunov exponent by L(u, A, E) to emphasize the dependence on w. The following theorem is proved in [4]. Theorem 1.1 (Bourgain-Goldstein). Suppose i n f ^ s L(w, A, E) > 0. Then for a.e. w and a.e. x the Hamiltonian Hx displays Anderson localization, as well as dynamical localization. N o t e : This does not explicitly require large A. If v{x) = cos(2nx) (the almost Mathieu operator), then Theorem 1.1 applies for A > 2 (by Herman's result L(o;, A, E) > log(A/2) for A > 2). Jitomirskaya [16] proved this result for the almost Mathieu operator and all Diophantine to and a.e. x. The proofs of both Theorem 1.1 as well as [16] are nonperturbative, and are based on the transfer matrix formalism. The following theorem is from [17]. Theorem 1.2 (Goldstein-S.). Suppose to is Diophantine (which means here that \\nuj\\ > nQogn)" forn> 1, a> I). If L(w, E) > 0 for all Ei < E < E2, then L(LJ, •) as well as the integrated density of states N(u>,-) are Holder continuous in E € [.Ei,^]N o t e : The integrated density of states (IDS) is the deterministic limit N{w, E) = lim — ^ — - # { 1 < j < 2N + 1 : EJN) < E} where E= Ej (x,w) are the eigenvalues of Hx restricted to [-N, N]. Thouless's formula relates the Lyapunov exponent L and the IDS N: L(u,E)=

f\og\E-E'\N(w,dE').

Sinai's work on large disorder [18] shows that the IDS can be no better than Holder-^ continuous, and Bourgain [3] obtained the exponent ^ — for the almost Mathieu operator and large

208

W . SCHLAG

A. In [17], the Holder exponent depended on the Lyapunov exponent. Bourgain refined [17] to obtain a Holder exponent that is uniform in L{E). Bourgain and Jitomirskaya [7] modified the method from [17] to show that L{u, E) is jointly continuous in w, E at every point (UQ,E), U>Q G K \ Q as well as continuous in E for every w (without assuming that L is positive). The skew-shift on T 2 is defined to be Tu(x,y) = (x + y,y + u>) (modZ 2 ). Note the quadratic dependence on n in the nth iterate T£(x, y) = (x + ny + n{n - l)w/2, y + nu>) (mod Z 2 ). It is conjectured to lead to L(w, X,E) > 0 for all A > 0. The following theorem is proved in [5]. Theorem 1.3 (BGS). Fix e > 0. Then there exist Ao(e) and Q(A,e) C T 3 such that meas[T 3 \ fi(A, e)] < e and for all X > Ao(e) and (w,x,y) € Q(\,e), {HUtXty%j))n := —tf)n+i ~ ipn-i + Xv(T™(x,y))ipn displays Anderson localization. This theorem is related to the following "quantum kicked rotor" model:

idtV(t, x) = ad^(t, x) + ibdx^{t, x) + V(t, x)V(t, x) where V(t,x) = KCOS(27TX) ^ n e Z <5(£ — n). Let its unitary evolution operator be denoted by S(t). Then the monodromy operator is defined as W — 5(1) = Uam-n{TTnx) + 4>n-m(Tnx) for m 7^ n, with c/> exponentially decaying and small (for K small). Therefore, H is a long-range version of the skew-shift Hamiltonian from Theorem 1.3. In analogy with Theorem 1.3, one can show that Anderson localization for H holds for small K and most a, b. Hence, W also possesses an orthonormal basis of exponentially decaying eigenfunctions (where the exponential decay refers to the Fourier coefficients). More precisely, Bourgain proved the following result in [2]. Theorem 1.4 (B). For small K and (a,b) outside a set of small measure, one has: Let W(t,x) = (S(t)ty(Q,-))(x) be the solution of the kicked rotor equation. 7/*(0, •) is a smooth function on T, then \&(i) is an almost-periodic H 1 (T)-valued function and sup t ||^(t)||jyi(T) < oo. In recent work, Bourgain and Jitomirskaya have obtained a version of Theorem 1.4 that applies to a.e. choice of the parameters (a, b). Theorem 1.3 is proved by means of the transfer matrix formalism, whereas Theorem 1.4 cannot be obtained this way because of long-range interactions. One uses instead an approach that originated in the proof of the following theorem from [6]. Theorem 1.5 (BGS). Let v : T 2 —> K be analytic, nonconstant on any vertical and horizontal line segment. Let (Hxyip)nm — —AZ2tpnm+Xv(x+nu)i,y+mui2)ipnm for(n,m) £ Z 2 . Then for all e > 0, A > Ao(e) the operator Hxy displays Anderson localization for all (x,y,u>i,uj2) 6 T 4 up to a set of measure at most e.

On discrete Schrodinger operators with stochastic potentials

209

2. Transfer matrix formalism As in the case of truly random potentials, it is of basic importance to control the probability that a given energy E is close to an eigenvalue of the Hamiltonian H restricted to a box. In other words, one needs to estimate meas [x € Td : dist(H^,E)

< e~p]

where A = [-N, N] and p is relatively large (say, » (\ogN)A). By self-adjointness, this is the same as bounding the norm of the Green function \\(H£ — (i? + i 0 ) ) - 1 | | = | | G A ( I , i£)||. By Cramer's rule, one has f[-N,n-l] (Z) E) f[m+l,N] (x, E) f[-NtN](x,E)

GA(x,E)(n,m) where /[p>Q](a;, E) = det{Hf

— E). Moreover, there is the well-known relation

Mn(x,E)

=

f[i,n](x,E) .f[i,n-i](x,E)

-f[2,n]{x,E) -/[3in](x,£)_ '

Recall that Kingman's theorem implies that jf log ||Mjv(a;, E)\\ —> L^(E) for a.e. x e T. A quantitative version of this statement are the following estimates on the deviations: For large N meas[x : | log \\MN(x,E)\\ - N LN{E)\ > Na] < e~NT (3) for some fixed 0
u{x) = 2_J log \e(x + nu) — 1| = / log \z — C,\ ^(d() n=\

^N

2 r

where z = e(x) = e ' « ) ^ = 5Z n=1 <$ e (_ nw ). The left-hand side can be thought of as a Riemann sum. It is standard to estimate the error between such sums and their mean via Koksma's inequality: Let S := {xn}„=1 C T; N

y}/(*n)- / f{x)dx


J

n=l

°

where DN(S) = s u p J C T | # { n : xn e J } — N\ J||. The problem here is, of course, that log \e(x) — 1| £ BV. A more direct approach is to observe that u(x) is the Hilbert transform of a sum of saw-tooth functions. More precisely, let fo(x) = — \ — x for — ^ < x < 0 and fo(x) = \ — x for 0 < x < \. Then, with Ji being the Hilbert transform, (x)=n(J2fo(-

+ nu>))(x)

N

\HBMO

DN({ncj}%=1).

< C n=l

210

W . SCHLAG

Since for a.e. u> and N large Djv({nw}^ =1 ) < N£, the John-Nirenberg inequality implies meas [x : \u(x) - (u)\ > Na] < exp [ -

Na~e].

Observe the analogy of this estimate with the large deviation theorem (3). The transition to the noncommutative case is typically accomplished by means of two devices: — subharmonic extensions of log ||Mjv(£, E)\; — the so-called "avalanche principle" (this requires positive exponents L(E) > 0). The former is well-known and straightforward: Since v is analytic, u(x) = log ||MJV(x, E)\\ extends to a neighborhood of the unit circle as a subharmonic function. Such functions have a Riesz representation: u{z) = J\og\z-C\[x{dQ

+ h{z)

with some measure /x > 0, and a harmonic function h. Here \\fi\\ ~ TV, but there is a lot of cancellation in the integral to ensure, for example, that ||u||BMO < (logN)A. This latter property can only be captured by means of invoking the underlying dynamics. More precisely, one obtains the structure of u as a sum of shifts of another function by means of the following Avalanche Principle (Goldstein-S.): n.

Let {An}»=1 Then

e S*(2,R). Suppose \\An\\ >fi>N, N-l log ||Mjv|| + J2 n=2

\\An+1An\\

N-l l0

S WAn\\ ~ E l 0 § l l ^ n + 1 ^ n=l

< y/JI\\An+1\\

||A„|| for all


where MN = Ajq • • • A\. In the work by Bourgain, Goldstein, and the author, this device has been used to establish the following properties: — positivity of the Lyapunov exponent for large disorder; — inductive proof of large deviation theorems; — Holder continuity of the integrated density of states. We now give a typical application of the avalanche principle. Write o MnN(x,E)= Y[ Mn(Tjnx,E), where n ~ (logN) A . j=N-l

Suppose, for 0 < a, r < 1 fixed, > na] < e""'

meas [x : \\og\\Mn(x,E)\\-nLn(E)\

(4)

and the same for 2n, where Aa > 100. Furthermore, suppose Ln(E) > L,2n{E) > 1 for all E, and Ln(E) - L2n{E) < 1/100. Then up to a set of measure < CNe~n" ~ N~", and for n large, one has for all j \\Mn(Tjnx)\\

> en-n°

> e n / 2 =: /x,

||M n (r°' + 1)n a;)M n (r^x)|| > e^L^{E)-n"

>

> ^\\Mn{T^+^nx)\\

e2nLn(E)-2n°

\\Mn{T'nx)\\.

On discrete Schrodinger operators with stochastic potentials

211

Prom the avalanche principle, one concludes that up to a set of measure < N " , N-2

\og\\MnN(x,E)\\

+ £

N-2

- Y, logHManCr^z.JSOII
\og\\Mn(T^x,E)\\

(5)

Typically, the sums are uniformly in x close to their means (since they represent averages over very long orbits). The conclusion then is that u(x) •= — log||MTlW(a;,JE;)|| =u0(x)+

ui(x),

where \\u0 - (u0)\\oo < CN~1+e =: e0, and ||ui||i < CN~90 =: ei. Provided the Riesz mass of the subharmonic extension of u(x) is < N20, the splitting lemma (see below) yields 20 \\U\\BMO < C(e0 + \/N Ei) < CN~1+e. By means of the John-Nirenberg inequality, this implies that meas[x

: | log ||M n i V (a;,£)|| -nNLnN{E)\

> (nN)a] < Ce" •cN"

which is (4) for N instead of n (provided T < a — s). The same argument applies to 2nN. Moreover, averaging (5) over x yields that \LnN(E)

+ Ln(E) - 2L2n(E)\

<

CN~\

which implies LnN(E) > 1 — 2{Ln(E) — L2n(E)) — CN~l. Continuing inductively leads to positivity of L(E) as well (4) for all n. The following result is the aforementioned splitting lemma from [5]: Lemma 2.1. Let u be subharmonic on a neighborhood of T with Riesz mass N. Suppose u = u0+ui onT with ||UO||L°°(T) = £o and ||MI||LI(T) = £i- Then ||U||BMO < C(e0 + -JWe[). To motivate this statement, consider N points Zj — e(xj) in T. Suppose that P(z) = Ylj-iiz — Zj) satisfies sup| z | = 1 \P(z)\ < eT. We claim that DN{{XJ}^=1) < C^/NT where -DJV is the usual discrepancy (see above). Proof. u(x) = log |P(e(x))| = W ( £ ? = i /o(- - a;,-))- Set F := £ f = 1 /o(- - Xj). Let KN be a smooth bump function KN{8) = K(N6), K>0,fK = l,supp/f c [-.01, .01]. Then

(KN*fo)(--jj)--<M-)<(KN*fo)(;

+ 1j) +

Using JTu = 0, ||u||i < CT, F = Ti~1u = —Hu, one now has \\F\\00<2\\H-1(u*KM)\\00

+

CN —

CN + ^

Setting M = \/N/T gives H-FHoo < CVNT easy to check that H-^Hoo ~ DN({XJ}JLI)>

(and thus also desired.

as

||U||SMO

+

<

CVNT).

CN —

Finally, it is •

212

W . SCHLAG

Note that this argument shows that if u(z) = flog\z - Qdfj,(C) with supp(/i) C T, sup T w < (u) + T (here (u) = 0 and thus ||u||i < r ) , then ||U||BMO < C\/IHI r - A small variation of this argument proves Lemma 2.1 above. Final remarks on the transfer matrix formalism: — For the shift and skew-shift, large deviation theorems for u(x) = log ||Mjv(a;,£)|| are obtained via (a) subharmonicity (b) almost invariance u(x) ~ u(Tx). The latter can be either be the simple invariance property sup x |u(a;) — u{Tx)\ < C (suffices for the shift) or the avalanche principle (needed for the skew-shift). The main difference between these cases is that the Riesz mass of u in the former is TV, in the latter iV2 (or Nc for higher-dimensional versions of the skew-shift). The avalanche principle requires positive exponents, and the first step (but only that one) is perturbative. This means that both the initial positivity of the Lyapunov exponent, as well as the large deviation theorem at the initial scale are obtained by making the disorder very large. It is conceivable that the required information at the first step could also be provided by a numerical calculation. We would like to emphasize that for the plain shift, large deviation theorems do not require positive exponents and are non-perturbative. This allows Bourgain and Goldstein [4] to prove localization on the basis of L(E) > 0 alone. — The avalanche principle gives positive Lyapunov exponents for large disorders under very general circumstances. But this either requires the Riesz mass of log ||Mjv(a:, E)\\ to grow at most like Nc (so that the necessary large deviation theorem can be obtained simultaneously), or one needs to prove the LDT by other means at all scales. One case where the Riesz mass grows exponentially rather than polynomially is the doubling map Tx = 2x (mod 1). For this model, one proves the LDT by means of well-known subgaussian bounds for sums of martingale differences, see [8]. — In recent work, Goldstein and the author have obtained large deviation theorems for the entries of the monodromy matrix rather than the entire norm. Recall that the entries are the determinants f^(x,E).

3. Localization The large deviation theorems from the previous section only control the probability of a single resonance at a fixed energy E, i.e., meas [xGT

: dist{E,H^'N])

< e~N"].

It is well-known that this does not suffice in order to prove Anderson localization. Rather, one needs to control the probability of double resonances. More precisely, let DCA,C C T denote the set of u> with \\ntj\\ > r4^ for all n ^ 0 and define 5N := meas[w € DCA,C : d i s t ^ s p e c ^ j 1 ' " 1 ) ) <

e~n\

G[k,k+N}{®,w,E) is bad for some E and \k\ ~

Nc].

Here n ~ (logiV)' 4 , and G[kik+Nj(x,oj,E) is good if both \\G[k:k+N](x,uj,E)\\ < eN" (a < 1 fixed) and \G[ktk+N](x,u,E)\(jJ) < e~~ 0 a

On discrete Schrodinger operators with stochastic potentials

213

lower bound on the Lyapunov exponents. One then needs to show that V . 82i < oo. By means of standard methods (resolvent identity plus the polynomial Shnol-Simon bound on generalized eigenfunctions) this ensures that for a.e. LO £ DCA,C the operators Ho,w display Anderson localization. Finally, letting c —> 0 in the Diophantine condition allows one to extend this to a.e. w £ T, thus proving Theorem 1.1. To estimate 5N, note that SN <

Yl

meas

[ w G DCA,c • log\\MN{k(j,Lj,E)\\

<

NLN(w,E)-Na, for some |fc|~JV c ].

Bespec(<-')

The set on the right-hand side is the projection onto the w-axis of the intersections fijv := U|fc|~Arc SN n Ik of the lines £k := {(w, ku) : UJ £ T} with SN:={(u,x)£DCA,cXT

: log \\MN(x,u>,E)\\ < NLN{u,E)

-

Na).

The measure of SN is very small by the LDT, but this by itself does not preclude QJV = T (consider the case where S^ is one of the lines Ik)- We need to know that the intersections of SN with any horizontal line consist of a small number (say Nc many) connected components. This property is given by the Milnor-Thom bound of dc on the number of connected components of semi-algebraic sets of degree d, at least if the potential v is a trigonometric polynomial (the case of general analytic v then follows by approximation). Indeed, if v is a trigonometric polynomial, then the horizontal sections of SN are contained in semialgebraic sets of very small measure and degree Nc, as can be seen from the fact that the entries of MN are polynomials in w of degree < Nc'. From this complexity bound and the LDT bound get Spj < Cn meas[fi/v] < N~T for some r > 0 by means of the following lemma from [4]. L e m m a 3.1. Let 5 c T for every x £ T. Then

2

be such that {u> £ T : (u>, x) £ <S} consists of at most M intervals

meas[w £ T : (u,ku) £ S for some \k\ ~ N] < CJVc(meas[«S])*

+CMN - l

Concluding remarks — Localization can be obtained by this method for shifts of any dimension, as well as the skew-shift. In the latter case the main difficulty is the LDT (only known for large disorders). — The long-range case as well as the Laplacian on Z 2 cannot be treated by the transfer matrix formalism, so no LDT or avalanche principle are available. In those cases one needs to develop estimates on the probability that a given Green's function is bad (in the spirit of Frohlich-Spencer's multiscale method), which again relies on subharmonic function arguments and reductions to smaller scales. — Establishing Holder regularity of the IDS requires a sharp LDT meas [x £ T : | log \\MN(x, E)\\ - NLN(E)\

> 5N] < e'^

N

.

214

W.

SCHLAG

This L D T is known, but only for the shift on T. Averaging the avalanche principle over x by means of this L D T yields \LN{E)

+ Ln[E) < CN~1+e

T h u s \L(E) - LN(E)\

\LN(E) Hence \L(E) - L(E')\

- 2L2n{E)\

< N~1+e

where n ~ log TV.

and

- LN(E')\

< C\E - E'\a,

< CN~1+£

+ ec*n\E

-

E'\.

for some a > 0, as desired.

Open problems — Holder regularity of IDS for shifts on Td with d > 2. Does t h e IDS become more regular as the number of frequencies increases? Currently, t h e results deteriorate with the number of frequencies. Related question: Is there a L D T of t h e form meas [x G T : | log \\MN(x,

—

— — —

E)\\ - NLN(E)\

> 6N] < e~Cs

N

for shifts in higher dimensions? Determine the Holder exponent for t h e IDS. More precisely, can one get Holder | — provided the potential has only non-degenerate critical points? This is known (Bourgain) for the almost Mathieu model and large disorders. Prove a version of Theorem 1.5 on Zd, d > 3. Positivity of the Lyapunov exponent for small disorders for the skew-shift. Positivity of the Lyapunov exponent and Anderson localization for all positive disorders with Tx = 2x (mod 1), see [8].

References 1. M. Aizenman, S. Molchanov, "Localization at large disorder and at extreme energies: an elementary derivation", Coram. Math. Phys. 157, 245-278 (1993). 2. J. Bourgain, "Estimates on Green's functions, localization and the quantum kicked rotor model", Ann. of Math. (2) 156, 249-294 (2002). 3. J. Bourgain, "Green's function estimates for lattice Schrodinger operators and applications", to appear in Annals of Mathematical Studies, Princeton. 4. J. Bourgain, M. Goldstein, "On nonperturbative localization with quasiperiodic potential", Annals of Math. 152, 835-879 (2000). 5. J. Bourgain, M. Goldstein, W. Schlag, "Anderson localization for Schrodinger operators on Z with potentials given by the skew-shift", Coram. Math. Phys. 220, 583-621 (2001). 6. J. Bourgain, M. Goldstein, W. Schlag, "Anderson localization for Schrodinger operators on Z 2 with quasi-periodic potential", Acta Math. 188, 41-86 (2002). 7. J. Bourgain, S. Jitomirskaya, "Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential", dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays, J. Statist. Phys. 108, 1203-1218 (2002). 8. J. Bourgain, W. Schlag, "Anderson localization for Schrodinger operators on Z with strongly mixing potentials.", Comm. Math. Phys. 215, 143-175 (2000). 9. R. Carmona, J. Lacroix, Spectral theory of random Schrodinger operators., Birkhauser, Boston 1990.

On discrete Schrodinger operators with stochastic potentials

215

10. E. I. Dinaburg, Y. G. Sinai, "The one-dimensional Schrodinger equation with quasiperiodic potential" M i . Anal. i. Priloz. 9, 8-21 (1975). 11. A. Figotin, L. Pastur, Spectra of random and almost-periodic operators, Grundlehren der mathematischen Wissenschaften 297, Springer, 1992. 12. J. Prohlich, T. Spencer, "Absence of diffusion in the Anderson tight binding model for large disorder or low energy", Comm. Math. Phys. 88, 151-189 (1983). 13. J. Frohlich, T. Spencer, P. Wittwer, "Localization for a class of one-dimensional quasi-periodic Schrodinger operators", Comm. Math. Phys. 132, 5-25 (1990). 14. H. Fiirstenberg, "Noncommuting random products", Trans. AMS 108, 377-428 (1963). 15. I. Ya. Goldsheid, S. A. Molchanov, L. A. Pastur, "A pure point spectrum of the stochastic one-dimensional Schrodinger equation", Funkt. Anal. Appl. 11, 1-10 (1977). 16. S. Jitomirskaya, "Metal-insulator transition for the almost Mathieu operator", Annals of Math. 150 (1999). 17. M. Goldstein, W. Schlag, "Holder continuity of the integrated density of states for quasi-periodic Schroinger equations and averages of shifts of subharmonic functions", Ann. of Math. (2) 154, 155-203 (2001). 18. Y. G. Sinai, "Anderson localization for one-dimensional difference Schrodinger operator with quasi-periodic potential", J. Stat. Phys. 46, 861-909 (1987).

Non-zero random Lyapunov exponents versus mean deterministic exponents for a twist like family of diffeomorphisms of the two sphere MICHAEL SHUB* (U.

Toronto)

We consider a monotone twist map of the unit two sphere in three space and consider the family of dynamical systems given by composing the twist maps with the group of orientation preserving orthogonal transformations of three space. We compare the Lyapunov exponents of random products in this family with the mean of the Lyapunov exponents for random products restricted to a small neighborhood of the individual elements. We prove these two numbers are asymptotic as the twist tends to infinity. Moreover, we present numerical evidence that the same is true for the mean of the Lyapunov exponents for the deterministic systems. If the experimental evidence is borne out, these families would present examples of an analytic family of volume preserving maps of the two sphere containing twist maps and with positive entropy. This is joint work with Francois Ledrappier, Carles Simo and Amie Wilkinson. We compare the results with results on a similar problem concerning random products of linear maps obtained with Jean-Pierre Dedieu.

1. Introduction What I am reporting on is joint work with Keith Burns, Charles Pugh, Amie Wilkinson [2], Jean-Pierre Dedieu [4] and Francois Ledrappier, Carles Simo and Amie Wilkinson [6]. Almost all of the contents here are taken from these three papers without specific attribution. The material may be found in much expanded form in these references. Any errors which may have crept in are mine alone. One of the major achievements of the uniformly hyperbolic theory of dynamical systems is the work of Anosov, Sinai, Ruelle and Bowen on the ergodic theory of uniformly hyperbolic systems. Anosov proved that smooth volume preserving globally hyperbolic systems are ergodic. Sinai, Ruelle and Bowen extended this work to specifically constructed invariant measures for general uniformly hyperbolic systems, now called SRB measures. The ergodicity of these measures asserts that although particular histories are difficult to compute the statistics of these histories, the probability that a point is in a given region at a given time, is captured by the measure. Much of the work in dynamical systems in recent years has been an attempt to extend the results of the uniformly hyperbolic theory to more general systems. One theme is to relax the notion of uniform hyperbolicity to non-uniform or partial hyperbolicity, and then to conclude the existence of measures sharing ergodic properties of the SRB measures. The Proceedings of the Seattle AMS Summer Symposium on Smooth Ergodic Theory and Its Applications contains much along these lines. In particular, you can find the survey of recent progress on ergodicity of partially hyperbolic systems [2] included there. Also there is the more recent [8]. 'Partially supported by an NSF Grant.

216

Non-zero random Lyapunov exponents versus!mean deterministic exponents ...

217

The work on the ergodicity of partially hyperbolic systems may be considered as an elaboration of the Hopf argument for the ergodicity of the geodesic flow on surfaces of constant negative curvature. The crucial point being, of course, the existence of at least some hyperbolicity so that stable and unstable manifolds exist and ergodic averages are a.e. constant along them. In this paper we attempt to study some mechanisms which will ensure the existence of some hyperbolicity in terms of the existence of some non-zero Lyapunov exponents.

2. Unitarily invariant measures on GL n (C) Let Li be a sequence of linear maps mapping finite dimensional normed vector spaces Vi to Vi+i for i £ N. Let v £ Vr0\{0}. If the limit lim £ log ||Lfc-i... £o(w)|| exists it is called a Lyapunov exponent of the sequence. It is easy to see that if two vectors have the same exponent then so does every vector in the space spanned by them. It follows that there are at most dim(Vo) exponents. We denote them Aj where j < k < dim(Vo). We order the Aj so that Aj > Aj+i Thus it makes sense to talk about the Lyapunov exponents of a diffeomorphism / of a compact manifold M a t a point m £ M, \j(f,m) by choosing the l sequence L; equal to T / ( / ( m ) ) . Given a probability measure fi on GL n (C) the space of invertible n x n complex matrices we may form infinite sequences of elements chosen at random from \x by taking the product measure on GL„(C) N . Thus we may also talk about the Lyapunov exponents of sequences or almost all sequences in GL„(C) N . Oseledec's Theorem applies in our two contexts. For diffeomorphisms / Oseledec's theorem says that for any / invariant Borel measure v, for v almost all m £ M, / has dim(M) Lyapunov exponents at m, Xj(f,m) for 1 < j < dim(M). For measures fi on GL„(C) satisfying a mild integrability condition, we have n Lyapunov exponents r\ > r • • • > rn > - c o such that for almost every sequence . . . gu • • • gi £ |A2|>--->|An|. The integrability condition for Oseledec's Theorem is g £
(*)

218

MICHAEL SHUB

Theorem 2.1. If /J, is a unitarily invariant measure on GL„(C) satisfying (*) then, for fc = 1 , . . . ,n, k

k

£ log ^(,4)1 d/i(^) >£*-;.

/

MeGL„(C)

i=1

i=1

By unitary invariance we mean fx(U(X)) = fJ.(X) for all unitary transformations U G U„(C) and all ^-measurable X G GL„(C). Thus non-zero Lyapunov exponents for the family, i.e. non-zero random exponents, implies that at least some of the individual linear maps have non-zero exponents i.e eigenvalues. For complex matrices we have achieved part of our goal. Later we will suggest a way in which these results may be extended to dynamical systems. Corollary 2.1.

f>g+1^(^)1 d»(A)>J2ri+•

f JAeGhn(C)

i=1

i=1

Theorem 2.1 is not true for general measures on GL„(C) or GL„(R) even for n = 1. Consider

and give probability 1/2 to each. The left hand integral is zero but as is easily seen the right hand sum is positive. So, in this case the inequality goes the other way. We do not know a characterization of measures which make Theorem 2.1 valid. In order to prove 2.1 we first identify the right hand summation in terms of an integral. Let Gn>fe(C) denote the Grassmannian manifold of k dimensional vector subspaces in C™. If A £ GL„(C) and Gn^ G G n| k(C), A\Gntk the restriction of A to the subspace G„,fc. Let v be the natural unitarily invariant probability measure on Gnife(C). The next theorem is a fairly standard fact. Theorem 2.2. If /j, is a unitarily invariant probability measure on GL n (C) satisfying (*) then k

f^n=

[

i=1

•M€GL n (C)./G„, fc eG„, fc (C)

[

\og\det(A\Gn,k)\dv(Gn,k)dn(A).

We may then restate Theorem 2.1. Theorem 2.3. If /i is a unitarily invariant probability measure on GL n (C) satisfying (*) then, for k = 1 , . . . ,n, k

f JA€GLn(C)

V l o g 1^(^)1 rf/x(A)> / ,_,

JAeGhn(C)

/

]og\deb(A\Gn,k)\MGn,k)dn(A). JGn,k£Gn,k(C)

Theorem 2.3 reduces to a special case. Let A G GL„(C) and // be the Haar measure on U n (C) (the unitary subgroup of GL n (C)) normalized to be a probability measure. In this case Theorem 2.3 becomes:

Non-zero random Lyapunov exponents versus mean deterministic exponents ... Theorem 2.4. Let A € GL n (C). Then, [

J2log\\i(UA)\dLi(U)>

JUGVn(C)

219

forl
log|det(>l|Gn,fe)|di>(Gn,fc)-

JG„,*eG„, fc (C)

i=1

We expect similar results for orthogonally invariant probability measures on GL„(R) but we have not proven it except in dimension 2. Theorem 2.5. Let fi be a probability measure on GL,2(R) satisfying g G GL 2 (R) ->log + (||5||) and log + (||5 _ 1 ||) are fi-integrable . a. If n is a §0>2(R) invariant measure on G L ^ R ) then, f ]oS\\1(A)\dn(A)= JAeGh+(R)

[ f logWAxWdS^x) JAEGL+(R) Jxes1

dfi(A).

b. If n is a S02(M) invariant measure on GLJ(M), whose support is not contained in R0>2(R), i.e. in the set of scalar multiples of orthogonal matrices, then f JAeGh^

\og\X1(A)\dn(A)> (R)

f

\og\\Ax\\dS1(x)dfi(A).

[ 1

JAeGl.2 (R) JxeS

Here G L ^ R ) (resp. GL^"(R)) is the set of invertible matrices with positive (resp. negative) determinant.

3. M e t h o d of proof A flag F in C" is a sequence of vector subspaces of C": F = (Fi, F 2 , . . . , Fn), with Fj C F J + I and dimF, = i. The space of flags is called the flag manifold and we denote it by F„(C). An invertible linear map A : C™ —> C" naturally induces a map A$ on flags by At(FuF2,...

,F n ) = (AFUAF2,...,

AF n ).

The flag manifold and the action of a linear map A on F n (C) is closely related to the QR algorithm, see [10] for a discussion of this. In particular if F is a fixed flag for A i.e. A$F = F, then A is upper triangular in a basis corresponding to the flag F, with the eigenvalues of A appearing on the diagonal in some order: X\(A, F), ..., Xn(A, F). Let V^ = {(U,F) G U n (C) x F„(C) : (UA)tF = F). We denote by IIi and n 2 the restrictions to VA of the projections U„(C) x F„(C) —> U n (C) and U n (C) x F n (C) —* F„(C). VA is a manifold of fixed points. We use the diagram

U„(C)

F n (C)

in order to transfer the right hand integral in 2.4 over F n (C) to an integral over U„(C).

220

MICHAEL SHUB

4. A n a l t e r n a t e m e t h o d of proof We give an alternate proof of Theorem 2.5 a in [6]. Let A € GL„(C) or GL„(R) and fi be the Haar measure on U„(C) or SO(n) respectively normalized to be a probability measure. Let G denote GL n (C) or GL n (R). Now we interpolate between random products and deterministic powers by changing /i. Let 5 > 0 and Gs be the delta neighborhood of the identity in G. For g £ G, GggA is a neighborhood of gA in GA. We normalize Haar measure restricted to Gs and push it forward to G$gA. Let us call this measure ns,g- Let ri (S, g) be the largest random exponent for this measure. Now it is not too difficult to see the following propositions. Proposition 4.1. lim^oT\{6,g) =

X\(gA).

Now let vs,g be the stationary measure on G n ,i induced by ns,gProposition 4.2. n(5,g) = / P e G ( n ) 1 ) log \\(A\P)\\ dv5,g. Let i/g = J„pG vs,g dfi. It follows that: Proposition 4.3. JgeG Xi(gA) d/j, = l i m ^ 0 / P e G („,i) l o S \\(A\P)\\

d

"s-

Now n = / p G G / n !) log ||(A|P)|| dv(G(n, 1)). So a comparison of / G G X\(gA) d/j and r\ can be achieved via an understanding of the relationship between i/(G(n, 1)) and i/g. These and the next proposition are proven in [6]. Proposition 4.4. For G = 50(2), i/(G(ra, 1)) = vs for all S > 0. Thus we have another proof of Theorem 2.5 a. The general situation is different. The inequality in Theorem 2.4 for k = 1 is strict when n > 2 unless A is an isometry. So for G = SU{n) the measures v& favor the expanding directions of A as S —* 0. Experimentally the same seems to hold for SO(n) when n > 2. We pursue this line of argument for a dynamical system analogue.

5. A dynamical systems analogue We now use notation in agreement with that in [6]. The content of this section is mostly drawn from [6]. The families we consider are not obtained from a specific set of equations, but from the following construction. Let / : S2 —> S2 be an area-preserving diffeomorphism of the round sphere, and let 50(3) be the isometry group of 5 2 . Let F={9°f

I 2 €50(3)}

be the left 50(3)-coset of / in Diff (5 2 ), and let v be the push-forward of Haar measure on 50(3) to T. Provided that / is not itself an isometry, the family T has nonzero random Lyapunov exponents with respect to v. The question [6] addresses is whether these random exponents can somehow be connected to the Lyapunov exponents of individual members of J7, at least on u-average.

Non-zero random Lyapunov exponents versus mean deterministic exponents ...

221

To test whether there might be such a connection, we chose / to be a twist map, all of whose Lyapunov exponents are zero. The resulting family T has similarities to the standard family on the 2-torus. The key difference is that the maps in T are "integrable", in the sense that the phase space is foliated by invariant curves, while the standard family is not integrable. The dynamics of the individual elements of J- and how they depend on parameters is an interesting topic, but in [6] we only study some key properties in the case of small twists. We mainly focus on two quantities, the random exponent R{v) and the average exponent k{v), which we now define. Let /z be Lebesgue measure on S2 normalized to be a probability measure. Suppose for now that v is an arbitrary Borel probability measure supported on a subset T of DiffM(S'2), the space of //-preserving diffeomorphisms of 5 2 . For / G T, the largest Lyapunov exponent of / at x G S2 is also found by computing the limit: hm-log\\Txr\\

= >
(1)

n-»oo n

which exists for //-almost every x by the subadditive ergodic theorem. We define the average exponent of f to be A ( / ) = / \1(f,x)d^(x) Js2 and the average exponent of v to be

(2)

A(v) = J\(f)Mf)-

(3)

Rather than iterate a single diffeomorphism / G J-, we might choose instead a sequence of diffeomorphisms {/i, / 2 l . , . } c f and form their composition: /

( n )

:=/n°/n-l"-°/l.

If the sequence is chosen independent and identically distributed with respect to v, then almost surely the limit lim - log \\Txf™ || =: R(x, (/Of, v)

(4)

n—»oo n

will exist, for /i-almost every x. (This too follows from the subadditive ergodic theorem, applied in the appropriate context). Further, the integral of R(x, (/i)i°,^) with respect to fi is almost surely independent of the sequence (fi)f- We define the random exponent of v to be this integral: R{y)=

f R(x,(fi)?>,v)dn{x) (5) Js2 (see Kifer for an introduction to the subject of random diffeomorphisms and their exponents) . The quantity A.(v) is mysterious from a computational perspective, but useful from a dynamical one. The quantity R(v) is relatively easy to estimate and is usually positive, unless v is fairly degenerate [3]. In [2] we asked: Question 5.1. Is there a positive constant C — perhaps 1 — such that A(v) > CR(v)l

222

MICHAEL SHUB

Some motivation for Question 5.1 can be found in the linear algebra results above. This is the question we now consider here. We return to the specific family of diffeomorphisms mentioned earlier. For e > 0, we define a one-parameter family of twist maps fe as follows. Express S2 as the sphere of radius 1/2 centered at (0,0) in R x C, so that the coordinates (r, z) G S2 satisfy the equation |r| 2 + |z| 2 = 1/4. In these coordinates define a twist map fe : S2 —> S2, for e > 0, by fe(r,z)

= (r,exp(27ri(r +

l/2)e)z).

Let F n (C) e be the orbit SO(3)fs. Let v be the push-forward of Haar measure on 50(3). We denote the resulting random and average Lyapunov exponents by R(e) and A(e), respectively. To summarize the numerical results obtained in [6], it seems that the inequality A(e) > R(e) is satisfied for large e and even that A(e)/R(e) —> 1 when e —• oo. But the inequality goes the other way for small e. We give heuristics to explain why the inequality is satisfied when "e = oo". We also present an outline of why the same inequality is not satisfied for small e. A partial analysis of the maps of the form g o fe is carried out in [6]. Our analysis of the case "e = oo" is carried out in analogy with the alternate proof of section 4. We interpolate the quantities R(e) and A(e) by a third quantity R(e, S), which measures the exponents of the "in-between" process in which each element of F n (C) e had added noise in a 5-ball. For each g e 50(3) we consider a 5 diffused process around gf. The induced process on T\S2 has a unique stationary measure mffi£i5,which is the unique fixed point of a "simple" linear operator. The measures m 9i£i 5 have the following properties: — ms,e,i5 is stationary for the 5-diffused process about gfe; — mg,e,i5 i s absolutely continuous with respect to m, with smooth density; — m s,e,s projects to Lebesgue measure /x on S2. The measure mE^ is defined by me,s = /

rng:e,5dg,

JSO{3)

where, for each g e 50(3), Jn S)£i j is the measure above. In analogy with the propositions of section 4 we prove that the random exponent R(e, S) satisfies limsupi?(e, 8) < A(e), R(e,6)= R(e)=

f

logHT/vlldm^t;),

[

log||r/i/||(fm(«).

JTiS2

Non-zero random Lyapunov exponents versus mean deterministic exponents . . .

223

If it is the case t h a t m€is —> m as 5 —> 0, then it would follow t h a t : A(e) > lim sup R(e, 5) «-»o = limsup/ 5—*0

= f =

JTxS2

log

\\Tfv\\dmeiS(v)

log\\Tfv\\dm(v)

R(e).

Hence the properties of this measure m£ts are potentially quite interesting with regard t o Question 5.1. For fixed 5 we show t h a t the 'simple" linear operator defining the measures mgtet$ have a limit as e —> oo and t h a t t h e fixed points at infinity have t h e required property. This is suggestive, b u t does not prove the theorem we would like.

References 1. D. V. Anosov, "Geodesic flows on closed Riemannian manifolds of negative curvature", Proc. Steklov. Inst. Math. 90 (1967). 2. K. Burns, C. Pugh, M. Shub, A. Wilkinson, "Recent Results about Stable Ergodicity", in Proceedings of Symposia in Pure Mathematics vol. 69, Smooth Ergodic Theory and Its Applications (A. Katok, R de la Llave, Y. Pesin, H. Weiss, eds.), AMS, Providence, R.I., 2001, pp. 327-366. 3. A. Carverhill, "Furstenberg's theorem for non-linear stochastic systems", Probability Theory and Related Fields 74, 529-534 (1987). 4. J. P. Dedieu, M. Shub, "On random and mean exponents for unitarily invariant probability measures on GL(n,C)", to appear in Asterisque. 5. I. Ya Gol'shied, G. A. Margulis, "Lyapunov indices of a product of random matrices", Russian Math. Surveys 44, 11-71 (1989). 6. F. Ledrappier, M. Shub, C. Simo, A. Wilkinson, "Random versus deterministic exponents in a rich family of diffeomorphisms", to appear in JSP. 7. V. I. Oseledec, "A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems", Trans. Moscow Math. Soc. 19, 197-231 (1968). 8. "Stable ergodicity", preprint. 9. D. Ruelle, Ergodic Theory of Differentiable Dynamical Systems, Publications Mathematiques de 1'IHES, volume 50, 1979, pp. 27-58. 10. M. Shub, A. Vasquez, "Some linearly induced Morse-Smale systems, the QR algorithm and the Toda lattice", in The Legacy of Sonia Kovalevskaya, Linda Keen ed., Contemporary Mathematics vol. 64, AMS, 1987, pp. 181-193.

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Equilibrium statistical mechanics Session organized by D.

BRYDGES

(Vancouver) and E.

OLIVIERI

(Rome)

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Dimensional reduction for isotropic and directed branched polymers JOHN Z. IMBRIE (U.

Virginia)

I will describe an exact relation between self-avoiding branched polymers in D + 2 continuum dimensions and the hard-core continuum gas at negative activity in D dimensions (joint work with David Brydges [1,2]). Our results explain why the critical behavior of branched polymers should be the same as that of the i
1. Introduction and main results In this article we describe some beautiful identities which connect branched polymer models with repulsive gases in lower dimensions. For ordinary (isotropic) branched polymers (BP) the reduction in dimension is 2, and our results [1,2] provide a rigorous version of [3] (in which the critical behavior of BP in D+2 dimensions is connected with that of the Yang-Lee edge in D dimensions). For directed branched polymers (DBP), the reduction in dimension is 1, and our results [4] parallel earlier work on directed lattice animal a (which have been related to dynamical models of hard-core lattice gases [5] and to the critical dynamics of the Yang-Lee edge [6,7]; see also exact results in [8-10] and the review [11]). Forest-root formulas provide a unified picture for dimensional reduction for both BP and DBP. They are used to interpolate in the extra dimensions. They have other applications as well: we show how the one-dimensional forest-root formula generalizes some formulas of [12], used for interpolating in cluster expansions. We now define generating functions for BP and DBP. Let T be a tree graph on { 1 , . . . , TV}. For BP, the kth monomer is at position yk S RD+2 and we write yk = (wk,xk) £ C x RD. For DBP, it is at yk = (tk,xk) e E + x S, where S is either RD or IP'. While we must use continuous coordinates for BP, we allow discrete spatial coordinates for DBP, and for some models the time coordinate is discrete as well. For a unified treatment with DBP we consider here rooted BP (in contrast to [1,2,13], where BP are defined mod translations. Fix the vertex 1 as the root, with y\ = 0. For each pair (i,j) = ij, we define = (*•• x - ) = f(Wi~wJ>Xi~xA> 1 (\ti — tj\,Xi — Xj),

forBP

> for DBP.

^x

Each link of T connects a vertex j to a vertex i where i is one step closer than j to the root along T. For DBP, we require tj > ti. a

Animals (for which loops are allowed) are generally believed to have the same critical behavior as BP (where loops are not permitted). Likewise for directed animals and DBP.

227

228

JOHN Z. IMBRIE

The weight associated with each configuration depends on a linking weight V(y) and a repulsive weight U(y). The generating functions are written as oo

N

»

v

N=l

'

oo

•MILXK

T

JV

^

)

ji£T

ji£T *< T

( 2 )

/•

(N ~ 1)! ^

V

+ xRD)s-i

^

.X^

Here each pair {i, j } appears exactly once, either in rij;gT o r m IljigT- We a s s u m e that U —* 1 as yji —> oo, so that the repulsion vanishes at infinity. For BP, {/ is evidently invariant under rotations in K-° +2 ; for DBP we require U(t,x) = U(t, —x). In order to get dimensional reduction, we require V(t)=2U'(t),

forBP,

V(t,x) = U'(t,x),

for DBP,

'

where prime denotes the i-derivative. Note, however, that in the BP case t denotes a squared-radius variable, whereas for DBP t is a "time" variable. For positive weights, we require that U and V are positive. Furthermore, we assume that V is an integrable function of y so that (2) is well-defined. We relate these generating functions to the density of a repulsive gas in D dimensions. Let A c S and define the grand canonical partition function ,N

r

/ N=0

'

JA

N

T\dxi »=1

Ki<j
where Uy = U(\xij\2|)2 \ for BP and Uij = U(0,Xij) for DBP. Then define the pressure and density: P

^

=

A

1

^ |A) 1 ° g Z H C ^ - ) ;

PKC

^

= Z

TzP^'

^

T h e o r e m 1.1. For all z such that the right-hand side converges absolutely,

PHc(z) = \~2*ZBP\to)' y—ZDBp{-z).

(6)

We can also prove a dimensional reduction formula for correlations. Let N

i

N

6

Xi

p( ) = Yl ^~ ^ t=l

where x,xt £ RD and y,y{ £ RD+2

p{y) = Yl5(y~Vi)>

(7)

i=l

for BP or R+ x S for DBP. Then the density-density

Dimensional reduction for isotropic and directed branched polymers

229

correlation functions of t h e three systems can b e written as

GBp(0,y;z) = £ ^

K * ~ '•)• Y A C X I " ) " -

N

°°

GDBP(0, jj; z)

P® II [<%^toil2)] II U{\Vji\2),

/

7^-TTTE 1

=£

1

jXi eX r

L L

• MT

r

y^^—^ £ /

G H c ( 0 , X; Z) = j h n (p(0) p{x))HCA

p(y) [ ] [<%n^)] II ^ ) .

(8)

.

Here (-)HC,A is the expectation in the measure for which Znc(z)

is the normalizing constant.

T h e o r e m 1.2.

GHc(0,

\-2Trfd2wGBp(o,y;-~), x\z) = < V X; Z) = / 0 °° dt GDBP{0, V, -z),

Here d2w is defined as dudv,

wl

where y = (w, x) e C x

RD,

where y = (t, i ) e R

S.

(9) +

x

where w = u + iv.

2. Examples A n a t u r a l B P example is obtained by letting U(t) = fl{t — 1), where $ is the usual step function. T h e n ^(|2/ij| 2 ) = 5(|j/y| — 1), and we have a model of hard spheres linked together with kissing conditions determined by t h e tree graph T. See figure 1. One can also consider soft repulsion of t h e form U(t) — e~v(-t\ This is particularly interesting when D — 0, because the sine-Gordon representation allows one to write oo

/

r

I

dip 2

(10)

J2TTV

exp -L (ze* + ±

Figure 1. A branched polymer in R2 (left) and a directed branched polymer in Z 2 (right).

230

JOHN Z. IMBRIE

(the t axis). Alternatively, if we let U(t, x) = ^{t + \x\ — 1), then we obtain models of hard diamonds (when |ar| = X^ a =i \xa) or hard double-cones (when \x\2 = Yla=i \x"a) distributed uniformly in contact with the positive surface of the monomer it is linked to (subject to the constraint of nonoverlap with other monomers). In D = 1, the pressure of the hard-sphere gas is computable, it is p{z) = LambertW(z) = -T(-z), where T(z) = that

Y1°N=I ZNNN~1/N\

(11)

is the tree generating function [2]. So Theorem 1.1 implies

ZBP(z) = - i - 0 H C ( - 2 7 r * ) = ] T

(2nz)NNN ^

N=l 00

ZDBP(Z)

= -PHC{-Z)

= Yl

(12)

zN]yN N\

•

N=l

Thus we have exact expressions for the volume available to BP and DBP of size N. One can also consider lattice DBP examples by taking U(t,x) = 1 — J(a;)'!?(l — t), where I(x) is the indicator function of a set of "neighbors" in the lattice, such as {x : \x\ < 1}. Since V(t,x) = I(x)8(t — 1), the set determines which sites a link can jump to, with t always increasing by 1. This model is closest to the standard examples of DBP. See figure 1 for a configuration in two dimensions. The factors rijieT U{yji) enforce nearest-neighbor exclusion for monomers on the same level (same value of t). But there is a subtlety when a monomer at level t has n neighbors at level t — 1 in the polymer. In this case, we can write the [/-factors as •# n - 1 (0), which should be interpreted as ~ = j •&n~1d,d (this can be seen by approximating t? with a smooth function and integrating over t). Monomers which are separated by more than one unit in the t direction do not interact, since U(t, x) = 1 for £>1. By Theorem 1.1, the generating functions of these models equate to the density of the nearest-neighbor exclusion models in D dimensions associated with the weights U(0,x) = 1 — I(x). In D = 1, the pressure can be calculated explicitly [16, equation 2.16] p(*)=lnQ + i v T + 4 ^ ,

(13)

so the generating function is d X •7 (\ < \ l ( ZnMz) = -zTzP(-z) = - [j—

^ V* [27V-1]!! 2 ^ - ^ - I) = E . Wl

(14)

which gives an explicit enumeration of the number of DBP with N monomers. 3. Critical e x p o n e n t s As these reduction formulas are valid out to the edge of convergence of the generating functions, one can deduce the values of the BP and DBP critical exponents by looking at solvable repulsive gas models in low dimension. Let us define an exponent OJHC from the

Dimensional reduction for isotropic and directed branched polymers singularity of the pressure of the hard-core gas, p(z) ~ (z-zc)2~aHC. exponents 7 B P and 7DBP can be denned from ZBP ( - ^ )

~ (z - z.)1-^,

231

Likewise, susceptibility

ZDBP(-z) ~ (z - zc)1-^.

(15)

Note that zc is negative. By Theorem 1.1, the singularities must be the same, so aHc(£>)=7Bp(-D + 2 ) = 7 D B p C D + l).

(16)

Closely related to the susceptibility exponents are the counting exponents. If one defines CJV, d/v from 7

ZBP(Z) =

Z

OO

OO

CNZN

T Y .

>

"•Z N=l

Z

°BP(2)

= T dNzN,

(17)

N=l

then #BP, #DBP a^e determined from the asymptotic behaviors CN

~(~S)~N

N 6BP

~

>

dN~(-zc)-NN-e™*.

(18)

The exponent #BP is usually denned from counting unrooted BP, the difference being a factor N which is produced by z-^. We see that the unrooted generating function Y^, CNZN is related to the pressure p(z) of the repulsive gas [1]. From (16) and the relations #BP = 3 — 7 B P ,

#DBP=2-7DBP,

(19)

one can determine #BP, #DBP from a n o In D = 0, we have «HC = 2 because Znc{z) = 1 + z, so the "pressure" has a logarithmic singularity. For D = 1, one can compute «HC = 3/2 from the solutions for the pressure (11), (13), which have a square root singularity. For D — 2 there is an exact solution for the hard-hexagon model [17], which has GJHC = 7/6 [5,18]. This model works as the starting point for one of our lattice DBP models (with D +1 = 3). Our construction for BP does not work as described above for the hard hexagon model. Nevertheless, one expects the same value of anc for hard spheres. Hence the exponents #BP {D + 2), #DBP {D +1) are determined for D = 0,1,2 (rigorously for D = 0,1). See the table below. Theorem 1.2 connects the Green's functions for the three models, so it is clear that the exponents for the divergence of the correlation length are all equal. For DBP this gives information only on the transverse correlation exponent ^fj B P , which describes the vanishing of the rate of decay in the x directions. The exponent vuc can be computed in D = 1 or more generally determined from a n c via hyperscaling (Dis^c = 2 — CCHC)- Theorem 1.2 also implies that rjnc = VBP [2], and one can compute ?7HC in D = 1, or more generally determine it from Fisher's relation (77 = 2 — 7/^). In D = 2 it can be obtained from the conformal field theory of the Yang-Lee edge [19]. The repulsive gas singularity at negative activity is believed to be in the same universality class as the Yang-Lee edge [16,20]. One way of seeing this is by writing down the sine-Gordon representation for the repulsive gas (see (10) for the D = 0 case). The interaction is elip, whose lowest order term at the critical point is iip3, the Yang-Lee edge interaction. The Yang-Lee edge exponent a can be equated with 1 — anc- This leads to the Parisi-Sourlas

232

JOHN Z. IMBRIE

relation 9BP(D + 2) = 2 + a(D) [3] and its analogue for DBP: 6DBp(D + 1) = 1 + a{D) [6,7,21]. The table below summarizes what we know about exponents for these models (the last line gives the presumed mean-field exponents and upper critical dimensions; see [22,23] for results on BP in high dimensions). OLUC

HC dim£>

DBP BP = 7BP dimD + 1 dim.D+2 = 7DBP

= #DBP

#BP

^BP

= ??BP

a =

— ^DBP

0

1

2

2

0

1

1

2

3

3 2

1 2

3 2

1 2

-1

l 2

2

3

4

7 6

5 6

11 6

5 12

4 5

1 6

MFTD>6

D>7

D>8

1 2

3 2

5 2

1 4

-1

0

1 2

4. Forest-root formulas The key ingredients for proving Theorems 1 and 2 are a pair oi forest-root formulas, which we use to interpolate in the extra dimensions. Let / ( t ) depend on a collection of non-negative real variables t = {£ij}i 0 when any tt —• oo. In the BP case, the t's are functions of another set of variables: i^ = \wi — Wj\2, tj, = |WJ| 2 , with each u>i G C. The forest-root formula is

The sum is over forests F and roots -R (i? is any subset of { 1 , . . . , N} and F is a loop-free graph on { 1 , . . . , N} such that each connected component or tree of F has exactly one root in R). See figure 2. The expression /( F ' f l ) denotes the iVth partial derivative of / with respect to *$,-, ij G F and U, i £ R. The simplest example is when N = 1, F = 0, i? = {1}, in which case (20) reduces to the fundamental theorem of calculus: /(0)=

f'(t)^

= -

f(t)dt.

(21)

In the DBP case the i's are the underlying variables, and t%j = |£« — t,-|. Our second forest-root formula is

/(°) = E / (F,fl)

,/M

n [-*»•] n [-^ -^)]/(F'fi)(t)+ r+fi

(22)

jiGF

As in (2), each link in a tree of F connects a vertex j to a vertex i which is one step closer to the root for that tree. The integration region is {tr > 0, r G R and £, > t,, ji G F } . For iV = 1, (22) is just /(0) = - /0°° /'(*) df.

Dimensional reduction for isotropic and directed branched polymers

233

tree tree

tree tree forest Figure 2. Example of a forest.

4.1.

P r o o f of t h e m a i n r e s u l t s

Let us see how these forest-root formulas lead to a proof of Theorem 1.1. Let N

f(t)=g(t1/e)l{g(eti)

J ] Utj,

(23)

l
where Utj is either f/(|yy| 2 ) = £/(*»j + \%ij\2) for B P or U(Uj,Xij) for D B P (we suppress the dependence on { x y } in / ( t ) ) . Here g is any smooth function which decreases to 0 and satisfies g(0) = 1. For each ji e F, Uji is differentiated and becomes t h e linking weight Vji (Vji/2 for B P ) . For each r € R a g is differentiated. One finds t h a t (up t o terms which vanish as e —> 0) all the trees of F decouple due to the large separation in the ^-direction (the distribution of tr for an n-vertex tree is essentially —(g(etr)n)'dtr or —(g(etr)n)'cPw/ir, which are very spread-out probability measures). One tree has its root fixed at 0 by a factor — (g(tr/e))'dtr or — (g(tr/e))'d2w/Tr, which converges to a <5-measure at 0. T h e others cancel with the normalization Znc(z), so t h a t

p z)

^ =}%»a

z c{z) l

oo

N

.

N

» ~ £ ML ndx^

-2-7rZBp 1 Similarly, one can show t h a t pnc(z) = ~^DBP(—Z). For each link there is a factor — ^ (BP) or —1 (DBP), and this leads to the scaling of Z D B P and Z B P and their arguments in Theorem 1.1. Further details of this argument are in [2]. Theorem 1.2 can be obtained by differentiating (6) with respect to a source at x (see [1] for details). 4.2.

Decoupling expansions

It is interesting to look at these forest-root formulas from the point of view of decoupling expansions. Complete decoupling occurs only at t = 00, so it is important to note t h a t the

234

JOHN Z. IMBRIE

trees of F still interact in (20),(22). However, if we start with a function h which depends only on {Uj}, and let / ( t ) = h(t) f l i = 1
Mo) = E / F

JK

II[-d(tj-*i)]>*(F)(t).

(25)

+ jieF

Note that if a tree has n vertices, the sum over its root gives a factor n. This appears in — (g(etr)n)' which, as noted above, is a spread out probability measure. Thus the root sums disappear (one still needs to select a root for each tree, however — take the one with the smallest label in { 1 , . . . , N}, for example). One is tempted to change variables here, to Sji = tj — ti for ji € F. Then h^ has to be evaluated at tki = \tk — U |, where tk is a height variable obtained as the sum of s parameters along the tree joining the root to k (tki = oo if k and I are not in the same tree). The result is a new decoupling formula which has some similarity with Theorem III.2 in [12]. In fact, the rooted Taylor forest formula of [12] can be obtained from (22) by a limiting procedure similar to what we did in making a lattice model of DBP. Let H(w) depend on {toy}, a set of decoupling parameters in [0,1] (no assumption that Wij = Wi—Wj). Make a change of variable Wij = $(1 — s^), where $ is a smooth, monotone approximation to the step function. Apply (22) with Sji =tj —t^ for ji £ F. The result of this is that for each kl £ F, s^i is determined as the height difference tki- As Sji € [1— £, 1+e] for ji € F, the height variables tj. become discretized and actually measure the number of steps from the root to k. Then if the height difference is 2 or more, Wki = 0, and if the difference is 1, all Wjk linking j to the next level down are equal to Wji, where ji G F. Thus we obtain

n dw ^ (F) ( w )>

tfw^z/

(26)

which is Theorem III.2 of [12]. We have switched from s-derivatives and integrals to wderivatives and integrals, noting that under a change of variables ®M- -ds is invariant (except for the loss of the minus sign because ^7 < 0). 4.3. The two-dimensional forest-root formula The two-dimensional forest-root formula (20) is proven using a supersymmetry argument in [1]. Replace each variable ti in / ( t ) with _ dwj A dvjj Ti = WiWi -\ — , 2m

(27)

and each t%j with y = WijWij +

dwi A diDi ——-. 2m

(28)

Then / ( r ) is defined by its Taylor series. A "localization" formula / / ( r ) = /(0) JcN

(29)

Dimensional reduction for isotropic and directed branched polymers

235

holds, which becomes the forest-root formula when expanded out. This can be proven by deforming the problem to the independent case (21), using ideas from [24]. See [14] for an alternative argument, which uses the linearity of (29) to reduce to the case where / is an exponential (a Gaussian calculation) [25]. See also [26]. 4.4. Proof of the one-dimensional forest-root formula As mentioned above, the TV = 1 case is just the fundamental theorem of calculus. For N = 2, consider / ( t i , /2, £12) and use subscripts 1, 2, 12 to denote partial derivatives. Then /»oo

/(0) = - / ds(f1(S,s,0) + f2(s,s,Q)). (30) Jo Apply the N = 1 formula to the fa term (integrating with respect to x2 — £1), and to the fa term (integrating with respect to x\ — x2). The result is /*oo

/(0) = /

/>oo

dtx\

Jo

/>oo

d(t2 - i 1 )(/ 1 , 2 + / M 2 ) + / Jo

Jo

/*oo

dt2

d(t2 - tiX/2,2 + J 2 , 12 ),

(31)

Jo

since ^- = 1 for t2 > t, and ^- = 1 for tx > t2. The two / i j 2 terms combine to form /E2 dtidt2fit2, which is the term R = {1,2} of (22). The other two integrals are the terms i?={l},{2}. We prove the general case by induction on N. Begin as above with f(0) = -

dflV/fe(s>...,s)0,...,0)I Jo

(32)

fc=i

where the integral is along the diagonal, t\ = t2 = • • • = t^. Consider one of these terms, say k = N, and apply (22) in the variables it = U — tjv, i = 1, • • •, N — 1, keeping £jv = s fixed:

fN(tN,...

,tN,0,... ,0) = 53 JNI (F,R)

R

+

l[[-dir} n [-d{ti - ii)\fg>k\t). r£R

(33)

jieF

Note that when computing the derivative of f^ with respect to ir, there will be a term /^)7and also a term JN,TN (coming from the dependence on trN = tr — t^ = ir). Thus each (F, R) on { 1 , . . . , N — 1} gives rise to 2'fll terms, each of which can be assigned a unique (F, R) on { 1 , . . . , TV}. R consists of N, together with each r £ R with an /jv,r term. F consists of F, together with rN, r € R when r gives rise to an /N,VN term. Observe that each root in R ceases to be a root in R if it is connected by a bond rN in F. We obtain in this way all (F, R) with N € R, and each satisfies the condition that each tree of F contains exactly one root. As a result, /(°)=E

E

k=l (F,R):keR

/„[-<***] n R

+

i€iJ\{fc}

H(
(34)

ji€F

It is evident that if we take the term (F, R) of (22), and consider the subset of the integration region for which tk = min r £ fli r , we obtain the term k, (F,R) of (34). This completes the proof.

236

JOHN Z. IMBRIE

Acknowledgments I t h a n k David Brydges and J o h n Cardy for discussions which improved my understanding of these problems. Research supported by N S F grant PHY-0244884.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

D. C. Brydges, J. Z. Imbrie, Ann. Math., to appear; arXiv:math-ph/0i07005. D. C. Brydges, J. Z. Imbrie, J. Statist. Phys. 110, 503 (2003); arXiv:math-ph/0203055. G. Parisi, N. Sourlas, Phys. Rev. Lett. 46, 871 (1981). J. Z. Imbrie, "Dimensional reduction for directed branched polymers", preprint. D. Dhar, Phys. Rev. Lett. 5 1 , 853 (1983). J. L. Cardy, J. Phys. A 15, L593 (1982). N. Breuer, H. K. Janssen, Z. Phys. B 48, 347 (1982). D. Dhar, Phys. Rev. Lett. 49, 959 (1982). P. Di Francesco, E. Guitter, J. Phys. A 35, 897 (2002); arXiv: cond-mat/0104383. Sumedha, D. Dhar, J. Phys. A 36, 3701 (2003); arXiv: cond-mat/0303450. M. Bousquet-Melou, Discrete Math. 180, 73 (1998). A. Abdesselam, V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions", in Constructive physics, Lecture Notes in Phys. vol. 446, Springer, Berlin, 1995, pp. 7-36; mp.arc: 94-291, http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=94-291 . J. Frohlich, "Mathematical aspects of the physics of disordered systems", in Phenomenes critiques, systemes aleatoires, theories de jauge, Part II (Les Houches, 1984), North Holland, Amsterdam, 1986, pp. 725-893. J. Z. Imbrie, Ann. Henri Poincare 4, 421 (2003); arXiv:math-ph/0303015. J. L. Cardy, J. Phys. A 34, L665 (2001); arXiv:cond-mat/0107223. S.-N. Lai, M. E. Fisher, J. Chem. Phys. 103, 8144 (1995). R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc., London, 1982. A. Baram, M. Luban, Phys. Rev. A 36, 760 (1987). J. L. Cardy, Phys. Rev. Lett. 54, 1354 (1985). Y. Park, M. E. Fisher, Phys. Rev. E 60, 6323 (1999); arXiv:cond-mat/9907429. H. E. Stanley, S. Redner, Z. R. Yang, J. Phys. A 15, L569 (1982). T. Hara, G. Slade, J. Statist. Phys. 59, 1469 (1990). T. Hara, G. Slade, J. Statist. Phys. 67, 1009 (1992). E. Witten, J. Geom. Phys. 9, 303 (1992); arXiv:hep-th/9204083. D. C. Brydges, J. Wright, J. Statist. Phys. 5 1 , 435 (1988); J. Statist. Phys. 97, 1027 (1999). J. L. Cardy, "Lecture on branched polymers and dimensional reduction"; arXiv:cond-mat/ 0302495.

Random path representation and Ornstein-Zernike theory of fluctuations DMITRY I O F F E

(Technion, Haifa)

Rigorous approach to the Ornstein-Zernike theory of fluctuations is based on a random walk/random line type representation of semi-invariants (high temperature models) and of phase boundaries (low temperature two-dimensional models) and on renormalization procedures which set up the stage for a probabilistic analysis along the lines of the thermodynamic formalism of one-dimensional systems associated to countable Markov shifts. Recent results include sharp asymptotics of correlation functions for sub-critical percolation models/high-temperature finite range Ising models in any dimension and an invariance principle for low temperature twodimensional interfaces.

1. Introduction Classical Ornstein-Zernike theory [12] leads to a sharp asymptotic description of point-topoint radially symmetric density correlation function G on R d for simple fluids away from criticality. The original computation hinges on three main ad hoc assumptions: A l The interaction between particles at different points of Kd is either direct or mediated by a direct interaction with a third particle: G(x) = F(x) + f

F(y) G(x - y) dy,

(1)

where F is the direct correlation function. A2 J F(x) dx < 1, which is an expression of non-criticality. A 3 The direct interaction F is a short range one. Then there exists a decay exponent £ > 0 and a number c < oo, such that the asymptotic behaviour of G is given by G(x) = _ £ _ e - « W ( l + o(l)).

(2)

In probabilistic terms (2) could be (in the case of non-negative F) understood as follows: Define the Laplace transforms (t G M.d and (•, •) is the corresponding scalar product) W(t)=

f

e{t'x)F{x)dx

and

jRd

G(t) = f JRd

e^G(x)dx.

(3)

By A 3 function F is defined and analytic on Rd. Furthermore, by A2 the G-integral in (3) has positive radius of convergence £ > 0 (no coincidence — it is the same £ as in (2)), and for every \t\ < £,

237

238

DMITRY IOFFE

Thus ¥(t) = 1 on {t : \t\ = £}. In particular, given x G Rd and the "dual" direction t0 = to(x) = £x/\x\, the function

foiv) = e^v)F(y) is a probability density on Rd. Let Vi.V^,... be the i.i.d. steps of an Revalued random walk, each V* being sampled from /o. Then the renewal formula (1) could be developed as oo

k

*iWG{x) = Y,®h{Vx fe=i

+ --- + Vk = x),

(5)

l

where ®i/o is the A;-fold convolution of /o- Because of the radial symmetry the expectation Eo^fc = VlogF(io) is co-linear with x. The OZ formula (2) follows now from classical local limit results and the Gaussian summation formula. It is important to stress that (2) is a consequence of the pre-assumed random walk type structure of the two-point function G. The purpose of this lecture is to explain recent attempts to recover the Ornstein-Zernike formula directly from the picture of microscopic interactions in various lattice models of Statistical Mechanics including weighted lattice self-avoiding walks [7,8], sub-critical short range Bernoulli percolation models [3] and sub-critical finite range ferromagnetic Ising models [4]. In all these models two-point functions enjoy random line type representations and the corresponding versions of the Ornstein-Zernike formula (2) could be reinforced with invariance principles for these random lines [5,6,10]. Similar results hold for phase separation lines in the nearest neighbour 2D Ising model below the critical temperature [6] and, apparently, pertain to a much larger class of low temperature 2D models in the Pirogov-Sinai regime [9]. It is remarkable that by and large the original assumptions A 1 - A 3 indeed perceive the mathematical structure of the twopoint functions. However, rigorous verification of an appropriate form of A3 comes as the major renormalization step involved in the proof and, furthermore, in general it appears to be natural to build the corresponding local limit analysis not on the results for sums of independent steps (as generated by the renewal assumption A l ) , but rather on the general thermodynamic formalism of one-dimensional systems generated by Ruelle's operator for shifts over countable alphabets (of irreducible random path-type objects).

2. Bernoulli bond percolation 2.1. The OZ formula Each nearest neighbour bound of Zd is independently open with probability p and closed with probability 1 — p. The two-point function is defined as Gp(x) = P p (0 <-> x),

Random path representation and Ornstein-Zernike theory of

fluctuations

239

where the event {0 <-> x} implies the existence of a connected chain of open bonds leading from the origin to x. Define now the (direction dependent) inverse correlation length ip{x) = - lim -

logG([nx\).

(6)

n—»oo n

A fundamental result of the percolation theory [2,11] implies that £p is an equivalent norm on Rd for every p < pc(d), where pc is the percolation threshold. Theorem 2.1. [3] For any d > 2 and for any p < pc(d) the asymptotic behaviour of G(-) is given by

G(z) = M ^ e - « > M (1+ 0 (1)), where n(x) = x/\x\ G S d _ 1 is the direction ofx and ^p is a positive locally analytic function onSd~l. The qualitative picture behind Theorem 2.1 is that under the conditional measure P p ( • |0 <-> x) the common connected cluster C0<x has an essentially forward one-dimensional structure: The meaning of the word "forward" is encoded in the geometry of £p: Being an equivalent norm on Rd it is a support function of a compact convex set K-P={t

: (t,y)
My e Rd) .

Alternatively the set K,p could be described as the closure of the domain of convergence of the series t

_> Y,e(t'v)G(y)-

(7)

y

Given x G Rd \ 0 a boundary point t0 e dJCp is said to be dual to x if £ p (x) = {t0,x). Given such tQ and a small enough 5 > 0 let us say that a vector y e Rd is forward if {to,y) > (1 - S)£p(y). Let C0 be the convex cone of forward vectors. The forward structure of the connection cluster is quantified on large enough renormalization scales K < oo. With such scale K fixed the irreducible decomposition of CotX is depicted on figure 1:

Hxiui

KxiUi)

(\ 'ill.

"I

T-Cx{u„) Figure 1. Irreducible decomposition Co,* = C L U Ci U ... U C„ U C R : u 0 = 0, un+i = x and u i , . . . , u„ are cut points of Co,x such that C, C KB(ui) +C$ for i = 1,..., n - 1 and CR C KB(un) + CY

240

DMITRY IOFFE

Namely, let us say that ti e Co,j is a cut point if 0 < (u, x) < (x,x) and Hx (u) D Co,x = u, where TLx{u) = {v : (x, v — u) = 0} and Co,x is embedded in Rd with all its open bonds. In the irreducible decomposition depicted on figure 1 un is the first (largest x-projection ) cut point of Co,x s u c n that C R = CQ>X d{v : (v — un) > 0} satisfies C R C KB(un) + Cs, where B(u) is the unit R d -ball centred at u. Accordingly, u„_i is the first cut point after un such that C„_i = Co,x H {v : (v — un) > 0} \ C R satisfies C„_i C KB{un^\) + Cs and so on. The collection U = {u$ = 0, u\,... ,un,un+\ = x} of irreducible cut points could be viewed as a trajectory of a Zd-valued random walk with steps Vj_, = ui, VR = x — un and Vi = V{Ci) = Uj+i — Ui\ i = 1 , . . . , n — 1 being the displacement along the end-points of the cluster Ci. The crucial OZ assumption A 3 on the short range nature of direct correlations finds its rigorous expression in the following Lemma 2.1. There exists a finite renormalization scale K = K(d,p,5) stants c\, C2 > 0 such that

and positive con-

P(max{diamCt,diamCi,...,diamCfl} > a \ 0 <-> x) < ci|x|e~ C2a ,

(8)

uniformly in x G Rd. In particular the left-most and the right-most clusters CL and C R could be ignored on the CLT scale (as \x\ —> oo). The remaining intermediate clusters C\,..., Cn-\ have identical geometric structure. In the sequel we shall refer to them as irreducible. Let to S dKp be a dual point to x, £p(x) = (to,x). Define:

QP(y)=

£

e(*°^Pp(C).

C irreducible : V(C)=y

It readily follows from Lemma 2.1, the irreducible decomposition of CojX on figure 1 and a characterisation of K.p in terms of the convergence of the series in (7) that Q p is a probability distribution on Z d with exponentially decaying tails. Similarly, the (in general nonprobabilistic) weights O L ( S / ) = Y,

e (t0,!/) P P ( C L )

and

QR(y) =

CL:0-»?/

£

e^Pp(CR)

CR:-y—0

also have exponential tails. In view of Lemma 2.1 the irreducible decomposition of CotX could be recorded (for generic directions of x) as oo

e*'<*>G(aO = o (e- C3 l*l) + £ Q L ( y ) Q R ( * ) £ y,z

n

(g)Q„ (V, + • • • + Vn = x - y - z),

(9)

n=l 1

and the statement of Theorem 2.1 follows by classical local limit computations. 2.2. Diffusive scaling of the connection cluster C0t\_Nxj The geometry of K.p is related to a local limit analysis based on the representation formula (9) through (7). In particular, it follows [3] that dK,p is locally analytic and, in addition,

Random path representation and Ornstein-Zernike theory of

fluctuations

241

at each point t0 G dK,p the principal curvatures Ki,... ,Kd-i are strictly positive. These principal curvatures explicitly come into picture on the level of the invariance principle: Let x G S d _ 1 . By the very construction all the steps VL, V i , . . . , Vn-\, VR. in the irreducible decomposition of Co^iVxj n a v e positive ^-projection. Let CM — £N{VL, VI, • • •, VL) be the polygonal interpolation through the points of U. By construction, the cardinality # { £ J V D Ti-hNx) — 1 for every h G [0,1]. Thus the polygonal line CN could be represented as a function on [0,1]. Define C^(h) = £jv l~l Ti-hNx a n d consider the following rescaled centred version of the latter:

N{h) = -1= (CN(h) - hNx). 'N

The random function w takes values in the tangent space Rd~1 of dJCp at to, where to is the dual direction to x. We shall parametrise 4>N{h) = (cpN(h),..., 2, under P( • | 0 <-> [-^J) is where B1,...,

Bd~l

any p < pc{d) and any x £Rd

the weak limit of 4>N

are independent Brownian bridges on [0,1].

Since by Lemma 2.1 the Hausdorff distance djy(Col[./vxj,£Ar) vanishes under the above diffusive scaling, Theorem 2.2 could be equally considered as an invariance principle for the whole connection cluster Co, [JVXJ .

3. Ferromagnetic Ising models 3.1. OZ formula Let J = {Jv}vezd> be a collection of nonnegative real numbers such that Jv = J^v and Jv = 0 if \v\ > R, where R is some finite number. To avoid trivialities we assume that {v : Jv > 0} generates Zd. The (formal) Hamiltonian of the corresponding finite range ferromagnetic Ising model on Zd is given by H

(°) = ~~ Yl

J

y-x°-xOy .

(10)

In the high temperature region (3 < (3C = f3c(3,d) there is the unique infinite volume Gibbs state ( -)p and, furthermore [1], the inverse correlation length £.p(x) = - lim -log(f7O0-|„xj) is an equivalent norm on Rd \ 0. In the sequel we shall use JCp to denote the convex set supported by £3. Theorem 3.1. In any dimension d>l and for any ferromagnetic model (10) the asymptotic decay of the two-point correlation function in the high-temperature region j3 < j3c{3,d)

242

DMITRY IOFFE

is given by

M^^0k^"-><-<(l+om),

(11)

where nx = x/\x\ € S rf_1 is the unit vector in the direction of x, and the function typ is strictly positive and analytic. As a byproduct of the proof of the theorem above we infer that dICp is locally analytic and strictly convex. In two dimensions Kp is reminiscent of the Wulff shape (by duality it is precisely the low temperature Wulff shape in the case of nearest neighbour interactions). By the analyticity of dK.p, no roughening transition is possible in such low temperature 2D models where the theory applies [5]. The relation between the curvature of dKp and the variance of fluctuations is discussed in Subsections 3.2. The proof of Theorem 3.1 is based on a careful analysis of the geometry of the following high temperature random line representation of the two point-function: q

{o-0O-x)/3 = Yl

^ '

(12)

A:0-»x

where the sum runs over a family of admissible self-avoiding paths, see e.g. [4,13] for details. As in the percolation case the crux of the matter is to develop an irreducible decomposition of such paths on a large enough renormalization scales and then to reinterpret (crocrx)p in terms of the associated random walk whose steps are just displacements along the successive irreducible pieces. The new difficulty arising now is that is not possible anymore to write down an OZ equation: Indeed, for the Ising model the weights do not factorize: qp(XUf) ^ qp(X) 9/3(7), where AII7 is the concatenation of two (compatible paths). Consequently local limit analysis of {o-Qax)p cannot be based on the results about sums of independent random variables. However, it happens that the weights qp possess a number of nice properties which give rise to essentially the same local limit structure as in the independent case (see e.g. §1 of [5]). The crucial role is played by the strict exponential decay of the two-point function and by the following exponential decoupling property: For any pair of compatible paths A and 77 define the conditional weight qp(\ | 77) = qp(X U rf)/qp{r}) where A H 77 denotes the concatenation of A and 77. Then, there exists C4 < 00 and 0 € (0,1) such that, for any four paths A, 77, 71 and 72, with A H 77II71 and AII77II72 both admissible,

qp{X\nUl2)

)

fr!

(13)

xGX

3/G71U72

As in the case of Bernoulli percolation the notion of irreducibility is spelled out in terms of the geometry of dICp. Given a finite renormalization scale K, a small number 5 > 0, a target point x € Z d and the direction to £ dKp dual to x, let us say that ui is a break point of a path A = (u\,..., un) if {ut} - \r\Hx{ui), and {ui+i,...,un}

cKB(ii)+C0,

Random path representation and Ornstein-Zernike theory of

fluctuations

243

where, as in the percolation case, Co is the cone of forward directions. A path A is irreducible if it does not contain break points. If 7 = Ai II • • • II An is the the irreducible decomposition of 7, 9lv ul c fe_/ i4

Y^ 5 2

~ ^ s0 .

( )

which, in view of the decoupling property (13), already implies uniform estimate on the ratios of conditional weights. As a consequence, the irreducible representation

(l

]C

g/?(AiII...IIA n )

A:0-*x A=AiII...IIA n Afcirreducible, k=l,...,n n-1

= J2 n>l

£ A:0-»x A=AiU...nA„ At irreducible, A;=l

q0(Xn)Hq0(Xk\Xk+1U---UXn)

(15)

fe=l n

can be interpreted in terms of thermodynamic formalism of one-dimensional systems associated to Ruelle's operators for shifts over countable alphabets (of irreducible paths) with uniformly Holder continuous potentials and, accordingly, classical local limit results come into play through the analytic perturbation theory of simple poles of the corresponding resolvent operators [4]. 3.2. Invariance principle for high temperature models As in the case of the Bernoulli percolation the irreducible representation (15) elucidates the effective random walk structure of the two-point function (o-Qcrx)p: Path weights q@ give rise to the probability measure Qx,p on the space of all connection paths 7 : 0 1—» x in their irreducible representation 7 = Ai I I . . . An; Qx,/3(7) = Qx,/3 (Ai I I . . . A„) =

rq0

(Ai I I . . . A„)

(O"0<Jx//9

Let Vi = V(Xi) be the displacement along the i-th irreducible piece Aj. Lemma 3.1. For any dimensional > 2, for any (3 < (3c(d) and for any positive number 5 > 0 there exists a finite renormalization scale K = K(d,P,S) and positive constants Ce,cr > 0, such that QX:/3 (max{diam(Ai), V 2 ,..., Vn} > a) < c 6 |x|e" C 7 a . (16) uniformly in x € Z d . Let x € S d _ 1 and consider the family of measures Q[Nx\,0 a s N —> 00. Let 7 = AiII... A„ be the irreducible decomposition of a connection path 7 from 0 to [Nx\. As in the percolation case we use CN = £jv [7] to denote the linear interpolation through the vertices 0,V1,V1+V2,...,V1

+ --- +

Vn=[Nx\.

By Lemma 3.1 there exists c 8 > 0 such that c 8l°gA0

244

DMITRY IOFFE

vanishes as N —> oo. Thus 7 and £#[7] are indistinguishable on scales much larger than log N, under the diffusive scaling in particular. As in the percolation case define: (j>N{h) = - = (CN(h) VN

hNx),

with 4>N being understood as a function from [0,1] to the tangent space to dtCp at the dual to x point to, £/3(x) = (t0,x). Theorem 3.2. [6] For any d > 2, any (3 < 0c(d) and any x £ S'* -1 the weak limit of 4>M under Q[Nxit0(-) is

(^/irlB\...,^r1Bd-1),

where B1,..., Bd~l are independent Brownian bridges on [0,1], and « i , . . . , KJ,-I are principal directions of curvature of dK,p at to. I would like to stipulate that the results above go beyond finite range ferromagnetic models and, in fact, reflect large scale limit properties of a whole class of random line type objects once a (12)-like decomposition of the corresponding partition function is available. A set of general explicit conditions on path weights is formulated in [5]. In particular, appropriate modifications of these techniques lead to a proof of the invariance principle for phase separation lines in the 2D Ising model up to the critical temperature [6] and, more generally, to a theory of interfaces in the two dimensional low temperature models in the Pirogov-Sinai regime [9].

Acknowledgments Part of the work reported in this paper has been accomplished during my stay at Isaac Newton Institute for Mathematical Sciences in the framework of the programme "Interaction and growth in complex stochastic systems".

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

M. Aizenman, D. J. Barsky, R. Fernandez, J. Stat. Phys. 47, 342 (1987). M. Aizenman, D. J. Barsky, Comm. Math. Phys. 108, 489 (1987). M. Campanino, D. Ioffe, Ann. Probab. 30, 652 (2002). M. Campanino, D. Ioffe, Y. Velenik, Prob. Theor. Rel. Fields 125, 305 (2003). M. Campanino, D. Ioffe, Y. Velenik, to appear in Advanced Studies in Pure Math. (2003). L. Greenberg, D. Ioffe, preprint (2003). O. Hryniv, D. Ioffe, to appear in Mark. Proc. Rel. Fields (2003). D. Ioffe, Mark. Proc. Rel. Fields 4, 323 (1998). D. Ioffe, Y. Velenik, in preparation. E. Kovchegov, to appear in Mark. Proc. Rel. Fields (2003). M. V. Menshikov, Sov. Math. Doklady 33, 856 (1986). L. S. Ornstein, F. Zernike, Proc. Acad. Set. (Amst.) 17, 793 (1915). Ch. E. Pfister, Y. Velenik, Comm. Math. Phys. 204, 435 (1999).

Construction of a 2-d Fermi Liquid HORST KNORRER

(ETH-Zentrum, EUGENE

Zurich), J O E L FELDMAN (U. of British TRUBOWITZ (ETH-Zentrum, Zurich)

Columbia),

The temperature zero renormalized perturbation expansions of a class of interacting many-fermion models in two space dimensions have nonzero radius of convergence. The models have "asymmetric" Fermi surfaces and short range interactions. One consequence of the convergence is the existence of a discontinuity in the particle number density at the Fermi surface. Here we describe the main results and highlight some of the strategy of the construction.

1. T h e r e s u l t s The concept of a Fermi liquid was introduced by L. D. Landau in [16-18] and has become the generally accepted explanation for the unexpected success of the independent electron approximation. An elementary sketch of Landau's well known physical arguments can be found in [4, pp. 345-351]. More thorough and technical discussions are presented in [1, pp. 154-203] and [22]. Roughly speaking, at temperature zero, the single particle excitations of a noninteracting Fermi gas become (almost stable) 'quasi-particles' in a Fermi liquid. The quasi-particle spectrum has the 'same structure' as the noninteracting single particle excitation spectrum and the quasi-particle density function n(k) still has a jump at the 'Fermi surface'. The quasi-particle interaction at temperature zero is encoded in Landau's /-function / ( k p , k ' F ) . It is well known that there are a number of potential instabilities that can drive an interacting Fermi gas away from the Fermi liquid state. See, for example, [21, §1.2,4.5]. One of the most celebrated is the BCS instability for the formation of Cooper pairs leading to superconductivity in 2 and 3 dimensions. This is a potential instability for any time reversal invariant system [15,19]. Another important instability is the Luttinger instability. There are solvable models in one space dimension that exhibit qualitatively different behavior from that of a three dimensional Landau Fermi liquid. In particular, the quasi-particle density function n(k) is continuous across the 'Fermi surface' but has infinite slope there. These systems are called Luttinger liquids. For a rigorous treatment of Luttinger liquids in one dimension, see [5] and the references therein. Anderson [2,3] suggested that a two dimensional Fermi gas should exhibit behavior similar to a one dimensional Luttinger liquid. [11, Theorem 1.4] rigorously shows that this is not the case for a specific class of models. In particular, we show that, at temperature zero, the density function rc(k) has a jump discontinuity across the Fermi surface [11, Theorem 1.5]. The existence of the Landau f-function and its basic regularity properties follow directly from [11, Theorem 1.7]. For results concerning Fermi liquids at strictly positive temperature, see [7,8,23,24]. The class of models that we consider is somewhat unusual in that the Fermi surface survives the turning on of all sufficiently weak short range interactions. To motivate the

245

246

HORST KNORRER, JOEL FELDMAN, EUGENE TRUBOWITZ

class, consider a gas of fermions with prescribed, strictly positive, density, together with a crystal lattice of magnetic ions. The fermions interact with each other through a two-body potential. The lattice provides periodic scalar and vector background potentials. As well, the ions can oscillate, generating phonons and then the fermions interact with the phonons. At the present time our result is restricted to d = 2 space dimensions. But we believe that the difficulties preventing the extension to d = 3 are technical rather than physical. Indeed, there has already been some progress in this direction [6,20]. To start, turn off the fermion-fermion and fermion-phonon interactions. Then we have a gas of independent fermions, each with Hamiltonian

#o = ;^-(iV + A(x)) 2 + £/(x). We assume that the vector and scalar potentials A, U are periodic with respect to some lattice r in R 2 . Note that it is the magnetic potential, and not just the magnetic field, that is assumed to be periodic. This forces the magnetic field to have mean zero. By convention, bold face characters are two component vectors. Because the Hamiltonian commutes with lattice translations it is possible to simultaneously diagonalize the Hamiltonian and the generators of lattice translations. Call the eigenvalues and eigenvectors e ^ k ) and ^ ^ ( x ) respectively. They obey flo<7^,k(x) = e„(k)0„,k(x), ^ , k ( x + 7) = e ' < k ^ , k ( x ) ,

V7er.

The crystal momentum k runs over R 2 / r # where r#

= {b G M2 | (b,7)

G 2TTZ for all 7 G T}

is the dual lattice to T. The band index v G N just labels the eigenvalues for boundary condition k in increasing order. When A = U = 0, £„(k) = ^ (k — k„,k) 2 ^ or s o m e b„, k e r # . In the grand canonical ensemble, the Hamiltonian H is replaced by H — /j,N where N is the number operator and the chemical potential fi is used to control the density of the gas. At very low temperature, which is the physically interesting domain, only those pairs v, k for which e„(k) « /J, are important. To keep things as simple as possible, we assume that £„(k) ss n only for one value v$ of v and we fix an ultraviolet cutoff so that we consider only those crystal momenta in a region B for which |e^0 (lc) — JJL\ is smaller than some fixed small constant. We denote E(k) = e„ 0 (k) — fi. When the fermion-fermion and fermion-phonon interactions are turned on, the models at temperature zero are characterized by the Euclidean Green's functions, formally defined by

(2) The action Aty,$)

= -fdk

(ik0 - £(k)) V^Vfc + V(V, V0-

(3)

247

Construction of a 2-d Fermi Liquid The interaction V will be specified shortly. We prefer to split A - J dk(iko — •E(k))^fcVfc a n d write


Q + V where Q

J/(M)e*^>n fc , g dV> fc , g d&, g

f eVWrf)

duett,®

where due is the Grassmann Gaussian "measure" with covariance C(k)

1 ik0 - E(k)'

We have here dropped some factors of 2ir. We will continue to routinely drop various unimportant constants throughout this article. We now take some time to explain (2). The fermion fields are vectors i>k

V'fe.T .V'fea.

Tpk = [>l>k,t

4>k,l]

whose components ipk,a,'4>k,(r, & = (fco,k) e K x B, a € {T,j}, are generators of an infinite dimensional Grassmann algebra over C That is, the fields anticommute with each other, (7)

(7)

(7)

(7)

We have deliberately chosen ip to be a row vector and ip to be a column vector so that

•p = V'fc.r^p.T + V'AuVva.

V'feV'p

ipk,-[ipPA V'fe.T^P.l .V'fe.iV'p.T V'fcj^pj.

In the argument fc = (fco,k), the last d components k are to be thought of as a crystal momentum and the first component k0 as the dual variable to an imaginary time. Hence the \ / - T in iko - E(k). Our ultraviolet cutoff restricts k to B. In the full model, k is replaced by (i/,k) with v summed over N and k integrated over Rd/T*. On the other hand, the ultraviolet cutoff does not restrictfcoat all. It still runs over R. So we could equally well express the model in terms of a Hamiltonian acting on a Fock space. We find the functional integral notation more efficient, so we use it. The relationship between the position space field ip(x), with x = (t,x) running over (imaginary) time x space, and the momentum space field tpk is really given, in our single band approximation, by

1>(x) = Jdke

:k0t~

Vo,k (x)lAfe-

We find it convenient to use a conventional Fourier transform, so we work in a "pseudo" space-time and instead define i>(x) Under suitable conditions on the real one.

I

dkellc-xil>k-

,,k(x), it is easy to go from the pseudo space-time ip(x) to

248

H O R S T KNORRER, JOEL FELDMAN, EUGENE TRUBOWITZ

For a simple two-body fermion-fermion interaction, with no phonon interaction, V = - -

^2

J

dtd^dyu{-x.-y)i>a{t,-x.)^}a{t,y)\[)T{t,y)i}T{t,y).

T h e general spin independent form of t h e interaction is 1 V("0,^) = -

4

f ]T T,r6{T,i}

m

Y[dxiV(xi,X2,X3,Xi)ll>T{X4) i=l

dki5{ki

+k

2

- h - k4) il>kli>k3 V>fc2Vfc4-

Spin independence is imposed purely for notational convenience. I t plays no role. T h e function V(xi,X2,X3,X4), or equivalently (fci, A^l^lfe, ^4), can implement b o t h t h e fermionfermion and fermion-phonon interactions. Its precise value does not concern us. We just assume H y p o t h e s i s 1 . 1 . The interaction is weak and short range. That is, V is sufficiently near the origin in QJ, which is a Banach space of fairly short range, translation invariant functions V(xi,X2,X3,X4). See [11, Theorem 1.4] for %3's precise norm. For some results, we also assume t h a t V is "fc 0 -reversal real", V(Rxi, where R(xo,x)

Rx2, Rxs, Rxi) — V(xi,x2,x3,X4)

(4)

— (xo, —x), and " b a r / u n b a r exchange invariant", V(-X2,-Xi,-X4,-X3)

= V(XI,X2,X3,XA).

(5)

If V corresponds t o a two-body interaction w(xj — X3) with a real-valued Fourier transform, then V obeys (4) and (5). Our goal is t o prove t h a t perturbation expansions for various objects converge. These objects depend on b o t h -E(k) and V and are n o t smooth in V when -B(k) is held fixed. However, we can recover smoothness in V by a change of variables. To do so, we split E(k) = e(k) — Se(V, k ) into two parts and choose Se(V, k ) t o satisfy an implicit renormalization condition. This is called renormalization of t h e dispersion relation. Define t h e proper self energy E(p) for t h e action A by t h e equation

(^o-e(p)-E(p))

S(P-q)=

J

e

^

)

W

k

a

T h e counterterm Se(V, k) is chosen so t h a t E ( 0 , p ) vanishes on t h e e(p) = 0 } . We take e(k) and V, rather t h a n t h e more natural data. T h e counterterm 5e will be an o u t p u t of our main theorem. Banach space £. While t h e problem of inverting t h e m a p e i-> E well understood on a perturbative level [13], our estimates are not so nonperturbatively. Our main hypotheses are imposed on e ( k ) .

d

^

•

Fermi surface F = { p | E(k) and V, as input It will lie in a suitable = e — Se is reasonably yet good enough t o do

Construction of a 2-d cFermi Liquid

249

Hypothesis 1.2. The dispersion relation e(k) isia real-valued, sufficiently smooth, function. We further assume that (a) the Fermi curve F = {k e R 2 | e(k) = 0} \s a simple closed, connected, convex curve with nowhere vanishing curvature. (b) Ve(k) does not vanish on F. (c) For each q € R 2 , F and —F + q have low degree of tangency. (F is "strongly asymmetric".) Here —F + q = {-k + q | k € F}. Again, for the details, see [11, Hypothesis 1.12]. It is the strong asymmetry condition, Hypothesis 1.2.c, that makes this class of models somewhat unusual and permits the system to remain a Fermi liquid when the interaction is turned on. If A = 0 then, taking the complex conjugate of (1), we see that £„(—k) = e„(k) so that Hypothesis 1.2.c is violated for q = 0. Hence the presence of a nonzero vector potential A is essential. We shall say more about the role of strong asymmetry later. For now, we just mention one model that violates these hypotheses, not only for technical reasons but because it exhibits different physics. It is the Hubbard model at half filling, whose Fermi surface looks like

This Fermi curve is not smooth, violating Hypothesis 1.2.b, has zero curvature almost everywhere, violating Hypothesis 1.2.a, and is invariant under k —> —k so that F = —F, violating Hypothesis 1.2.c with q = 0. To give a rigorous definition of (1.2) one must introduce cutoffs and then take the limit in which the cutoffs are removed. To impose an infrared cutoff in the spatial directions one could put the system in a finite periodic box Rd/LT. To impose an ultraviolet cutoff in the spatial directions one may put the system on a lattice. By also imposing infrared and ultraviolet cutoffs in the temporal direction, we could arrange to start from a finite dimensional Grassmann algebra. We choose not to do so. Our goal is to prove that formal renormalized perturbation expansions converge. The coefficients in those expansions are well-defined even without a finite volume cutoff. So we choose to start with x running over all R 3 . We impose a (permanent) ultraviolet cutoff through a smooth compactly supported function U(k). This keeps k permanently bounded. We impose a (temporary) infrared cutoff through a function ve(kQ + e(k) 2 ) where f £ («) looks like

£

250

HORST KNORRER, JOEL FELDMAN, EUGENE TRUBOWITZ

When e > 0 and v£{k,Q + e(k) 2 ) > 0, \ik0 — e(k)| is at least of order e. The coefficients of the perturbation expansion (either renormalized or not) of the cutoff Euclidean Green's functions G2n;e{x\,(T\,

• • • ,yn,Tn)

= ( J \ VV* (^i)V'n (Vi)

where SfW,$)ev«>>i»drc.M) (/)£

"

. , , ^ M

J^^li^MJ)

Cf(k)„e(fcg + e (k) 2 )

H r )

' ' ~ ifc0-e(k) + Je(k)

are well-defined. Our main result is Theorem 1.1. [11, Theorem 1.4] Assume that d = 2 and that e(k) ,/iii/iis Hypothesis 1.2. There is — a nontrivial open ball B C 93, centered on the origin, and — an analytic function V G B — i > 5e(V) € £, that vanishes /or V = 0, such that: — for any e > 0 and n G N, the formal Taylor series for the Green's functions Ginfi converges to an analytic function on B; — as £ —> 0, G2n\e converges uniformly, in x\,...,yn and V G B, to a translation invariant, spin independent, particle number conserving function Gin that is analytic in V. If, in addition, V is kg-reversal real, as in (4), then <$e(k; V) is real for all k. Theorem 1.2. [11, Theorem 1.5] Under the hypotheses of Theorem 1.1 and the assumption that V G B obeys the symmetries (4) and (5), the Fourier transform G2(fco,k) = j(teoAe'(-fe»I»+k'x)

G 3 ((0,0, T), (x0,x, T))

= J dx0 d"xe«-fco*o+k-*)

ik0 - e(k) - E(fc)

G 2 ( ( 0 ; 0 j i ) ; (a . 0) X j i } )

when U(k.) = 1

of the two-point function exists and is continuous, except on the Fermi surface (precisely, except when fco = 0 and e(k) = 0). The momentum distribution function n(k) = lim [ ^

eik°r

is continuous except on the Fermi surface F. IfkG

G2(k0,k)

F, then lim n(k) and lim n(k) exist e(k)>0

and obey lim n(k) -

k—+k e(k)<0

lim n(k) = 1 + 0{V) > \ .

k—*k e(k)>0

2

e(k)<0

Construction of a 2-d Fermi Liquid

251

Theorem 1.3. [11, Theorem 1.7] Let Gi{k1,k2,h,ki) (spin dropped from notation) he the Fourier transform of the four-point function and 4

G^{ki,k2,k3,ki)

= Gi(ki,k2,k3,k4)Y\ /=1

l > G2{ke)

its amputation by the physical propagator. Under the hypotheses of Theorem 1.2, Gf has a decomposition G±(ki,k2,k3,ki)

=

N{ki,k2,kz,ki) +

l . / ^ i + fe k3 + ki

2^1

2

'

2

1 T/k3

+ k2 ki+k4

'fc2 ~ ^V \

with — N continuous, — L(qi,q2,t) continuous except at t = 0, — lim L(qi,q2,t) continuous, to—>0

— lim L(qi,q2,t)

continuous.

Think of £ as a particle-hole ladder

2. Blocking t h e Cooper channel We now discuss further the role of the geometric conditions of Hypothesis 1.2 in blocking the Cooper channel. When you turn on the interaction V, the system itself effectively replaces V by more complicated "effective interaction". The (dominant) contribution

to the strength of the effective interaction between two particles of total momentum t Pi+P2=:qi + q-i is

stuff

dk-r ' [tfc0 - e(k)] [i(-ko +10) - e ( - k +1)] ' / Note that [ik0 - e(k)] = 0 <=» k0 = 0, e(k) = 0 4=^ k0 = 0, k € F, [i(-k0 + k) - e(-k + t)] = 0 *=> k0 = t0, e ( - k + t) = 0 <=> fc0 = t 0 , k G

t-F.

252

HORST KNORRER, JOEL FELDMAN, EUGENE TRUBOWITZ

We can transform jf^z^^j locally to jfc0Lfcx by a simple change of variables. Thus jg~^y^ is locally integrable, but is not locally L2. So the strength of the effective interaction diverges when the total momentum t obeys to = 0 and F = t — F, because then the singular locus of ik -efkl coincides with the singular locus of i , k + f y_ e /_ k , t-,. This always happens when F = —F (for example, when F is a circle) and t = 0. Similarly the strength of the effective interaction diverges when F has a flat piece and t / 2 lies in that flat piece, as in the figure on the right below. t - F

F

On the other hand, when F is strongly asymmetric, F and t — F always intersect only at isolated points. A "worst" case is illustrated below. There the antipode, a(k), oik € F, is the unique point of F, different from k, such that the tangents to F at k and a(k) are parallel.

k + a(k) - F / * " * " Y k{

) a(k)

For strongly asymmetric Fermi curves, [iko - e(k)j [i(-ko +10) - e ( - k + t)] remains locally integrable in k for each fixed t and strength of the effective interaction remains bounded.

3. Power counting and nonperturbative bounds The proofs of Theorems 1.1, 1.2 and 1.3 are quite technical. The whole construction is given in a series of papers [10]. In the first paper [11] of the series, the main difficulties and our strategies to overcome them are described. Here, we concentrate on one aspect, namely the need to use both position space and momentum space arguments and the problems created by the interplay between them. The Green's functions G2n are constructed using a multiscale analysis and renormalization. The multiscale analysis is introduced by choosing a parameter M > 1 and decomposing momentum space into a family of shells, with the j t h shell consisting of those momenta k obeying \iko — e(k)| w 1/MJ'. Correspondingly, we write the covariance as a telescoping series C(k) = J27Lo C^(&) where, for j > 1,

253

C o n s t r u c t i o n of a 2-d Fermi Liquid

is the "covariance at scale j " . By construction C^\k) vanishes unless y/k% + e(k) 2 is of order M~J, and ||C (j) (fc)|| L ~ ss MK We consider, for each j , the effective interaction at scale j

Wj(0,0, V, VO = log i - 1 e^C+v(V.+C^+C) d/XCw_, (C) c-}) where the source term J( = f dx <j)(x) ((x) + ip(x)((x) and the partition function Zj is chosen so that VVj(0,0,0,0) = 0. The coefficients in the expansion of Wj(, <j>, 0,0) in powers of 4> are the Euclidean Green's functions G2n.M-j • The effective interactions are controlled using the recursion relation

Wi+1(4>, <£, ,$ = log i y e*J<+w>«<*'*+M+Sd»cU)

(C, 0 -

(6)

The recursion relation (6) is the renormalization group map. The main difficulties in controlling it already arise when (j) =