Fractional Statistics And Quantum Theory, Second Edition

). Hence the angular equation reduces to

( * ^ + «) W ) = AF,(tf).

(3.3)

Ch 3.

Quantum Mechanics of Anyons

45

Since the problem is being looked upon as that of two interacting bosons, hence the eigenfunction Fi (4>) must satisfy the boundary condition (3.4)

Fl( + * ) = Fl{).

Clearly, the solution of Eq. (3.3) consistent with the boundary condition (3.4) is Fi () = e"* ,1 = 0, ± 2 , ± 4 , A = (I - a)2 .

(3.5)

Even in this simple example we can see a clear difference between bosons (a = 0), fermions (a = 1), and anyons (0 < a < 1). In particular, whereas the eigenvalue A is two-fold degenerate for bosons (except when I = 0) and fermions, it is non-degenerate for any other value of a. On substituting the eigenvalue A in the radial equation, we get

[ _ * ( | L + I « ) + J*L((_O)>1 „„_*!,(„. ,3.6,

[ m \ or2 r or J mr2 Several comments are in order at this stage.

J

(1) The net effect of the anyonic statistics is to replace angular momentum / by I — a in the radial Schrodinger equation for the relative motion of the two particles. This is true not only for this particular case but is valid in any two-anyon problem. This justifies our assertion in the last chapter that all particles, except bosons, experience a centrifugal barrier which prevents intersecting trajectories. This also justifies the hard-core requirement which excludes coincident points while calculating the coordinate space of N identical particles in the last chapter. (2) Eq. (3.6) is easily identified as the Bessel equation with the solution n2k2 R(r) = J\i-a\(kr),

E =

(3.7)

where, as expected, the spectrum for two free anyons, is continuous. For r —> 0, the nonsingular solution (3.7) behaves like R(r —• 0) ~ r1'"0"1

(3.8)

which can also be directly verified from Eq. (3.6). Thus, for particles other than bosons, the relative wave function vanishes at the origin so that the hard-core requirement, which excluded the possibility of coincident particles, is justified a posteriori. This behavior also suggests

46

Fractional Statistics and Quantum Theory

that apart from bosons, all other particles (0 < a < 1) satisfy some kind of Pauli exclusion principle. (3) Prom Eq. (3.8) it is also clear that in the ground state (I = 0), the repulsion between two anyons monotonically increases as one goes from a = 0 (bosons, which experience no repulsion) to a = 1 (fermions, which experience maximum repulsion). Put differently, one can regard anyons as bosons with an extra repulsive interaction or as fermions with an extra attractive interaction. (4) What happens if instead of being free, the two anyons experience some potential? Prom Eq. (3.6) it is clear that if the potential is a function of the relative distance r alone i.e. V = V(r) then the solution of the angular equation is unchanged from Eq. (3.5), while in the radial Eq. (3.6) one merely has to add V(r). The obvious question is: what are the various forms of V(r) for which the radial equation is analytically solvable? As is well known, the only potentials for which the eigenvalues and the eigenfunctions can be written down analytically in a closed form for all the partial waves are the oscillator and the attractive Coulomb potentials. On the other hand, in the repulsive Coulomb case, the phase shifts can be obtained in a closed form for all the partial waves. In both the cases one can also add the potential g2jr2 and the problem is still analytically solvable. Apart from these cases, there are a few other potentials for which the solutions are obtained by solving a transcendental equation. The best known examples of this kind are anyons in a circular box with hard walls and anyons with hard-disk repulsion. In addition, the examples of two-anyons in a uniform magnetic field or two-anyons experiencing a uniform magnetic field plus oscillator potential can be reduced to the oscillator case and hence are exactly solvable. Apart from these, there are a few examples which are only quasi-exactly solvable. The most interesting among these is that of two like-charged anyons in a uniform magnetic field. We shall discuss some of these examples in the next few sections. Two anyons in a circular box with hard walls Let us now consider the problem of two free anyons in a circular box of radius R with hard walls [1]. Since these are free anyons for r < R, we can immediately borrow the results obtained above. Thus the wave function for two anyons can be decomposed into a product V>(r,R)=^(R)V>2(r)

(3.9)

Ch 3.

Quantum Mechanics of Anyons

47

where V'i(R-) = e l K R is the free particle center of mass wave function while the relative wave function ip2(r) is a s given by Eqs. (3.5) and (3.7) i.e. V>2(r) = e"*J|,_ a |(fcr).

(3.10)

Here the energy corresponding to the relative motion is related to k by E = h2k2/m, and / = 0, ±2, ±4.... If we now consider free anyons in a circular box (of radius R) with hard walls, then the relative wave function must vanish at r = R i.e. J\i_a\(kR) = 0, so that the allowed bound state energy eigenvalues are TP

f^\

K,i(a

=

H2Jj2

\l-a\,n

frz— mR2

,„ 1 r ^

(o.ll)

where j v > n represents the values at which the Bessel function Jv is zero i.e. Jv{jv,n) = 0. The zeros of the Bessel function have been well tabulated [2] from where we find that for anyons (0 < a < 1) there is no degeneracy in the spectrum, while for bosons and fermions the spectrum is two-fold degenerate. Further, there are no crossing of energy levels. I might add here that historically, the second virial coefficient of the noninteracting anyon gas was first calculated by using these eigenvalues [1].

3.3

Two Anyons in an Oscillator Potential

We now discuss the interesting example of two anyons in an oscillator potential [3]. As in the case of two free anyons, the problem is separable in the center of mass and relative coordinates i.e. \mu2{r\ + r|) = muj2(R2 + ^r 2 ).

(3.12)

Unlike the free particle case, the spectrum corresponding to both the center of mass and the relative motion is now discrete and we shall see that the energy levels of a two-anyon system cannot be obtained as a sum of single anyon energy levels (except of course when a = 0 or 1). Notice that the center of mass motion is again independent of the statistics of the particles and it just corresponds to the quantum mechanical problem of a single particle of mass 2m in a two dimensional harmonic oscillator. As for the relative part, the eigenfunctions in the <j> variable are

48

Fractional Statistics and Quantum, Theory

again given by Eqs. (3.3) to (3.5) while the radial equation takes the form [dr

r2

r dr

Ah2

h2 J

v

v(3.13) ;

'

On using the ansatz R(r) = r l ' - Q l e - T ^ x ( O ,

(3.14)

it is easily shown that xir) satisfies the differential equation „. . f(2|Z — o; +1)

X»+ [—

;

mw 1 , , . \mE ,,,

.

.mui~] , ,

- -^-rj X'(r)+ ^ - (\l -a\ + 1 ) — j x (r) = 0. (3.15)

On further substituting ^ ^

(3 16)

= ^

-

we find that x(p) satisfies

PX"(P) + [\l ~ a\ + 1 - p}X'(p) + \ [ ^ - (\l ~ a\ + 1)] X(P) = 0 (3.17) which is the differential equation satisfied by the associated Laguerre polynomials. Using any standard book on Quantum Mechanics [4], we can immediately write down the energy eigenvalues for the problem Enrti(a)

= (2nr + \l-a\

+ l)hu

(3.18)

where as before / = 0, ±2, ±4,... while the corresponding eigenfunctions are W,*(^,«) = e^r-K-le-^i^l (^r 2 )

(3.19)

with L\ being the associated Laguerre polynomial. Several comments are in order at this stage. (1) Not surprisingly, for a = 0,1 we get the usual energy levels for the bosons and the fermions respectively. (2) The energy eigenvalue spectrum and the corresponding degeneracy can be compactly written in the form En(a) = (2n + 1 + a)hw , deg. = n + 1 for I < 0 = (2n + 1 - a)hu , deg. =n

for I > 0

(3.20)

Ch 3.

Quantum Mechanics of Anyons

49

where n — nr + | = 0,1,2,... since I is an even integer. We shall be using these energy levels and the corresponding degeneracy in the next chapter to compute the second virial coefficient of an anyon gas. (3) What is the origin of the large degeneracy that one finds for the spectrum of two anyons in an oscillator potential? It turns out that there is a SU{2) symmetry in the problem which is responsible for this degeneracy [5]. It has also been shown that in this problem one has two independent supersymmetries [6].

Fig. 3.1 Spectrum of two anyons in an oscillator potential.

(4) A plot of the energy levels (in units of tkJ) as a function of a is given in Fig. 3.1 [3]. The degeneracy of the levels are also mentioned. Prom the figure it is clear that the energy eigenvalues of the two anyon system are a monotonic function of a, and change continuously as one goes from the bosonic (a = 0) end to the fermionic (a = 1) end. In particular, notice that the ground state energy of two-anyons monotonically increases as one goes from a = 0 to a = 1; it being least for bosons and maximum for fermions. This is understandable, since the repulsion between two anyons monotonically increases as a goes from 0 to 1, being maximum for fermions. We also note that the energy levels do

50

Fractional Statistics and Quantum Theory

not cross each other as a function of a. We shall however see that for some other two-anyon problems and for almost all known multi-anyon problems (N > 2), there are always level crossings. (5) The energy spectrum is not equi-spaced except when a = 0, 1/2 or 1, and even at a = 1/2 (semions), the spacing between the energy levels is half of that for the bosons and the fermions. There is, however, one curious fact about the spectrum which is worth noting. In particular, notice that for 0 < a < 1, the alternate energy levels are equi-spaced with spacing 2fkv while for a = 0,1, the whole spectrum is equi-spaced with spacing 2hu>. (6) The energy eigenfunctions corresponding to the center of mass motion are easily written down. They are simply obtained from those of the relative motion by putting a = 0, replacing l,<j> by L, $ respectively, and also changing the mass m by Am. Thus the energy eigenvalues of the center of mass Hamiltonian are given by Ecm = (ni + \L\ + l)hu .

(3.21)

On combining the energy eigenvalues corresponding to the center of mass and the relative motion it is easily seen that except for a = 0 and 1, the two-anyon energy levels bear no simple additive or combinatoric relation to the levels of the one anyon system. The same thing is also true of the eigenfunctions. For example, the ground state wave function for the two-anyon system is given by 1/>(R, $ , r, <j>) OC r»ei»e-^(R2+r2/4)

=

r,e-m,(fi

2

+r

2

/4) _

( 3 .2 2 )

Written in terms of the original two particle coordinates, the ground state wave function of the two-anyon system is thus

^(n,r 2 ) oc (n -T2)°e-n*Lrt+'$K

(3.23)

Except for a = 0,1, this two-anyon wave function does not bear any simple relationship to the single anyon wave function. Thus even the two-anyon wave function cannot be written as the product of the single particle wave functions. A similar thing can also be shown in the other cases, including that of two free anyons discussed above. This in essence is the root cause of the difficulty, and that is why, till today, the problem of the statistical mechanics of an ideal anyon gas has remained an unsolved problem.

Ch 3.

51

Quantum Mechanics of Anyons

(7) Prom Eq. (3.19) we also notice that as in the free anyon case, the two anyon-relative wave function vanishes as r —> 0 (a ^ 0), thereby again justifying the hard core assumption made in the last chapter. (8) The other two-anyon problem which can be analytically solved for all partial waves is when V(r) = —e2/r. It is easily shown [5,7] by using Eq. (3.6) that the energy eigenvalues are (n — nr +1 = 0,1, ±2, ±3,...)

^"W(|,,^|

+

i)»-

(3 M)

'

The corresponding eigenfunctions are also easily written down. Unlike the oscillator case, the energy here is a nonlinear function of a. However, E~1/2 is linear in a. Besides, as in the oscillator case, there is a large degeneracy in the problem. In particular, for positive n, one has \ (^^0 degenerate states if n is even (odd) while for negative n the degeneracy is -^ ( 2 ') depending on if \n\ is even (odd). What is the origin of this large degeneracy? It turns out that there is a 0(3) symmetry in the problem which is responsible for this degeneracy [5]. It has also been shown that there is a supersymmetry in this problem [5,6]. 3.4

Two Anyons in a Uniform Magnetic Field

Let us now consider the interesting problem of two anyons in a uniform magnetic field . This problem is not only of academic interest but is also of physical relevance since the only known place where the anyons do seem to play a role is the fractional quantum Hall effect, where the charged particles experience an external uniform magnetic field. Let us consider two spin-less, charged, identical anyons (charge —e, mass m) moving in the x-y plane under the influence of a static and uniform magnetic field B, which is perpendicular to the x-y plane (B = Boz). As a first step, we neglect the Coulomb repulsion between the anyons. The effect of Coulomb repulsion will be considered later in the chapter. The solution to this problem is very similar to the usual Landau level problem, with some interesting twists arising from the anyonic statistics. Hence we shall first solve the problem of two spinless, charged identical particles under the influence of an orthogonal uniform magnetic field and then come back to the anyon problem and point out the interesting twists. The Hamiltonian for two charged particles in a uniform magnetic field

52

Fractional Statistics and Quantum Theory

is given by [4]

^ = ^ [ ( P i - e A 1 ) 2 + (p 2 -eA 2 ) 2 ]

(3.25)

AJ = J B X r*

(3.26)

where

is the symmetric gauge vector potential of the particles. Without loss of generality, we assume BQ > 0 since the spectrum can be shown to be independent of the sign of Bo. In terms of the center of mass and the relative coordinates R and r denned in the last chapter, the Hamiltonian (3.25) is easily separated i.e. H — Hcm + Hrei, where #cm = ^ ( P - 2 e A c r o ) 2

fireZ = - ( p - ^ A r e J ) 2 . m

(3.27)

(3.28)

2

Here A c m = \BX

R; Arel

= ^B x r .

(3.29)

Thus the center of mass (relative) problem is effectively that of a single particle of mass and charge TTIR = 2m, qR = —2e (mr = m/2, qr = — e/2) i.e. both are the Landau level problems whose solutions are known from the early days of quantum mechanics [4]. For example, consider the Schrodinger equation for the relative motion in cylindrical coordinates i.e. HrenP{r,c}>)=ETJ){r,4>).

(3.30)

^(r,0)=ea*iJ(r)

(3.31)

On making the ansatz

where I = 0, ±2, ±4, ...(±1,±3,...) in the case of bosons (fermions), the radial Schrodinger equation can be shown to be (3.32)

Ch 3.

53

Quantum Mechanics of Anyons

which is very similar to Eq. (3.13). Hence the energy eigenvalues are immediately obtained (see Eq. (3.18)) as En,i = (2n + \l\ - I + l)huc

(3.33)

where u>c = ^ ^ is the cyclotron frequency, n = 0,1,2,..., while I — 0, ±2,... or ± 1 , ± 3 , . . . depending on if the charged particles are bosons or fermions respectively. The corresponding radial eigenfunction is (see Eq. (3.19)) ^.«(r)=r"'exp(-^r»)L"!"(^r»)

(3.34)

where L\ is the associated Laguerre polynomial. The solution of the center of mass problem can similarly be written down. It is worth noting that in both the cases, every state (including the ground state) is infinite-fold degenerate, corresponding to the all possible positive integer values of I. However, for samples of finite area, this degeneracy is finite since in this case the angular momentum is bounded from above. Let us now solve the two-anyon problem in a uniform magnetic field [8]. As seen in the last two sections, the center of mass problem is unaltered and hence the same as Eq. (3.27), while as far as the relative motion is concerned, one simply has to replace the angular momentum I by I—a in the relative radial Eq. (3.32) while the angular eigenfunction is still exp(il). On replacing I by I — a in Eq. (3.32), it follows that in this case the radial Schrodinger equation will be [dr2

r dr

r2

2h

16h2

h2 \

(3.35)

Hence a la Eqs. (3.18) and (3.33), it follows that the energy eigenvalues for the relative motion of two-anyons in a uniform magnetic field are given by Enj = (2n +\l-a\-(l-a)

+ l)huc

(3.36)

where 0 < a < 1. We thus find that the energy eigenvalues fall into two classes that we label / and 77 : Efo)

= (2n + \)hwc, I = 2,4,...

££ 7 (a) = (2n + 2\l\ + 2a + l)fnoc ,1 = 0, - 2 , - 4 , . . . .

(3.37) (3.38)

Notice that every energy level in the class / is infinite-fold degenerate, corresponding to the fact that for every positive even value of l(= 2,4,6...)

54

Fractional Statistics and Quantum Theory

the spectrum is the same. On the other hand, if we define n\ = n + \l\, (I = 0, —2, —4,...) then it easily follows that the degeneracy of the levels in the class / / is ^^i^1^) if n is even (odd). The corresponding energy eigenfunctions are also immediately obtained from Eqs. (3.31) and (3.34) by replacing I by I — a in the later equation <;(r,^a)=e^V-«exP(-^r2)x^(^r2),

l>0

(3.39)

(3.40) A few comments are in order at this stage (1) One important difference between the class / and the class / / states is that, whereas the energy of the former is independent of a, the latter's energy is not. Further, whereas the class I states (which includes the ground state with energy tiujc) are infinite-fold degenerate, the class / / states have only finite degeneracy for 0 < a < 1. (2) The ground state energy is hioc which is independent of the anyon parameter a and further for every value of a, this state is infinitefold degenerate. This happens because of the complete cancellation between the statistical frustration (caused by the long range repulsive interaction between the anyons) and the external uniform magnetic field. (3) For each value of a (0 < a < 1), one has alternatively, Landau levels with infinite-fold degeneracy and intermediate states with finite degeneracy. Only in the case of a = 0 (bosons) and a = 1 (fermions), all states are infinite-fold degenerate and the spectrum is equi-spaced with the spacing of 2twjc. (4) Unlike a = 0,1, for 0 < a < 1 the spectrum is not equi-spaced (except for a = 1/2 when the spacing is however hujc). However, again as in the oscillator case, the alternate levels are indeed equi-spaced with the spacing of 2huic while for a = 0,1 the whole spectrum is equispaced with the spacing of 2tkoc. Thus, again for 0 < a < 1, it is not possible to build the two-anyon wave function by simple product or Slater determinant of the single particle basis functions. (5) One problem which has not been satisfactorily answered as yet is about the origin of the infinite-fold degeneracy for the alternate anyon levels.

Ch 3.

55

Quantum Mechanics of Anyons

Two Anyons in Magnetic Field with Harmonic Attraction Let us now consider the problem of two-anyons experiencing a uniform magnetic field as well as harmonic inter-particle attraction [9,8], i.e we add V(r) = \muj2r2 to the relative Schrodinger Eq. (3.35). Proceeding as above, the energy eigenvalues and eigenfunctions for this problem are (n = 0,l,2,...;Z = 0,±2,±4,...) E n ,z(a) = (2n + \l-a\ + \)hut T {I ~ ce)Juoc

*B,z(r,*,a) = W - l exp ( - ^ r

2

) ^ - ! (^r2)

(3.41)

(3.42)

where w2 = ui2, + LO2 with wc being e\Bo\/2m. Note that in this case the spectrum depends on the sign of Bo i.e. — (+) sign in Eq. (3.41) corresponds to BQ > 0 or Bo < 0 respectively. A few comments are in order at this stage. (1) From Eq. (3.41) we observe that the infinite-fold degeneracy of the pure magnetic field problem has been lifted because of the addition of the oscillator potential . However, so long as cjc/ut is a rational number, the spectrum always has some degeneracy for any a (0 < a < 1). (2) The ground state energy depends sensitively on whether Bo > (<)0. This is easily seen by writing the energy Enj(a) as EnJ(a) = (2n+l + l-a)fkutT{l-a)fkuc, = (2n+\l\+a

+ l)hut±{\l\+a)hu)c,

1 = 2,4,... 1 = 0,-2,....

(3.43)

Prom here we immediately see that for Bo > 0 the levels starting from the bosonic and the fermionic ground states meet when (3 - a)fojjt =F (2 - a)hujc = (1 + a)fkot ± ahvc

(3.44)

i.e. when a = a* = (1 — ^ £ ) . Thus for Bo > 0, the ground state energy monotonically increases as one goes from a = 0 to a = a* and then it decreases as a increases to one from a*. In other words, unlike the other two-anyon problems, the ground state energy of two-anyons is not maximum at the fermionic end but is maximum at a = a* < 1. In Fig. 3.2 we have plotted the ground state energy (E/Tvjjt) of two anyons as a function of a when Bo > 0 and x = u>c/^t = 0.75. Note,

56

Fractional Statistics and Quantum Theory

Fig. 3.2 Ground state energy of two anyons in an oscillator potential plus a uniform magnetic field Bo (> 0).

however, that for Bo < 0 the ground state energy is E0(a) = hwt + ah(ut - wc)

(3.45)

which monotonically increases as one goes from a = 0 to a = 1. (3) Whereas in the special case of u>t = wc (i.e. u> = 0), there is a complete cancellation between the statistical frustration (caused by the repulsive long range interaction between the anyons) and the external electromagnetic field, there is only a partial cancellation between the two when an external harmonic interaction is added. That is why, whereas the ground state energy in the former case is same no matter what a is, in the latter case the ground state energy is a function of a and is always more in the case of fermions compared to bosons. Note also that whereas the ground state is infinite-fold degenerate in case to = 0, it is non-degenerate in case w > 0.

Ch 3.

3.5

57

Quantum Mechanics of Anyons

Magnetic Field with Coulomb Repulsion

In Sec. 3.4 we considered the problem of two like-charged anyons in a uniform magnetic field. However, we neglected the Coulomb repulsion between the two anyons. In this section we consider the full problem taking into account the Coulomb repulsion [10] i.e. we add a potential V(r) = +e2/r to the relative Hamiltonian as given in Eq. (3.35). An explanation is in order at this point. Since our world is three dimensional, while anyons could exist only as quasi-particle excitations in certain two-dimensional systems (like in fractional quantum effect), hence we take the Coulomb potential to be e 2 /r rather than lnr. In the case of a uniform magnetic field plus repulsive Coulomb potential, the relative radial Schrodinger equation has the form

[|i+i» < ^+ ^ [dr2

r dr

r2

2h

a )

_ ^ _ ^ l 16h2

h2r

+ ^ l2 s ( r ) = 0. h \

(3.46) Because of the confining harmonic potential coming from the uniform magnetic field, the spectrum will be purely discrete even in the presence of the repulsive Coulomb potential. On splitting off the asymptotic behavior as r —* oo and r —* 0, we look for solutions of the form R(r) = r^^e-T^xW •

(3.47)

On substituting this ansatz in Eq. (3.46) and after a little algebra we find that x{v) satisfies the differential equation yx"(y) + (a- 2y2)x'(y)

+ (gy - d)x(y) = 0

(3.48)

where lmujc ,, , , me2 I 2h y= \h^rr> a = 2l-a + 1 , d=—rJ , V 2h h2 V muic 9 = r— + 2(1 - a) - 2\l - a\ - 2. nu)c

(3.49)

In general, analytic power law solutions do not exist to this equation except when the magnetic field BQ has special values. To get these solutions, let us write x(y) m a power series form oo

x(y) = J2hykfc=0

(3-5°)

58

Fractional Statistics and Quantum Theory

We then find that Eq. (3.48) reduces to a three term difference equation (k + 2){k + 1 + a)bk+2 - dbk+1 + (g- 2k)bk = 0 .

(3.51)

Clearly, if xiv) n a s to terminate at some finite value of k, say k = n, so that the highest power of y in x(u) is yn(n > 1), then the following two conditions must be satisfied g = 2n,

bn+1 = Q.

(3.52)

The condition g = 2n gives the energy eigenvalues as En,i = [n+\l-a\

+ l-(l-a)]hioc.

(3.53)

On the other hand, the condition bn+i = 0 leads to a relation between LOC, a and I i.e. for a given magnetic field one is able to obtain eigenvalues for at most one value of 1. For example, for n = 1 one finds that an exact polynomial solution to Eq. (3.48) is X(y)

= 1 + -j

(3.54)

provided g = 2 and d2 = 2a i.e. if A

-^r = {2\l-a\ + l)hwc nr and the corresponding energy eigenvalue is E=[2+\l-a\-(l-a)}huc.

(3.55)

(3.56)

The above \{v) IS nodeless and so this is an exact ground state solution but only for a specific value of I for which Eq. (3.55) is satisfied. Similarly, exact solutions (again onlyfora specific value of I) can also be derived for n = 2,3,.... Thus even though the entire spectrum is discrete, for a given magnetic field at most a few levels can be obtained analytically while the rest of the spectrum has to be obtained numerically.

3.6

Scattering of Two Anyons in Coulomb Field

So far we have discussed several problems in most of which the two anyon spectrum is purely discrete. It may be worthwhile to discuss a few examples where the spectrum is continuous and to see how the scattering cross-section changes as a function of the anyon parameter a. To that effect we now discuss the problem of two like-charged anyons experiencing the Coulomb

Gh 3.

Quantum Mechanics of Anyons

59

repulsion [7] i.e. V(r) = +e 2 /r. As explained above, we have in mind a three dimensional system in which the motion of the charged particles is confined to a plane so that anyons exist as quasi-particle excitations in certain two-dimensional systems. In this example, we also hope to study the effect of the interference between the planar Coulomb and the statistical interaction on the differential cross-section. It is worth pointing out here that scattering in two dimensions is peculiar since at low energy there is no effective-range expansion due to the presence of logarithmic singularity. As shown in the previous sections, the two-anyon center of mass motion is again that of a free particle while the relative radial Schrodinger equation in the presence of the repulsive Coulomb potential is given by

f»L + I » _ ( L ^ ! _ ^ + ^ U ) = o 2

[or

r or

2

2

r

2

h r

(3.57)

h \

where the total relative wave function is as before ip(r,<j>) — ell(^R(r), with I = 0, ±2, ±4,... in the bosonic basis. Since the motion in this case is purely continuous hence we make the ansatz R(r) = r"- a 'e' f c r X (y)

(3.58)

where E = h2k2/m, and y = —2ikr. On substituting this ansatz in Eq. (3.57) we find that x{y) satisfies the confluent hypergeometric equation [2]

yX"(y) + [(2\l-a\ + l)-y]X'(y)-^\l-a\

+ ^j + l^y(y) = 0 . (3.59)

Hence the regular solution of the Coulomb problem can be expressed as a linear combination of the partial waves with appropriate coefficients C; and is given by

^ ( r ) = J2 Cleu*r»-a\eik\F1{\l-a\ + ±+l^2\l-a\ l= — oo

^

+ l-,-2ikr\. '

(3.60) Here i-Fi(a, b; z) is the confluent hypergeometric function and (3 = me2/h2 is the inverse Bohr radius. The procedure for extracting the scattering amplitude from here is lengthy and tedious but straightforward and similar to the three dimensional Coulomb problem discussed in books on quantum mechanics [4]. We therefore omit the details but simply mention that the coefficients C\ are chosen in such a way that when the planar Coulomb distorted asymptotic plane wave is subtracted from the solution (3.60),

60

Fractional Statistics and Quantum Theory

only an outgoing phase shifted spherical wave is left, modulated by the scattering amplitude fk{4>) with <j> being the scattering angle. We obtain fk(4>)= £

_1

e «*( e X p[2*7 7 | J _ a |+t7r(|i|-|/-a|)]-l)

(3.61)

where

e^-. =

££^±i±|).

(3.62)

As expected, this expression for the phase shift reduces to the correct well known limits for (i) anyonic scattering [11] when j3 = 0, and (ii) two dimensional Coulomb scattering (e 2 /r) when a = 0. Note that in order to match our expressions with those of [11], we have to replace (f> by TT — (f> in Eq. (3.61). We now calculate the differential cross-section by summing over I in Eq. (3.61). Actually, being in the bosonic basis, we should only sum over all even integer values of I. But instead we sum over all integral values of I but take care of the bosonic basis by taking the sum of fk () and fk (
(3.63)

It turns out that the expressions are especially simple for a = 0,1/2,1 and can be written in a compact form. For example, in the bosonic and the fermionic cases, the differential scattering cross-section can be shown to be given by

da

0

1 ^ +

hf^\\

1

# = I*2 tanh {Tk ) U ? f c ^ I

• 4cos(^ln(tan2f))l ± (3 64)

W

J '

where the +(—) sign is for bosons (fermions). On the other hand, for semions (a = 1/2) we get da

-

P

cothf^f

l

+

l

| 4sin(J:lIuWf))l

(3.63)

A few remarks are in order at this stage: (1) The fermionic (a = 1) cross-section vanishes for <j> = TT/2 which is well known to be true for spin-less fermions.

Ch 3.

Quantum Mechanics of Anyons

61

(2) Notice that fk{4>) a s given by Eq. (3.61) is invariant under 4> —> —

—a. This corresponds to the combined parity and time reversal transformation PT. (3) From Eq. (3.61) it is also clear that fk( — TT) = fk( —> cj) + TT i.e. x —* —x and y —> —y.

Fig. 3.3 The Mott anyon-anyon differential cross section as a function of the scattering angle 6° (in degrees), for a = 0 (—), a = \ (• • •), a = | ( ), a = | ( ), and a = 1( ). The upper set of curves correspond to the low energy scattering with (3/k = 4 and the lower set of curves correspond to high-energy scattering with /3/fe = 0.5, and we have displayed only the a = 0, | , 1 curves, and divided the cross section by 10 for the latter set.

(4) The dimension-less differential cross-section /3*j? is a function of the scattering angle , a and the dimension-less parameter (3/k. In Fig. 3.3 [7] we denote scattering angle by 6 (rather than ) and have plotted / 3 ^ | on a logarithmic scale as a function of the scattering angle 6 for

62

Fractional Statistics and Quantum Theory

various values of a in case | = 4 as well as when (3/k = 0.5. Prom the figure we see that but for a = 0,1, the cross-section is clearly asymmetric about <j> = TT/2. Since in two dimensions the parity transformation may be taken to correspond t o i - > —x, y —> y (or x —> x, y —> —y) i.e. 4> —» 7T — 4> hence this asymmetry in the cross-section signals parity violation for 0 < a < 1. The presence of the parity conserving Coulomb term is very crucial here since the parity violation is brought about by the interference between the parity conserving Coulomb potential and the parity violating statistical interaction. In particular, the parity violating statistical interaction term alone cannot exhibit parity violation as is evident from the expression obtained in [11]. (5) The other interesting effect is the appearance of a pronounced dip in the forward angle region for low energy scattering of two anyons. For bosons (a = 0) there are clear symmetrical dips in the cross-section on both sides of <j> = ^. Now as a is increased, the forward angle dip shifts more and more towards cp = TT/2, approaching the null cross-section for a = 1. This effect is particularly spectacular for low energy scattering and is of statistical origin. Thus we have at least one example in two-anyon scattering where there is extreme asymmetry as well as pronounced dip in the differential scattering cross-section. This could be taken as one of the possible signature of anyons but it is not clear if this signal could be exploited experimentally. 3.7

Scattering of Two Anyons with Hard-Disk Repulsion

Another example of the scattering problem which can be solved analytically is that of two-anyons interacting via a hard-disk two-body potential [12]. For bosons and fermions in two dimensions, the hard-disk scattering problem has been discussed long time ago in connection with virial coefficients [13,14]. The two-anyon center of mass motion is again that of a free particle. As far as the relative motion is concerned, note that the hard-disk potential in two dimensions is given by V(r) = oo , r < a , = 0, r>a,

(3.66)

where r is the relative distance between the two anyons. Thus the relative

Ch 3.

63

Quantum Mechanics of Anyons

radial Schrodinger equation in the presence of hard-disk potential is given by

[|1 + I * \_dr2

( i ^ _ ^ r2

r Or

H2

+

! ^ W ) = 0,

(3.67)

h2 J

where the total relative wave function is as before ip{r, a, the general solution is given by

Mr A) = J2 eil*laiJ{i-a\(kr) + &jiV|,_Q|(fcr)],

(3.68)

where Jv and Nv are cylindrical Bessel and Neumann functions respectively. Since the hard-disk potential diverges for r < a, hence the wave function must vanish in this region. As a result, the boundary condition ipk(a, <j>) = 0 yields the phase shift for the l'th partial wave given by

Hence, the asymptotic wave function takes the form (3.69)

, , .n

^

(

2

V/2

ipk(r,4>) = > aA —— ^ \irkrj

e

^

d

cos kr cos<5|,_a| V

K -«|TT

.

n\

\- 5\i_a\ - - . 2 4 7 (3.70) From here it is easy to show that the net phase shift suffered by ipk (r, ) with respect to a plane wave is Aj(fc) = (J|_a + | 7 r , l>0 = Sw+a-^n,

l<0,

(3.71)

where ) is then given by

r ^ v27rt*;

(3.72)

As a result, in the bosonic basis, the differential cross section is given by d

^ = \h{)+h{-n)\2.

(3.73)

64

Fractional Statistics and Quantum Theory

By rearranging Eq. (3.72), we can split up fk{) into the pure anyonic (i.e. Bohm-Aharonov) part f{kA)() [11] and the hard-disk part f(kHD){4>):

M) = fiA)(4>) + f{kHD)(4>),

(3.74)

where .

r

f{kA\
OO

OO

-i

^ V ^ e ™ - 1) + "£e-ilHe-ia7r

- 1) ,

(3.75)

and HD)

fk

i

(0) = - _ =

r

ia7r e

°°

] P e"*(e

2W|

ia

— - 1 ) +e~ * ^

co

e-

a

*(e

2Wl

I

+« - 1 ) . (3.76)

The pure anyonic part can in fact be summed to give [11] e^sm(^

{A)

fc V

'

V2riksm(4>/2)

(3.77)

On using Eqs. (3.76) and (3.77) in Eq. (3.73) one can immediately obtain the differential cross section. As in the Coulomb scattering case (see the last section), one finds that for a = 0 (bosons) and a = 1 (fermions), the scattering cross sections are symmetric about = TT/2. Further, the fermionic cross section goes to zero at

= TT/2 due to the interference between the parity conserving hard-disk and the parity violating statistical interaction. For the special case of semions (a = 1/2) the phase shift takes a particularly simple form since the cylindrical Bessel functions reduce to the spherical Bessel functions. Hence, in that case Eq. (3.70) can be rewritten as

tan

WW = g g

(3-78)

where ji and n; are the usual spherical Bessel and Neumann functions respectively. On using the well known properties of the spherical Bessel and Neumann functions, one can show that at very low energies (fc —> 0), the semion differential cross section is given by ^ -> \\2ka dcp

irk

+ e^cosec(0)|2 ,

(3.79)

Ch 3.

65

Quantum Mechanics of Anyons

which clearly shows that the asymmetry arises from the interference term alone.

3.8

N Anyons in an Oscillator Potential

So far we have discussed several problems about two anyons experiencing various different potentials. In almost all the cases we were able to solve the problem completely and could see the interpolation as one goes from the bosonic to the fermionic end. Let us now discuss the nontrivial three and multi-anyon problems where for the first time we shall encounter the braiding effects arising from the long ranged three-body potential between anyons. As a first example, we discuss the case of JV-anyons in an oscillator potential [15,16,17,18,19]. Following the discussion in the last chapter, the TV-anyon Hamiltonian experiencing an oscillator potential is given by

H = -L VV? + mVrf] - ^- T, ^ + t~ Y £ 1T?

(3-8°)

where r^ = r» - r,- while Lij is the relative angular momentum given by (3.81)

Lij = (ri-rj)x(Pi-pj).

We now introduce the complex coordinates z = x + iy and z = x — iy and further measure the distances in the unit of 1/^mu) and also put h= 1. In this case the Hamiltonian (3.80) can be written as

f -»S*4 + ii:'*-f E ^ - f O + T H F ^ <3-82> i=l

i^tj

i=l

x

J

J

'

%^],k

J

J

where di = -J^-, dij = di — dj,Zij = Zi — Zj etc. It is worth pointing out that the total angular momentum J (as well as the relative angular momentum L) commutes with the Hamiltonian and is given by N

J = ^{zidi

- Zidi).

(3.83)

In order to find the exact solutions of the energy eigenvalue equation H*(zi,Zi) = E*(zi,Zi),

(3.84)

66

Fractional Statistics and Quantum Theory

we make the following ansatz for the energy eigenfunction * = 0(zi,Zi)x(zi,^)*,

(3.85)

(zi,zi)= \X\a,

(3.86)

where

/ 1N \ X(zi,zi)=expl--^2zizi\.

(3.87)

Here X = Y[ zij while $ is an arbitrary function. This type of trial wave function was first used by Calogero [20] in his classic study of JV particles in one dimension experiencing singular two-body centrifugal interaction. This ansatz can be understood as follows. As zt —> oo, Zij —> oo and the oscillator term dominates over the a—dependent terms and hence the factor x- The other asymptotic region is \zij\ —> 0 for any pair of particles, say i and j . Since the a-dependent terms are singular, we expect the wave function to vanish sufficiently fast in this limit and hence the factor <j>. On substituting the ansatz (3.85) in the Schrodinger equation, One can show that $ satisfies

[ - 2didi + zA + zidi-2af2^]^=\--NL

t<j

Z

ij 1

lU

~N(N - 1)1 $ . (3.88) ^

J

Notice that because of our ansatz, the three-body interaction term has disappeared from the equation for <]>. We have to choose $ to be single valued, and totally symmetric. Further, notice that $ could also have some singularities when the particles approach each other since what is only required is that ^ should be regular on the configuration boundary. We now have two types of exact solutions. (a) Suppose $> is a function of z» alone so that only the scaling operator Zidi contributes and let —j (j < 0) be the degree of scaling, then the energy eigenfunctions are E = \N - j + ^N(N

- 1)L,

$ = $(zO .

j<0

(3.89) (3.90)

It is worth adding that in this case 4* is also an eigenfunction of the total angular momentum operator (3.83) with the eigenvalue j < 0.

Ch 3.

Quantum Mechanics of Anyons

67

(b) Suppose $ is a function of t = J ^ ZiZi i.e. let \I> be a polynomial of degree n in t. Then the energy eigenfunctions are given by

r E=

i

a

2n + N + -N(N

- 1)\u

$ = L»(t)

(3.91) (3.92)

where i ^ is the associated Laguerre polynomial. Notice that the second solution is necessarily bosonic since t is symmetric. On the other hand, the first solution needs explicit symmetrization or anti-symmetrization of the wave function in terms of z\ to obtain the bosonic or the fermionic wave function. This is of course always possible and hence the degeneracy of the solutions as given by Eq. (3.89) is identical for both the fermionic and the bosonic type solutions for any given total angular momentum j (< 0). We can now combine the two solutions to obtain further j < 0 solutions. These are given by EnJ(a) =\2n-j + N+ ^N(N

- 1)L

$ = f(zi)Ll>\+a(t).

(3.93) (3.94)

Apart from these solutions it is also possible to have solutions for which cf> is given by (p=\X\~a

(3.95)

while x is again as given by Eq. (3.87). In this case, $ satisfies the equation

\-2dA + Zidi + zA + 2a Y ^ 1 $ = [- - N + ^N(N - 1)1 *. (3.96) With this choice, $ has to be of the form Xdf(zi) with d > 2a so that * vanishes if the two particles coincide. Further, as in the previous case, one can combine it with the j — 0 solutions so that the energy eigenvalues are given by Entj(a)=

[2n + j + J V - | j V ( J V - l ) L .

(3.97)

The corresponding eigenfunction can be easily written down. Caution must be exercised in choosing the value of j for these solutions since the wave function is not square integrable for all values of j . In fact, on demanding

68

Fractional Statistics and Quantum Theory

that the wave function be square integrable over the whole domain of a(0 < a < 1), one finds that j > (N - 1){N - 2)/2. The energy eigenvalues for the two solutions as given by Eqs. (3.93) and (3.97) can be compactly written in the form

EnJ(a) = Un + N + \j - ^N(N - 1)|L.

(3.98)

To date, these are all the exactly known well behaved solutions of the Nanyon problem in an oscillator potential . It is worth noting that for all the known exact solutions, the energy eigenvalues have a linear dependence on a while the corresponding eigenfunctions are finite order polynomials (apart from the factor of \X\a and the Gaussian factor). In fact, using a simple scaling argument, one can show that if the solution <j> of Eq. (3.88) is a polynomial (i.e. has a finite degree), then the corresponding eigenvalue must be linear in a. We have derived here the energy eigenvalues for the full iV-body Hamiltonian (including the center of mass) because we feel that this derivation is rather elegant and avoids all the complications that occur while deriving the exact solutions using the relative Hamiltonian. For an alternative derivation using the relative coordinates, see for example [16,21]. As emphasized before, the anyons do not appear in the center of mass and hence it is easy to convince oneself that the energy eigenvalues of the relative TV-body Hamiltonian will be given by En
= {2n + N-l

+ \l- ^N(N

- l)\)u.

(3.99)

Here I is the eigenvalue of the total relative angular momentum and the energy has been decreased by one unit to take care of the center of mass motion. For N = 2 this expression is identical to the exact spectrum (obtained by combining Eqs. (3.18) and (3.21)) so that for N = 2 these exact solutions constitute the full class and there are no missing states. However, these solutions do not include the non-holomorphic homogeneous boson and fermion states and as a result several iV-boson and ./V-fermion states (N > 2) are missing from the spectrum. For example, the Af-fermion ground state (N > 2) is missing from these exact solutions. Hence the anyon states which are close to the N-fermion ground state are also missing from these exact solutions. In fact, it turns out that these exact solutions form only a small subset among the full set of solutions of the A^-anyon

Ch 3.

69

Quantum Mechanics of Anyons

problem and the majority of solutions of the TV-anyon problem (N > 2) are not analytically known even till today. That is why, as we shall see in the next chapter, not much progress has been made regarding the question of the statistical mechanics of an ideal anyon gas. There is a basic reason for this. The point is that for N anyons, there are N - 1 relative coordinates, whereas there are N 2~ pairs of particles. These two numbers match only if N = 2. for TV > 3, the various pair-separation coordinates are not independent of each other since their number is more than N — 1. It is also worth noting that the effect of the three-body potential arising from the nontrivial braiding effects can only be felt if N > 2. What is the nature of the missing states? Significant information has been obtained in this respect by studying the three-anyon problem in perturbation theory, computing three and four-anyon spectra numerically and AT-anyon spectrum using perturbation theory. Let us therefore study in some detail the simplest nontrivial problem of three-anyons in an oscillator potential with particular emphasis on the nature of the ground state. 3.9

Three-Anyon Ground State

In order to discuss the three anyon problem it is best to separate the center of mass (which is independent of anyons) since the relative problem can then be discussed in terms of the hyper-spherical coordinates [22] p, 9,
=

Zl

+ Z2~ v6

2zs

= ^(cos s sin 4> + i sin 6 cos $)^

(3.100)

.

(3.101)

The ranges of (9,
70

Fractional Statistics and Quantum Theory

interval for 8 and into the rest of the interval. These symmetries are {9,<j>,1>}->{0 + n, + n,4}, {0,(p,ip} -> {6» + 7r,0,^ + 7r},

{6>,^}-l-0+|,0+|,V>+|}.

(3.102)

Since the relative wave functions are really functions of ui and u2, we must demand that the wave function be invariant under these three transformations which leave u\ and 112 invariant. This essentially says that the whole wave function is denned by its values in the fundamental region of the angular coordinates. Now u\,u2 have quite complicated properties under the cyclic permutation (c.p.) of the three particles :{xi,x 2 ,x 3 } -^ {x 2 ,x 3 ,Xi}. For u\ and u2 this transformation becomes

{"1,1*2} ^

j ^ « 2 - l-uu -\u2 - ^Ul J .

(3.103)

The remaining generator of the permutation group xi <-> x 2 on u\ and u2 is {«i,«2}-^{-ui,«2}-

(3.104)

Using the transformations (3.100) and (3.101), it is easily shown that the real advantage of the angular coordinates is that both the generators of the permutation group act simply on them i.e.

{#,,} ^ { M + ^ + TT}

(3.105)

{6,ci>,iP}^{-8,-4>,iP + n}.

(3.106)

Using P123 and P12, it is easy to construct projection operators which form fully (anti) symmetric wave functions for (fermionic) bosonic three-particle systems P± = ^[1 + ^123 + (Pl23) 2 ]i(l ±Pl2).

(3.107)

Radial Excitations It is worth pointing out that the variable ijj gives the overall orientation of the three anyon system. It is thus clear that its conjugate variable, the total (relative) angular momentum is conserved. Further, as expected, the total scale of the system, p, decouples from the statistical interaction. In

Ch 3.

71

Quantum Mechanics of Anyons

particular it is easily shown that the three-anyon relative Hamiltonian with harmonic interaction can be written in these coordinates as [24,23] H = H0 + -^(aH1+a2H2) mp2

(3.108)

where H

° = T~ ( " 1T2 ~ -pdp #" + T + 2m \ dp2 p2

m

W

)

= HurQad+^TO/0 2 . T

(3.109)

Here - A 2 is the Laplacian on the three-dimensional sphere i.e. A

2_

d2

~ dO2

2 sin 20 d

d2

1

cos 26» 86 + cos 2 20 d2

2*n26_d_d_ ,_J_d^ cos2 26> d di> cos2 20 <9^2'

(3.110)^

while the anyonic interactions i / i and H2 are given by - cos 20 sin 2(j)-§s + sin 26 cos 2 ^ + (cos 2<j> + cos 20) ^ 1

~

+

i cos 20(1 +cos 20 cos 20) _ 3(2 -cos 2 26) 2 ~ ( 4 - c o s 2 20)(l + cos20cos2«/))

+ C

^

°'P' (3.111) ^

*

j

Here c.p. means circular permutation 0—»0,0—> + ^,ip —> ip + ir. It is worth noting that only the angular part of the Hamiltonian is affected due to the anyons while if™ d ^s unaffected due to anyons but for the coefficient of the l/p2 term. As a result, the radial equation for bosons, fermions and anyons is the same but for the coefficient of the l/p2 term. This immediately implies a family structure for both the linear and the nonlinear anyon states. This can be seen as follows. Let us say that one has solved the angular eigenvalue equation (H™9 + Hi + H2)^ = 5(5 + 2 ) *

(3.113)

where, in general, the eigenvalue 5 is a complicated function of the anyon parameter a. One then has to solve the radial equation H Ud

o + ~^1} RM = ER(P) trip

( 3 - 114 )

j

where HQad is as given by Eq. (3.109). This is the well known radial Schrodinger equation for an oscillator potential whose solution is known to

72

Fractional Statistics and Quantum Theory

be (3.115)

En = (2n + 5 + 2)w

where n = 0,1,2,.... This immediately implies that for every solution of the anyon equation with n = 0, there are always solutions with radial excitations i.e. radial nodes for which n = 1,2,... whose energy is more by 2,4,... units respectively from the basic node-less state. Note that this argument is independent of the exact form of 5 i.e. irrespective of whether energy is linear or nonlinear in a, one always has this family structure. Further, if we replace the harmonic oscillator potential with any other p-dependent potential V(p), the effect of the anyons will remain the same in both the cases i.e. the eigenvalue S of the angular Eq. (3.114) is always the same and only the radial equation will be different for different V(p). We shall utilize this fact in Sec. 3.11 and consider solutions in case the anyons experience some other potential. The above arguments are also valid in the case of N-anyons [25,26]. In particular, the iV-anyon relative problem is best discussed in terms of the radial coordinate p and 2N — 3 angles and even in that case the decomposition as in Eqs. (3.108) and (3.109) is still valid except that now —A2 is the Laplacian on the (2N — 3) dimensional sphere and instead of - •$- one now has -—-—- -^- in HQ. Thus the family structure also exists for N-anyon case (in an oscillator potential) i.e. given any basic radial node-less state, there always exists a tower of states differing in energy by 2,4,... units. Besides, for a given V(p), the radial equation is the same for any a but for the coefficient of the 1/p2 term. Further, even in the iV-anyon case, only the angular part is affected due to anyons and hence the effect of anyons is the same for all potentials which only depend on p. Let us first consider the three-boson or the three-fermion spectrum. In that case a can be taken to be zero and further the spectrum of —A2 is k(k + 2) with k = 0,1,2,.... The resulting radial equation as obtained from Eq. (3.109) is nothing but the Schrodinger equation for a particle in four dimensions experiencing an oscillator potential . Thus the spectrum of three bosons or three fermions in an oscillator potential is given by Entk = (2n + k + 2)OJ . The corresponding fermionic (bosonic) eigenfunctions ip^

VfiL« = F^ip)Yiftl{6,

,i>) •

(3.116) ul

are given by

(3.117)

Ch 3.

Quantum Mechanics of Anyons

73

Here the four quantum numbers n, k, v, I are associated with the four coordinates p, 9, (f>, t/}. In particular, n = 0,1,2,... is the radial quantum number while k,v,l are the angular quantum numbers. Whereas k = 0,1,2,..., the other two quantum numbers (y, I) have the same parity as k and are restricted by \v\,\l\ < k. The condition that v and I have the same parity, comes from the boundary conditions on the fundamental region at ip = ±?r and = ±7r/2. They arise by noticing that points on different boundaries of the fundamental angular domain correspond to the same points u\,U2There are further restrictions depending on the statistics. In particular, to obtain fully symmetrized or anti-symmetrized states, the wave functions must be eigenvectors of the projection operator constructed in Eq. (3.107). This requires that v = 3q with q being an integer. Lastly, v > 0 can be assumed to avoid double counting. The normalized radial eigenfunction with measure (p3dp) is given by Fn,k{p) = Nn,k+1exP(-(/2)(k/2Lkn+l(0

(3.118)

where C, and Nn>k+i are defined by C = rn.p\Nn,k+1=m^v-^^/

(3.119)

The normalized angular eigenfunction with measure (cos 26d9d(j)dip) is given by

y£5 = ^ [ l ± ( - l ) ' + n ' ] < M , Qfon/= 0, Yk% = ^\^v,l)±{-l)l+n'(k,-v,l)}

forz,>0,

(3.120) (3.121)

and (k,v,l) is defined by (k, v, I) = C felV ,,e^(sin 29)eiv+eil* .

(3.122)

The numbers n',a,/3 are given in terms of the angular quantum numbers k, v, and I by 2n' = k-max{v,

\l\},a=

) - \ v + l\,(3=

\\v-l\.

(3.123)

The function O^f is defined in terms of the Jacobi polynomials P"P by G^Or) = (1 - x)a'2{l + xf'2P^{x)

(3.124)

74

Fractional Statistics and Quantum Theory

while Ck,u,i is the normalization factor for the angular eigenfunction Ck

_

1

/n'!(n' + a + /3)!(fc + l)

'"' ~ V 2 " + / V + <*)!(«' + /*)!

(3.125)

where for v = 0, Cfc,,,,/ has to be divided by an extra \/2We now use the above formalism to calculate the three-anyon ground state energy to O(a2) around the fermionic ground state since this state is missing from the exactly known anyonic spectra as given by Eq. (3.99). It is worth adding here that the energy of the three-anyon state around the bosonic ground state is already exactly known and it is also known that the first order in a correction to the three-boson ground state energy reproduces this exact energy [24,27]. To avoid the confusion, we use a = 0 (0 = 1) to denote boson and 0 = 0 (a = 1) to denote fermion where 0 = 1 — a. Thus the three-anyon Hamiltonian around the fermionic basis will be denoted by (see Eq. (3.108)) H = H0 + - ^ ( / 3 i f i + P2H2)

E{Q0)=4UJ.

(3.126) (3.127)

The corresponding, normalized, radial eigenfunction which follows from Eqs. (3.118) and (3.119) is F0,2(P) = ~Ce-^.

(3.128)

In order to obtain the corresponding angular eigenfunction, notice that v = 3<7 < k = 2 and hence v = 0. It then follows from Eqs. (3.120) to (3.123) that I = 0 and hence n' = 1 and a = /3 = 0. Thus the normalized angular wave function of the fermionic ground state is given by

^,0,0 = ^ 5 ^ 2 0 .

(3.129)

The first order (in 0) correction to the three fermion ground state energy i.e. (V'o^oo\fiH± iWi^oo) *s n o w easily calculated and shown to be zero. As far as the second order (in 0) correction is concerned, there are two contributions, one from H\ and the other from H2 and both are easily

Ch 3.

Quantum Mechanics of Anyons

75

calculated. The second order (in (3) contribution from H2 is obtained at first order in perturbation theory i.e. E£\/3)

= /^(VfeAolWSo.o) •

(3-130)

On using Eq. (3.117) it becomes

4 1} (/3) = p2(F0>2(p)\±\F0,2(p)){Y2^0\p2H2\Y2^0).

(3.131)

Using FQ:2(P) as given in Eq. (3.128) and the integration measure p3dp, we obtain {FO,2(P)\\\FOMP))

(3.132)

= \rnw. A

P

The angular integral has the measure cos 28d0d(f>dtp and hence using Eqs. (3.112) and (3.129) the matrix element is given by

(Yto>2H2\Y2^o) = ^ [i - In (£)] .

(3.133)

On combining Eqs. (3.132) and Eqs. (3.133) we then find that the second order (in /?) contribution to the three anyon ground state energy from H2 is positive and given by

41)03) = ^ [ l - 2 I n ( | ) ] .

(3.134)

The second order (in /?) contribution from Hi is obtained at second order in perturbation theory. Since the three-fermion ground state is nondegenerate, hence this contribution has the simple form ,

,

n,k,v,l

(-^0,2 — hn,k) '

'

(3.135)

In the summation, only the I = 0 states have nonzero matrix elements because both ^0,2,0,0 and Hi are ^-independent. This also implies that k and v are even integers since k, v, I have the same parity. Thus we define k = 2p(p = 0,1,2,...) and u - 2q(q = 0,3,6,...). The constraint v < k implies that q < p. On separating the matrix element (V>i~2p 2q,o l-^i I ^02 0 0) into a radial and an angular part, it easily follows from Eqs. (3.118), (3.119) and (3.128) that

(FnM\—2\FoAp))=^;^n

+P

-'l

(3-136)

76

Fractional Statistics and Quantum Theory

As far as the angular part is concerned, we first observe that the three terms obtained by cyclic permutations give equal contributions. Further, only the d/dd term in Hi contributes since Y^QQ is a function of 6 alone. In this way, we find that the angular matrix element vanishes trivially due to ^-integral for states where p — q is an odd integer. Similarly it also vanishes when v = 2q = 0. On the other hand, when p — q is an even, non-negative integer (with q > 0) then the angular matrix element is given by [23] (^.olV^il^.o.o) _ 6(-l) V6(2p + l)(p - g)!(p + g)l + 1 , g _! P 2«(p+l)! P-9 (0) ~

(3 137)

-

where P™@(x) is the Jacobi polynomial. On combining Eqs. (3.135) to (3.137) we find that the O(/32) contribution to the three anyon ground state energy from H\ is given by

2

2

oo

oo 3r
4 V) = -i8c/? £ £ £

[pni^m2

n=0p=3 r=l

( 2 p + l ) ( p - 3 r ) ! ( p + 3r)!(n + p - 2 ) ! ( n + p - l ) ! 2 6 -[(p-l)!] 2 (2p + n + l ) ! ( n ) ! " (3.138) Note that in the sum over r(q = 3r), it is required that p — 3r be an even integer. A numerical summation including p < 51 and n < 10000 gives E^\/3) = -(0.6118 ± 0.0006)w/32 .

(3.139)

Subsequently, Chou [16] has analytically evaluated the second order (in /?) contribution from Hi to the three anyon ground state energy and shown that

E^W

= - ^ [l - 3 In ( j ) ] < ^ 2 = -0.6163a;/32

(3.140)

thereby showing that the numerical estimate has been pretty good. On comparing Eqs. (3.138) and (3.140) we have a remarkable formula for the complicated triple summation over the Jacobi polynomials. It would be nice if some one could directly prove this triple summation formula. We can now combine Eqs. (3.127, (3.134) and (3.140) to write the total three-anyon ground state energy near the fermionic end, which is valid to the second order in (3: E0{p)=

|"4 + 4 . 5 1 n r ^ / 3 2 + O(/3 3 )|w^[4 + 1.29/32 + O(/33)]a;. (3.141)

Ch 3.

Quantum Mechanics of Anyons

77

Remarkably enough, we find that the O(/32) contribution to be positive i.e. near the three-fermion ground state, the three-anyon effective interaction is repulsive (and not attractive as one would have naively expected from the two-anyon problem). Note that there is a repulsion between two anyons which monotonically increases as one goes from the bosons to the fermions. This, I believe, is one of the most profound and least understood result that has emerged so far from the anyon dynamics. This highly nontrivial result obviously has to be attributed to the braiding effect arising from the three-body interaction between the anyons. Had there been no three-body interaction, then, near the fermionic end, the O{02) contribution coming purely from the two-body interaction would have been negative. Recall that in the two-anyon problem, one does find that the ground state energy monotonically decreases as one goes from the fermionic to the bosonic end. In other words, one has obtained the most intriguing and highly nontrivial result that near the bosonic as well the fermionic ground states, the threeanyon effective interaction is repulsivel It may be noted that the energy of the exact three-anyon eigenstate which starts from the bosonic ground state (at E = 2u>) is given by (see Eq. (3.99)) E(/3) = [2 + 3(1 - p)]w .

(3.142)

On comparing the three-anyon ground state energies starting from the bosonic and the fermionic ground states as given by Eqs. (3.141) and (3.142), we see that the two curves cross at (3* ~ 0.29 i.e. a* = 0.71. In other words, the three-anyon ground state energy is maximum at a* = 0.71 (and not at a = 1 as one would have naively expected) and at this point the ground state is two-fold degenerate. Thus the perturbative calculation from either the fermionic or the bosonic end will break down beyond a*. Summarizing, the three-anyon ground state energy monotonically increases from 2u> at a = 0 and reaches its maximum at a = a* = 0.71. As a further increases from a* to 1, the ground state energy monotonically decreases to 4w at a = 1. This is shown in Fig. (3.4).

3.10

General Results for N Anyons

The highly nontrivial result for the three-anyon ground state raises several questions.

78

Fractional Statistics and Quantum Theory

(1) Are there similar crossings in the ground state of the multi-anyon systems (N > 3)? (2) Are there similar crossings in the excited states? (3) Is the ./V-anyon effective interaction (N > 3) near the fermionic end always repulsive? (4) Are there avoided crossings in the ./V-anyon spectra? (5) Is the ./V-anyon system integrable?

Fig. 3.4

Spectrum of three-anyons in an oscillator potential.

Over the years, significant progress has been made in answering these questions at least qualitatively. For example, the three-anyon low lying spec-

Ch 3.

Quantum Mechanics of Anyons

79

trum in an oscillator potential has been numerically calculated [28,29]. In Fig. 3.4 [29] we have plotted the low lying spectrum of the relative Hamiltonian (in units of ui) as a function of the anyon parameter 6 (= it a). The angular momentum assignments of the bosonic states emerging from energy 2,4,5,6,7 (in units of to) are given by [0], [0,±2], [±1,±3], [02,±2,±4] and [±1 2 , ±3 2 , ±5] respectively with the upper power referring to the degeneracy of the angular momentum states. From the spectrum, one finds that there are two kinds of energy levels in the spectrum. In particular, apart from the exactly known linear states for which energy changes by ±3w as one goes from the bosonic to the fermionic end, there are several nonlinear states (i.e. where E is a nonlinear function of a) for which energy changes by ±u> as one goes from the bosonic to the fermionic end. Whereas all the linear states are analytically known, not even one exact solution is known as yet involving the nonlinear states. In fact this is the main difficulty in the three and multi-anyon problems. For example, the nonlinear state starting from the three-fermion ground state at 4ui, has an energy of 5UJ at the bosonic end. Note that the perturbative calculation if valid till /3 = 1 (note (3 = 1 — a) would predict the energy at the bosonic end to be 5.29u> rather than 5w, but of course the perturbation theory is expected to breakdown beyond/?* (=0.29). There are plenty of true crossings, but no avoided crossings in the threeanyon spectra. All energy levels are monotonic, i.e. dE/da does not change sign for 0 < a < 1. The numerical calculations also confirm the perturbative result about the cross-over between the ground states (and also the fact that there is only one cross-over). In fact both the calculations agree upto first two decimal places regarding the value of a*(= 0.71) at which the crossing occurs thereby showing the reliability of the perturbative calculation. Several pairs of states cross at the semionic point (a = i ) . Indeed, all the nonlinear states and those exactly known linear states which have slope —3u> have another state in the same family which is related to it by a mirror reflection around a = 1/2 [30] and hence most of the semion spectrum is at least two-fold degenerate. It would be interesting to explore the origin of this extra degeneracy in the semion spectrum. Let us now discuss the four-anyon spectrum which has been numerically calculated [31]. In Fig. 3.5 [31] we have plotted the low lying spectrum (in units of u>) of the relative Hamiltonian in an oscillator potential as a function of the anyon parameter a. The angular momentum assignments of the bosonic states emerging from energy 3,5,6,7,8 (in units of to) are

80

Fractional Statistics and Quantum Theory

Fig. 3.5 Spectrum of four-anyons in an oscillator potential.

[0], [0, ±2], [±1, ±3], [03, ±2 2 , ±42] and [±1 3 , ±3 2 , ±5] respectively. Here the upper power refers to the angular momentum degeneracy of the states. Several conclusions can be drawn from the figure. (i) There are both linear and nonlinear states in the spectrum out of which all the linear states are analytically known (see Eq. (3.99)) while not a single nonlinear state is analytically known as yet. (ii) Whereas the energy changes by ±6w for the linear states as one goes from the bosonic to the fermionic end, it changes by ±4u>, ±2w or 0 for the nonlinear states as one goes from the bosonic to the fermionic end.

Ch 3.

Quantum Mechanics of Anyons

81

(iii) Contrary to the three-fermion ground state (which is nondegenerate), the four-fermion ground state is three-fold degenerate with angular momenta I — 0, ±2. Whereas the I = 0 state has zero slope, the I = ±2 levels have nonzero slopes of equal magnitude but opposite sign (with I = — 2 being pushed down and I — +2 being pushed up). This fact persists for all energies. (iv) As in the three-anyon case, there is one cross-over in the four-anyon ground state. In particular, the linear state beginning from the bosonic ground state (at 3w) and the nonlinear state beginning from the fermionic ground state (at 7u> with I = —2) cross approximately at a* ~ 0.55. Thus at this point the four-anyon ground state is two-fold degenerate (just as the three-anyon state is at the cross-over point of a* = 0.71). (v) One very important difference between the three and the four-anyon spectra is that unlike the three-anyon case, in the four-anyon problem, the ground state energy monotonically decreases as one goes from the fermionic to the bosonic ground state. In particular, whereas the effective threeanyon interaction near the fermionic end is repulsive, the effective fouranyon interaction near the fermionic end is attractive and not repulsive. (vi) Another surprising result is the observation of a Landau-Zener type (or so called avoided) crossing at E = 12w between states with a nonlinear a dependence and the same angular momentum thereby suggesting that the many-anyon system may be non-integrable. We shall discuss the issue of integrability of the 7V-anyon system at the end of the chapter. By now the results about the three and four anyon spectrum (in an oscillator potential), have been generalized to the 7V-anyon case and the following conclusions can be drawn in the iV-anyon case: (1) In the 7V-anyon spectrum, there are both linear and nonlinear states and there are tower of states differing in energy by 2,4,... units from each basic state having zero radial node. All the linear states are analytically known with the spectrum being given by Eq. (3.99). For these states the energy changes by ±N(N2~1>)LJ as one goes from the bosonic to the fermionic end. On the other hand, for the nonlinear states, the energy (in units of w) changes by ^ ^ - 2, ^ [ = 1 1 - 4,..., - M ^ i ) + % While no general proof exists for this fact, it is supported by the WKB analysis [32,25] and is consistent with the numerical calculations for three and four anyons. (2) For any iV(> 3) there is always a cross-over in the ground state. In fact, it has been shown [33] that for large N, there are at least -
82

Fractional Statistics and Quantum Theory

number of crossings in the ground state and hence the ground state is only piecewise continuous. The fact that there must be a cross-over between the ground states for TV > 2 can be seen as follows. The Nanyon state starting from the bosonic ground state has energy (TV— l)u> at a = 0 with angular momentum I — 0 (see Eq. (3.99)). As a goes to one, this anyonic state has energy E = [(TV — 1) H— 2~ 1^ a n d angular momentum 2 • Clearly this is not the TV-fermion ground state but is an excited state. The TV-fermion ground state is obtained by filling the one particle oscillator levels from bottom to top. One can show that the total angular momentum of the /V-fermion ground state is always less than | ^ | — ^ | and its energy (excluding the center of mass motion) is <

=

r_2

+

(2TV_ t r) V i +

8(;v_r)_

^ ^

Hence there must always be at least one crossing in the ground state of the TV-anyon system (TV > 3). Here r denotes the number of electrons in the fc'th shell while (k — 1) shells are completely filled so that TV = r + fc(fc~1} (note that k = 2,3,... and hence TV - r - 1,3,6,10,...). Another way to see the crossing is to note from Eq. (3.143) that for large TV, the ground state energy of TV-fermions goes like TV3/2 while the state starting from the bosonic ground state has energy going like TV2 at the fermionic end. (3) The TV-anyon interaction around the fermionic ground state depends crucially on the value of TV. For example, when TV = 3,6, • • • (i.e. when V8TV + 1 is an integer) then at the fermionic point one has a closed shell so that the TV-fermion ground state is unique with zero angular momentum. In this case one can rigorously show that the effective TV-anyon interaction near the fermionic ground state is always repulsive. On the other hand, when TV does not correspond to these magic numbers, then the effective TV-anyon interaction near the fermionic ground state is always attractive. This can be proved in perturbation theory by noting that in these cases the TV-fermion ground state is degenerate and states with I — ±p(p being an integer) have non-zero and opposite slopes so that all states with I = — p will be pushed down because of the anyonic perturbation. However, we are not aware of any non-perturbative proof of this fact (It is of course clearly true for TV = 2 and 4). The fact that the effective TV-anyon interaction near the TV-fermion ground

Ch 3.

Quantum Mechanics of Anyons

83

state (in an oscillator potential) depends so sensitively on the value of TV (i.e. if shells are closed or not), raises doubts about the validity of the approximate schemes like mean field theory because they do not take into account this important fact. (4) There are several true crossings and a few avoided crossings in the excited state spectrum of the TV-anyon system. 3.11

TV Anyons Experiencing a TV-body Potential

As we have seen, so far only a class of exact solutions have been obtained in case TV-anyons (TV > 3) experience a harmonic oscillator potential. Further, all the known exact solutions are such that the energy eigenvalue spectrum is linear in the anyon parameter a. Besides, there is a cross-over in the ground state for any number of anyons (TV > 3). It is clearly of interest to enquire whether one can also obtain exact solutions in case N-anyons are experiencing some other potential and whether in those cases also the energy vary linearly with a. Further, is there a crossover in the ground state and if yes then at what value of a does it occur? The purpose of this section is to present one such example. In particular, we obtain a class of exact solutions in case TV-anyons are interacting via the JV-body potential [34] V(xi,x2)...,XAr) =

,

6

==• y jV Y^i<ji i ~ x j ) K

(3.144)

Note that even though this TV-body potential may appear to be very complicated, we shall see that, in terms of the hyper-spherical coordinates, it is just the Coulomb-like potential — e2/p and as is well known, the only two radial potentials for which the Schrodinger equation can be solved for all partial waves are the oscillator and the Coulomb potentials. In fact, it has recently been shown [35] that the -/V-body problem in one dimension with the inverse square interaction is analytically solvable when the TV particles are also interacting via the TV-body potential of the form as given by Eq. (3.144). The interesting point is that unlike the oscillator case, the exactly known energy spectrum here is not linear in a. However, (E)~1/2 is linear in a. Further, a la the oscillator case, these exact solutions include the ground state of TV-bosons but not the ground state of TV-fermions (TV > 3). We therefore perturbatively calculate the ground state energy of three-anyons near the fermionic end and show that for this potential

84

Fractional Statistics and Quantum Theory

also there is a cross-over between the ground states. We shall show that a similar cross-over must also occur in the case of N anyons (N > 4). The 7V-anyon relative problem is best discussed in terms of the hyperspherical coordinates in 2N — 2 dimensions i.e. in terms of the radial coordinate p and 27V — 3 angles and even in that case the decomposition as in Eqs. (3.108) and (3.109) is still valid except that now —A2 is the Laplacian on the (2N — 3) dimensional sphere and instead of | -§- one now has (2iV~3) A. in Ho. On using the fact that (i) only the angular part of the Hamiltonian is affected due to the anyons, (ii) the angular part is independent of the radial potential V(p) between the anyons and (iii) the radial equation for TV-bosons, TV-fermions and /V-anyons is the same but for the coefficient of the 1/p2 term, one can immediately write down a class of exact solutions for TV-anyons experiencing the TV-body potential (3.144). In particular, let us first note that the relative Hamiltonian for N particles can be formally written as H = Hrad +

mp2

(3.145)

where

V

(3.146) *<J

As remarked before, Hana is the same as in the oscillator case and only jjrad j s different. Now suppose that we have solved the angular eigenvalue equation Hangy =

^

+ 2]V

- 4)*

(3.147)

where S is obviously the same as in the oscillator case which in general is a complicated function of a. As a result, the radial Schrodinger equation in this case is (3.148)

This is a standard radial equation for the Coulomb problem whose solution is immediately written down [4]. In particular the energy eigenvalues are given by (3.149)

Ch 3.

Quantum Mechanics of Anyons

85

where E = me4e/2. It is worth repeating that as such this solution is not of much use unless one can determine 5 as a function of a. On using the fact that S is the same as for the exact solutions in the oscillator case, it is straightforward to figure out 5 and write down the exact solutions. In particular, on comparing with the oscillator exact solutions as given by Eq. (3.99) it is easily seen that 6 = I and hence the energy eigenvalues for the exact solutions are given by £nM) =

-in+\l-m^a\+N-=W

(3J50)

where I is the eigenvalue of the relative angular momentum operator. Several comments are in order at this stage. (1) Unlike the oscillator case, the exactly known spectrum here is not linear in a. However, (—e)~xl2 is indeed linear in a for all these solutions. (2) It is worth pointing out that for N = 2, the expression (3.150) gives the complete spectrum. For N > 3 however, it does not give the complete spectrum. For example, a la oscillator case, the ./V-fermion ground state is missing from these exact solutions and as in the oscillator case, one can show that a level-crossing must occur in the true ground state of the A^-anyon system (N > 2). (3) What is the nature of the missing states in the iV-anyon spectra? It turns out that whereas for the exact solutions (—E)~~xl2 is linear in a, for all the missing solutions (—E)~xl2 will have nonlinear dependence on a. Further, while all those states for which (—E)~1/2 varies linearly with a are known analytically, not even one of the states, for which (—E)~xl2 varies nonlinearly with a is known analytically. (4) Even though the nonlinear states are still not known, it follows from Eq. (3.149) that both the linear and the nonlinear states have a family structure i.e. once a basic solution with zero radial node is there, then one also has solutions with 1,2,... nodes and for them (—e)"1/2 is more by 1,2,... units respectively from that of the corresponding node-less solution. (5) Regarding the missing nonlinear states, it is clear that once we can obtain some result for them in one potential, we can always translate the results to the other case. In particular the entire discussion given about the N-anyon spectrum in the oscillator case applies here. For example, whereas for all the analytically known states (—e)"1/2 changes by ±N(N—l)/2, for the missing states it will change

86

fractional i

N(N-l)

o

Statistics and Quantum Theory N(N-l)

.

N(N-l)

,

n

-ii

<•

,i

by —^2—L - 2, —^—L - 4,..., ^—'- + 2 as one will go from the bosonic to the fermionic end. Further, by borrowing the results from the oscillator case, the energy of the missing low lying nonlinear states can be easily obtained in case iV = 3 or 4. (6) It is easily shown that as in the oscillator case, in this example too there are crossings in the 7V-anyon ground state. The easiest way to see this is to note that the 7V-anyon state starting from the bosonic ' 2 ~ 1 '. Clearly this is not the ground state has angular momentum A^-fermion ground state but is an excited state since one can show that the total angular momentum of the ./V-fermion ground state is always less than W*- 1 )!. (7) For the case of iV = 3, we can immediately borrow all the known results about the nonlinear states in the oscillator case and obtain corresponding conclusions in our case. For example, in the three-anyon case, (-e)" 1 / 2 changes by ±3 in case of the exactly known solutions, while it changes by ±1 in case of the missing states as one goes from the bosonic to the fermionic end. In particular, the three-fermion ground state at e = —4/49 will interpolate to the bosonic state at e = —4/81 and near the fermionic end, the energy of the corresponding nonlinear anyonic state will be given by (see Eq. (3.141)) £=

" [ 3 . 5 + 4.51n(|)(l-a) 2 ]2-

(3 151)

"

On the other hand, the anyonic state starting from the bosonic ground state at e = —4/9 is given by (see Eq. (3.150)) £=

-(TTW-

(3 152)

-

The two curves cross at a = 0.71. It is a curious numerical fact that for both the oscillator and the iV-body case, the cross-over occurs at almost the same point. (8) Finally it is worth pointing out that the degeneracy of the exact energy levels coming from the angular part is same for both the oscillator and the N-body potential (note the same factor \l ' 2~ ' a\ occurs in both the cases). This will in fact be true for any anyon potential which depends only on p. However, the degeneracy coming from the radial part is different in the two cases since whereas in the oscillator case one has the factor 2n + \l— N(-N2~1'a\, in our case the corresponding factor is n + \l - N(N2~l^a\. As a result, compared to the oscillator case, here

Ch 3.

Quantum Mechanics of Anyons

87

the degeneracy is much more. In particular, for a given energy, both even and odd angular momentum states are present in the spectra. As a result, if one plots (—e)"1/2 as a function of a, then one will find that one will not only have those levels which are present in the oscillator spectrum but there will be few extra states in our case which are not there in the oscillator case. For example, in the three-anyon case we have an extra state for which (—e)"1/2 changes linearly from 1.5 to 4.5 as one goes from the bosonic to the fermionic end. (9) Are there other potentials for which a class of exact 7V-anyon eigenstates can be found? We believe that the answer is no since only the Coulomb and the oscillator problems are analytically solvable in N dimensions (N > 2) for all partial waves. Of course in both cases one can always add the potential V(p) = g2/p2 and still the problem will be solvable analytically but now the degeneracy will be much less. All other potentials are at best quasi-exactly solvable and hence for them, eigenstates could be analytically obtained, if at all, for only specific values of the angular momentum I.

3.12

N Anyons in a Uniform Magnetic Field

The problem of ./V-anyons in a uniform magnetic field is more than just an academic problem. In the only known physical application of anyons (i.e. in the fractional quantum Hall effect) and in most of the other proposed realizations of the fractional statistics, the anyons feel an effective external magnetic field. Let us therefore discuss this problem [36,37,38]. Later on we will also consider the case of ./V-anyons experiencing a uniform magnetic field as well as an external harmonic interaction. The Hamiltonian for ./V-anyons experiencing a uniform magnetic field B is given by

H^^-jrL-a^^^-eAir,)}2

(3.153)

where A(TJ) = ^ z x r j is the external vector potential in the symmetric gauge. Without any loss of generality we choose B > 0 since the spectrum in this case is independent of the sign of B. After some algebra one can

88

Fractional Statistics and Quantum Theory

rewrite this H as 1

F

v^fr

v^zxiijl2

a

2

2

221

T

= ^ E { [ P « - E ^ ^ j +m u,yij-ucJ+

N(N-l)

2

Va

(3.154)

where J is the total angular momentum as given by Eq. (3.83) and u>c = e\B\/2m is the cyclotron frequency. We see that the Hamiltonian in the uniform magnetic field case is the same as in the oscillator case except for a constant term proportional to N(N — 1) and a term proportional to the total angular momentum J. Hence we can immediately write down the exact solutions for this case since the exact wave function is also an eigenfunction of the angular momentum operator J. Clearly the eigenvalues will be shifted from the oscillator eigenvalues by a constant as well as a term proportional to the eigenvalue of the total angular momentum. In particular, the exact energy eigenvalues (including the center of mass) for the problem of iV-anyons in uniform magnetic field are given by (n = 0,1,2,...) Enj(a) = \^n+N+\j-

N{N

~

l)

a[] u>c- (j-

^

^

1}

a ) LOC . (3.155)

The extra term (i.e. the last term on the right hand side) compared to the oscillator case (see Eq. (3.98)) changes the spectrum non-trivially. In particular, observe that the ground state energy is given by E0(a)=Nuc,

(3.156)

which is in fact independent of a and is infinite-fold degenerate a la the conventional Landau levels. Note that no matter what j is, so long as j > N(N — l)/2 the ground state energy is always Nuc. Further, because of the family structure, one has radially excited levels with n nodes having energy En(a) = (2n + N)wc, all of which are also infinite-fold degenerate. On the other hand, for j < 0 the energy eigenvalues take the form EnJ{a)

= [2n + N + 2\j\ + N(N - l)a] uc.

(3.157)

All these levels have only finite degeneracy and for all of them, the energy for a given n,N, and \j\, monotonically increases as one goes from the bosonic to the fermionic end. It may be noted that as in the oscillator case, one has only been able to obtain analytically, a class of exact energy eigenstates for all of which the energy eigenvalues vary linearly with a.

Ch 3.

Quantum Mechanics of Anyons

89

However among the huge class of nonlinear states not even one is known analytically as yet. Finally let us consider the case when the anyons experience an external harmonic interaction in addition to the uniform magnetic field. Following the oscillator problem as well as the above discussion, it is easy to see that H for the combined system is given by

(3.158) where

J^ = Jl + <J.

(3.159)

Hence the energy eigenvalues for the exact solutions (including center of mass) are now given by

EnJ(a)= L + A r + l i - ^ ' ^ a l l ^ T ^ - ^ " ^ ^ ^ (3.160) where the upper (lower) sign corresponds to B > 0(B < 0). We shall see below that the results now crucially depend on the sign of B. It is interesting to note that the infinite-fold degeneracy of the pure magnetic field case has been lifted by the addition of the external harmonic potential. Further, the ground state energy is now a function of a and its value crucially depends on whether B > (<) 0. Since the discussion for arbitrary N is quite complicated, hence, for illustration, we now discuss the case of three-anyons in some detail [39]. Three-Anyons in a Uniform Magnetic Field Plus an Oscillator Potential For N = 3, the exact energy eigenvalues (3.160) take the form Enti(a) = (2n + 2 + \l - 3a|)wt ^ (I - 3a)wc .

(3.161)

Note that here we have written down only the exact energy eigenvalues for the relative Hamiltonian since, as we shall see shortly, this will facilitate the discussion about the nature of the ground state of the three-anyon system. To motivate the discussion, let us first consider the spectra of threebosons or three-fermions in an oscillator potential plus uniform magnetic field. Using Eq. (3.116) it follows that the spectrum for the bosons or

90

Fractional Statistics and Quantum Theory

fermions is given by [39]

Eniktv,i = (2n + k + 2)a;tTlUc

(3.162)

in terms of the quantum numbers in the hyper-spherical coordinates. Difference between the fermionic and the bosonic spectra occur because different constraints are imposed on the values of n, k, u, I by the statistics. For example, when u> = 0 (i.e. u>t = u>c), the infinite-fold degenerate bosonic ground states with energy 2uc are \n = 0, k = 2p, v = 0,1 = ±k) and |0, k, v ^ 0, ±fc) while the infinite-fold fermionic ground states with energy hojc are |0, k, v ^ 0, ±fc). Similarly all the excited states can be written down, all of which are infinite-fold degenerate. As the harmonic interaction is introduced, this infinite-fold degeneracy is lifted and each level has only finite degeneracy. When u / 0 , then clearly u>t > coc and Eq. (3.162) implies that the unique bosonic ground state is |0, 0,0,0) with energy 2u>t- On the other hand, the fermionic ground state depends crucially on whether B > 0 or B < 0 and the ratio u>c/u)t. In particular, the fermionic ground state is |0,3, 3,3)(|0, 3,3, -3)) with energy 5wt - 3wc if B > 0 (B < 0) and u>t < 3wc, while it is |0, 2,0,0) with energy 4u>t if u>t > 3wc (see the discussion in Sec. 3.9). In particular, at tut — 3uic the doubly degenerate fermionic ground states are |0, 2,0,0) and |0,3,3,3) (or |0, 3,3,-3)) with energy 4wt(= 12wc). Let us now work out the three-anyon ground state in the various cases by making use of the exact solutions with energy as given by Eq. (3.161) and the perturbative results around the fermionic end as discussed in Sec. 3.9. First of all, notice that the three-boson ground state is |0,0,0,0) with energy 2u>t and it interpolates to the state |0,3,3, —3) in the fermion spectrum at a = 1 with energy 5ut ± 3toc (if B > or < 0). We thus see that the results about the anyonic ground state will crucially depend on the sign of B. Let us therefore discuss the two cases separately. (i) B < 0 : The energy of the three-anyon exact solution interpolating to the three-boson ground state |0,0,0,0) is given by (see Eq. (3.161)) £n=o,;=o(oO = (2 + 3a) ut - 3a wc.

(3.163)

For a = 1, its energy is 5wt — 3wc which corresponds to the energy of the fermionic state |0,3,3, —3). This state will be the fermionic ground state for values of the magnetic field for which it has lower energy than the |0,2,0,0) state (whose energy is 4u>t). Thus we conclude that the exact solution as given by Eq. (3.163) is the exact three-anyon ground

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Fig. 3.6 Ground state energy of three-anyons in oscillator potential plus uniform magnetic field in case B < 0 and (a) 0 < x = uic/^t < 1/3 (b) 1/3 < x < 1.

state for all values of a so long as 5u>t — 3wc < 4u>t i.e. u>t < 3wc. In this case, the ground state energy (3.163) monotonically increases as one goes from the bosonic to the fermionic end. On the other hand, when ujt > 2>u)c then the state |0,2,0,0) is the true three-fermion ground state. Hence, in this case, there must be at least one cross-over in the ground state of threeanyons. The value of a at which the cross-over occurs can be determined perturbatively using the results obtained in Sec. 3.9. These results were actually obtained for the case of an oscillator potential alone, but are trivially extended to the case at hand, since the fermionic ground state |0, 2,0,0} has zero angular momentum. The ground state energy of three-anyons near the fermionic end is given by (see Eqs. (3.127), (3.134) and (3.140) with UJ being replaced by u>t)

Eo(a)= [ 4 + ^ ( l - a ) 2 l n Q ) L t + O(l-a) 3 .

(3.164)

On comparing the two energies as given by Eqs. (3.163) and (3.164) we see that the two curves cross at , ,, - ( 1 - x) + [(1 - xf + 2(1 - 3x) ln(4/3)]V2 1-<*!(*) = ^ ^

(3-165)

where x = uc/u}t. Prom here we see that the cross-over point a\ (x) increases from 0.71 to 1 as x increases from 0 to 1/3. Note that beyond x = 1/3, the state |0,2,0,0) is no longer the ground state. In Fig. 3.6 [39] we have plotted the three-anyon ground state energy (in as a function of a in case B < 0 and (a) x = ujc/ujt = 0.25(< 1/3), and (b) x = 0.6(> 1/3).

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(ii) B > 0 : The three-anyon state which interpolates to the three-boson ground state is again the state with n = I = 0 but the energy of that state is now given by En=0,i=o = (2 + 3a)cot + 3aojc .

(3.166)

At a = l,this state interpolates to the state |0, 3, 3, —3} and has energy 5u>t + 3wc. This energy is always greater than the energy of the two possible fermionic ground states |0, 2,0, 0} and |0,3,3,3) which have energy 4cot and but — 3LUC respectively. Hence, in this case, there is at least one cross-over in the ground state. The three-fermion ground state will be |0, 2,0, 2) or |0,3,3,3) depending on whether ut > 3wc or ut < 3wc respectively and the interpolation is different in the two cases. For wt > 3wc (i.e. when |0,2,0,2) is the three fermion ground state), the cross-over value of a is obtained by equating the two energies as given by Eqs. (3.164) and (3.166). We find that the two curves cross at (x ~ uic/u>t)

1 - a2(x) = -(l+*HV(l+f

' + 2(1 + 3*)m(4/3) _

(3.167)

o In 4y o

Prom this expression it is easy to see that the cross-over point c*2(x) increases from 0.71 to 0.8 as x increases from 0 to 1/3, the point beyond which |0,2,0,2) is no longer the ground state. What is the cross-over point in case W{ < 3wc i.e. when |0,3,3,3) is the true fermionic ground state? This can be obtained by making use of another exact three-anyon eigenstate which interpolates from an excited bosonic state |0,6, v, 6) with energy 8u>t — 6wc to the fermionic state |0,3,3,3) with energy 5wt — 3wc. In particular, the energy of the corresponding anyonic state is given by E(a) = (2 + |6 - 3a\)wt - (6 - 3a)wc .

(3.168)

The two exact solutions as given by Eqs. (3.163) and (3.168) cross at at3 = l - x .

(3.169)

We observe that for 0.33 < x < 0.48, a^ is greater than a2 as given by Eq. (3.167). This implies that whereas there is one cross-over (at a = a^) when 0.48 < x < 1, there are two cross-overs in the ground state in case 0.33 < x < 0.48. The second cross-over point is easily obtained by comparing the

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two energies as given by Eqs. (3.164) and (3.168). We find

1

--<*>-

(1 - x) - J(l-l) 2 -2(3i-l)to(|)

- ^ w m

<3-™>

from which we see that the cross over point 0:4(0;) decreases from 1 to 0.55 as x increases from 0.33 to 0.48. To conclude, we find that for three-anyons in an oscillator potential and a uniform magnetic field with B > 0, there are at least two cross-overs in the ground state in case 1/3 < x < 0.48 while there is at least one cross-over otherwise. In Fig. 3.7 [39] we have plotted the three-anyon ground state energy as a function of a in case B > 0 and (a) x = uic/ujt = 0.25 (< 1/3), (b) x = 0.40 (1/3 < x < 0.48), and (c) x = 0.60 (> 0.48). What about the case of JV anyons (N > 3) experiencing both the oscillator potential and a uniform magnetic field? Unfortunately no general conclusions can be drawn because even though a class of exact energy eigenstate are again known, the ground state energy of ./V-fermions cannot be easily written down in this case. However in the special case when N = 3,6,10,15,..., there is a fermionic state containing only closed shells i.e. having I = 0. This state generalizes the |0, 2,0, 0) three-anyon state. Clearly, this I = 0 state has the same energy in an oscillator potential plus magnetic field as it would have with a purely harmonic oscillator of strength u)t. Its energy is given by

EfN= |"^vT+8JvU,

(3.171)

for all those values of N > 3 for which y/1 + 8N is a positive integer (i.e. N = 3,6,10,15,...). Following the arguments in the three-anyon case, it is then clear that when B > 0, the n = I = 0 exact 7V"-anyon energy eigenstate interpolates from the bosonic ground state with energy (N — l)uit to a fermionic state with energy greater than that of the 1 = 0 state as given by Eq. (3.171). Hence, there is at least one cross-over in the ground state when B > 0. Similarly, for B < 0 the same situation occurs when

(3.172) i.e. there is at least one cross-over in the ground state so long as uc

[3(Ar + l)-2y / rT8]V]

(3.173)

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Fig. 3.7 Ground state energy of three-anyons in an oscillator potential plus uniform magnetic field in case B > 0 and (a) 0 < x < 1/3 (b) 1/3 < x < 0.48 (c) 0.48 < x < 1.

Summarizing, we see that the multi-anyon (N > 3) problems are highly complicated due to the nontrivial braiding effects which give rise to the long ranged three-body potential.

Ch 3.

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95

Pseudo-Integrability of N Anyon System

In this chapter we have studied the TV-anyon problem in several different cases like in an oscillator potential, in an TV-body potential as well as in a uniform magnetic field and in all the cases we found a class of exact energy levels for all of which the energy (or (—E1)"1/2 in the iV-body potential case) varies linearly with a. Besides, numerically one has obtained several energy levels for which energy varies nonlinearly with a. The nonlinear spectrum displays many level crossings, some of which are of the LandauZener type (i.e. avoided crossings) [31]. This has led to the conjecture that the multi-anyon system (N > 3) may be non-integrable or even chaotic. On the other hand, if one looks at the classical Lagrangian for ./V-anyons in an oscillator potential, then one finds that the statistical interaction due to anyons is a total time derivative (see for example Eq. (2.32)). Hence the Euler-Lagrange equations of motion are the same as those of an Noscillator system which is known to be integrable. The puzzle then is, why do the numerical results suggest a non-integrable system? Secondly, what is the reason for the existence of a class of exact solutions in all these cases which is somewhat uncommon for a generic many-body problem? The point is that even though the system is naively integrable (the anyonic interaction being a total derivative), as shown in the last chapter, the classical relative configuration space on which the corresponding Lagrangian is defined is not simply connected but multiply connected and hence is topologically nontrivial [40]. In particular, the relative configuration space of N-particles is R^ — A, where A is as defined by Eq. (2.17) and denotes the generalized diagonal with set of all diagonal points removed. As a result, all the orbits which pass through these diagonal points are therefore modified. Hence, even though the number of constants of motion is the same as the number of degrees of freedom so that one has a potentially integrable system, it is not generically integrable via the action-angle variables. For example, even though one has 2N constants of motion in involution, there exist invariant surfaces which do not have the topology of a 2./V-dimensional torus. This bears a resemblance to the billiard system of Richens and Berry [42] although the reasons for the failure of integrability via the action-angle variables appear to be different. Following the arguments of [42], one can show that the system is only pseudo-integrable and this could be the possible origin of the level repulsion seen in the four-anyon spectrum. Further support for this picture comes from the study of the so called

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nearest-neighbour spacing distribution in the energy spectrum [43]. It has been shown that the spectrum exhibits features of both regular and irregular spectra as a function of the statistical parameter a. As far as we are aware off, this is probably the first example of a many body pseudo-integrable system. As far as the reason for the existence of a class of exact solutions is concerned (say in an oscillator potential), we have seen that in all the cases, the relative Lagrangian has a partial separability. In particular, we saw that (i) only the angular part of the Hamiltonian is affected due to the anyons, (ii) the angular part is independent of the radial potential V(p) between the anyons and (iii) the radial equation for iV-bosons, iV-fermions and ./V-anyons is the same but for the coefficient of the 1/p2 term. One can show that in this case the Hamiltonian has two collective degrees of freedom and the remaining are the relative degrees of freedom. Further, One can show [41] that the subset of exactly solvable solutions arise from the quantization of the collective coordinates (after the trivial center of mass is removed). It may be noted that these solutions do not carry any information about the internal dynamics of the system, which may be frozen as far as these solutions are concerned. Thus these solutions incorporate somewhat trivial aspects of anyon dynamics. The real anyon dynamics therefore resides in the solutions for which the energy eigenvalues depend nonlinearly on a. As we shall see in the next chapter, these trivial (linear) states conspire to cancel the divergent parts in the equation of state for an ideal anyon gas while the nonlinear states give the finite part which then defines the equation of state. Further, the spectrum shows evidence of many level crossings and level repulsions thereby indicating that the system is not integrable via action-angle variables but is only a pseudo-integrable system.

3.14

Quantum Computing and Anyons

In this section we shall briefly discuss about the possible role of anyons in quantum computation, a topic which is different from the rest of the material covered in this chapter. The idea of computational device based on quantum mechanics emerged when scientists were wondering about the question of fundamental limits of computations. It was Feynman [44] who produced the first abstract model in 1982 that showed how a quantum system could be used to perform

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computations. He also explained how such a machine would be able to act as a simulator for quantum physics. Soon afterwords, Deutsch [45] wrote an important paper where he showed that any physical process, in principle, could be modeled perfectly by a quantum computer so that a quantum computer would have capabilities far exceeding those of any traditional classical computer. The next major development was a paper by Shor [46] in which he proposed a method for using quantum computer to solve the important problem of factorization in number theory which has potentially significant application in the area of encryption. Subsequently, remarkable algorithms have been found in three other areas: searching a data base [47], simulating physical systems [48] and approximate but rapid evaluation of invariants of three dimensional manifolds [49]. Let us recall that in a classical computer the basic task is to interpret and manipulate an encoding of binary digits (called bits) into a useful computational result. Thus bit is the building block of information, classically represented as 0 or 1 in conventional classical digital computers. On the other hand, in a quantum computer, the building block of information is called a quantum bit or qubit. It should be noted that qubit is not binary in nature. This is because of the profound difference between the laws of classical and quantum physics. In particular, a qubit can not only exist in logical state 0 or 1 as in a classical bit, but also in states corresponding to a superposition of these classical states with a numerical coefficient representing the probability for each state. Thus by performing a single operation on the qubit, one would have performed the operation on two different values. Using this parallelism and right kind of algorithm, it is possible to solve certain problems in a fraction of the time taken by a classical computer. For example, a system of 500 qubits, which is impossible to simulate classically, represents a quantum superposition of as many as 2500 states! Just as information is manipulated through Boolean logic gates arranged in succession, in the classical computers, a quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single or pair of qubits. By applying these gates in succession, a quantum computer can perform a complicated unitary transformation on a set of qubits. Even though quantum computing holds tremendous promise, it must be noted that the quantum computing technology is still in its infancy. At present, we cannot clearly envisage what the hardware of that machine will be like. But one thing is certain. Any practical quantum computer

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must incorporate some type of error correction into its operation. This is because, quantum computers are far more susceptible to making errors than conventional digital computers and clearly some method of controlling and correcting these errors will be needed to prevent a quantum computer from crashing. The most formidable enemy of quantum computer is decoherence. The point is, a quantum system inevitably interacts with the environment. The information stored in the computer decays, resulting in errors and the failure of computation. Note, however, that decoherence is not the only enemy of quantum computers. Even if we can somehow isolate the quantum computer from environment, we could not expect to execute quantum logic gates with perfect accuracy. Small errors in quantum gates can accumulate over the course of computation, eventually causing failure and it is not obvious how to correct these small errors. The prospects for quantum computing received a tremendous boost from the discovery that quantum error correction is really possible in principle [50,51]. But it is important to realize that this is not enough. To carry out a quantum error-correction protocol, we must encode the quantum information we want to protect and then repeatedly perform recovery operations that reverse the errors that accumulate. Secondly, to operate a quantum computer, we must not only store quantum information but must also process the information. Thus we must be able to perform quantum gates, in which two or more encoded qubits come together and interact with one another. And we must design gates to minimize the propagation of errors. Incorporating quantum error correction will surely complicate the operation of a quantum computer. Because of this necessary increase in the complexity of the device, it is not a priori obvious that error correction will really improve its performance. This brings us to the novel concept of fault tolerant quantum computation [52,53]. A device is said to be fault tolerant if it works effectively even even when its components are imperfect. One of the best example of this concept is human body itself! It is a very illuminating example of imperfect hardware as well as hierarchical architecture with error correction at all levels. It may be noted here that similar issues also arose in the theory of fault tolerant classical computation. In 1956, Von Neumann [54] suggested improving the reliability of circuit with noisy gates by executing each gate many times, and using the majority voting. One shortcoming of his analysis was that he assumed perfect transmission of bits through the wires connecting the gates. In 1986, Gacs [55] was able to go beyond this assumption. In the last few years the fault tolerant methods have been

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developed sufficiently so that there is a feeling that it is now possible in principle for the operator of a quantum computer to actively intervene to stabilize the device against noisy (but not too noisy) environment. These techniques cope with sufficiently small errors. However, the error magnitude must be smaller than some constant (called an accuracy threshold) for these methods to work. According to rather optimistic estimates, this constant lies between 10~5 and 10~3, beyond the reach of current technologies. An obvious question is if one can design quantum gates that are intrinsically fault tolerant so that active intervention by the computer operator will not be required to protect the machine from noise? In 1997, Kitaev [56] suggested a novel way for achieving it-by anyon based computation. The central idea here to store and manipulate quantum information in a global form that is resistant to local disturbances. A fault tolerant gate should be designed to act on this global information so that the action it performs on the encoded data remains unchanged even if we deform the gate slightly, that is even if the implementation of the gate is not perfect. This is achieved by anyonic interactions which are topological in nature and which are immune to local disturbances. The idea here is to make use of braid group properties in order to do quantum computation. One can show that the accidental, uncontrolled exchanges are rare if the exchanged particles are widely separated and further if thermal anyons are suppressed. While the original proposal of anyon based quantum computation was by Kitaev [56,49], the first concrete model was given by Ogburn and Preskill [57,53] for anyons in the group A&, the even permutations of five elements. This work has subsequently been generalized by Mochon [58]. It must be pointed out at this stage that while all these schemes are theoretically possible, it is still not clear if we will be able to build an anyon based computer. The point is that even the simplest model with group A5 has 60 elements. Physically, this would require a sixty-component spin residing at each lattice link! Over the years, it has been realized that if quantum computer has to perform interesting computations then it must employ nonabelian anyons, as only they will be able to build up complex unitary transformations by performing many particle exchanges in succession. Kitaev [56] has described a family of simple spin systems on a square lattice with local interactions in which the existence of quasi-particles with nonabelian anyons can be demonstrated. The models are sufficiently interesting and yet they give rise to highly entangled state with infinite range quantum correlations.

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Unfortunately, they require four-body interactions. One remarkable thing coming out of whole thing is that the gauge phenomenon can emerge as collective effect in systems with only short range interactions. Could it be that the gauge symmetries in nature have a similar origin? Another possible anyon based model for quantum computation makes use of fractionally quantized quantum Hall effect. Unfortunately, mostly abelian anyons seem to play a role here though there is speculation that in some cases even nonabelian anyons could play an important role. Few years back, Averin and Goldman have proposed a device based on topological ideas [59]. In their model, anyons group around anti-dot holes of 0.2/XTO diameter made in a two-dimensional electron sheet. The holes are separated by O.Ol^zm wide gates. Individual anyons are then moved between the anti-dots in a way that allows for controlled braiding. They have demonstrated how a two-qubit controlled-NOT gate and single qubit gates could be constructed this way, thus showing their scheme implemented universal computation. They also discussed the decoherence mechanisms that would affect their model and provided some estimates of the dissipation and decoherence rates which showed that the device would not be significantly better, in this respect, than, say, schemes based on quantum dots. Summarizing, I feel that this field is still in its infancy and it is not completely clear if anyon based quantum computation will be feasible in practice or not. In any case, it is highly unlikely that a quantum computer will be in actual operation in foreseeable future, even the optimists do not expect one in coming fifteen-twenty years! References [1] D.P. Arovas, R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B251 (1985) 117. [2] Handbook of Mathematical Functions, eds. M. Abramowitz and LA. Stegun (Dover, New York, 1970). [3] J.M. Leinaas and J. Myrheim, Nuovo Cim. B37 (1977) 1. [4] L.D. Landau and L. Lifschitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, Oxford, 1977). [5] A. Comtet and A. Khare, Institute of Physics, Bhubaneswar Preprint IOPBBSR/91-11 (Unpublished). [6] R. Chitra, C.N. Kumar and D. Sen, Mod. Phys. Lett. A7 (1992) 855. [7] J. Law, M.K. Shrivastava, R.K. Bhaduri and A. Khare, J. Phys. A25 (1992) L183.

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[8] M.D. Johnson and C. Canright, Phys. Rev. B41 (1990) 6870. [9] A. Comtet, Y. Georglin and S. Ouvry, J. Phys. A22 (1989) 3917. [10] A. Vercin, Phys. Lett. B260 (1991) 120 ; J. Myrheim, E. Halvorsen and A. Vercin, Phys. Lett. B278 (1992) 171. [11] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [12] A. Suzuki, M.K. Srivastava, R.K. Bhaduri and J. Law, Phys. Rev. B44 (1991) 10 731. [13] R.L. Siddon and M. Schick, Phys. Rev. A9 (1974) 907. [14] W.G. Gibson, Mol. Phys. 49 (1983) 103. [15] Y.-S. Wu, Phys. Rev. Lett. 53 (1984) 111; 53 (1984) E1028. [16] C. Chou, Phys. Rev. D44 (1991) 2533 ; Phys. Rev. D45 (1992) E1433 ; Phys. Lett. A155 (1991) 245. [17] R. Basu, G. Date and M.V.N. Murthy, Phys. Rev. B46 (1992) 3139. [18] A. Polychronakos, Phys. Lett. B264 (1991) 362. [19] G. Date, M. Krishna and M.V.N. Murthy, Int. J. Mod. Phys. A9 (1994) 2545. [20] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197 ; 12 (1971) 419. [21] A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics, Lecture Notes in Physics m 14 (Springer-Verlag, Berlin, 1992). [22] J.E. Kilpatrick and S.Y. Larsen, Few Body Systems 3 (1987) 75. [23] A. Khare and J. McCabe, Phys. Lett. B269 (1991) 330. [24] J. McCabe and S. Ouvry, Phys. Lett. B260 (1991) 113. [25] M. Spoore, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B389 (1993) 645. [26] J. Grundberg, T.H. Hansson, A. Karlhede and E. Westerberg, Phys. Rev. B44 (1991) 8373. [27] D. Sen, Nucl. Phys. B360 (1991) 397. [28] M.V.N. Murthy, J. Law, M. Brack and R.K. Bhaduri, Phys. Rev. Lett. 67 (1991) 1817 ; Phys. Rev. B45 (1992) 4289. [29] M. Sporre, J.J.M. Verbaarschot, and I. Zahed, Phys. Rev. Lett. 67 (1991) 1813. [30] D. Sen, Phys. Rev. Lett. 68 (1992) 2977; Phys. Rev. D46 (1992) 1846. [31] M. Sporre, J.J.M. Verbaarschot, I. Zahed, Phys. Rev. B46 (1992) 5738. [32] F. Iluminati, F. Ravndal and J.Aa. Ruud, Phys. Lett. A161 (1992) 323; J. Aa. Ruud and F. Ravndal, Phys. Lett. B291 (1992) 137. [33] R. Chitra and D. Sen, Phys. Rev. B46 (1992) 10 923. [34] A. Khare, Phys. Lett. A 221 (1996) 365. [35] A. Khare, J. Phys. A29 (1996) L45 , 6459. [36] G.V. Dunne, A. Lerda and C.A. Trugenberger, Mod. Phys. Lett. A6 (1991) 2819; Int. J. Mod. Phys. B5 (1991) 1675; G.V. Dunne, A. Lerda, S. Sciuto and C.A. Trugenberger, Nucl. Phys. B370 (1992) 601. [37] K. Cho and C. Rim, Ann. Phys. 213 (1992) 295. [38] A. Karlhede and E. Westerberg, Int. J. Mod. Phys. B6 (1992) 1595. [39] A. Khare, J. McCabe and S. Ouvry, Phys. Rev. D46 (1992) 2714. [40] G. Date and M.V.N. Murthy, Phys. Rev. A48 (1993) 105. [41] G. Date, M.V.N. Murthy and R. Vathsan, corad-mai/0302019. [42] P.J. Richens and M.V. Berry, Physica 2D (1981) 495.

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G. Date, S.R. Jain and M.V.N. Murthy, Phys. Rev. E51 (1995) 198. R. Feynman, Int. J. Theor. Phys. 21 (1982) 467. D. Deutsch, Proc. Roy. Soc. London A400 (1985) 97. P. Shor, Proc. 35th Annual Symposium on Foundations of Computer Sc., IEEE Computer Society Press, Los Alamitos, CA (1994). L. Grover, Phys. Rev. Lett. 79 (1997) 325. S. Lloyd, Science 273 (1996) 1072. M.H. Freedman, A. Kitaev, M.L. Larsen and Z. Wang, quant-ph/0101025. P. Shor, Phys. Rev. A52 (1995) 2493. A.M. Steane, Phys. Rev. Lett. 77 (1996) 793. P. Shor, Proc. 37th Annual Symposium on Foundations of Computer Sc, IEEE Computer Society Press, Los Alamitos, CA (1996). J. Preskill, quant-ph/9712048. J. von Neumann, in Automata studies, ed. C.E. Shannon and J. McCarthy (Princeton Univ. Press, Princeton, 1956). P. Gacz, J. Comp. Sys. Sc. 32 (1986) 15. A. Kitaev, quant-ph/'9707021. R.W. Ogburn and J. Preskill, Proc. of QCQC 98, edited by C.P. Williams (Springer-Verlag, Berlin, 1999). C. Mochon, Phys. Rev. A67 (2003) 022315. D.V. Averin and V.J. Goldman, corad-mat/0110193.

Chapter 4

Statistical Mechanics of an Ideal Anyon Gas Be patient, for the world is broad and wide

— E.A. Abbott in Flatland 4.1

Introduction

In this chapter, we shall discuss the statistical mechanics of an ideal anyon gas. By an ideal gas we mean a gas of particles with no interaction apart from the statistical interaction. It is worth emphasizing here that the study of an ideal anyon gas is a kind of bench-mark study which is a must. Without a proper understanding of this problem, no worthwhile progress can be expected in the more realistic interacting anyon systems. Recall that a similar study in the case of an ideal Bose gas and an ideal Fermi gas was done right in the early days of quantum statistical mechanics. Such a study was possible because the wave function for TV non-interacting bosons (or fermions) is merely the product of the single particle wave functions with appropriate symmetry (or antisymmetry) factors. No such equivalent rule is known for anyons. As seen in the last chapter, even the two-anyon spectrum is not related to the single particle spectrum. Of course, this is because, the non-interacting anyon gas is not really non-interacting but is equivalent to an interacting Bose (or Fermi) gas which is known to be a notoriously difficult problem. In the absence of the standard path, perhaps the only other way to attack the problem of the non-interacting anyon gas is to try to obtain the spectrum of ./V-anyons in some potential and to use this spectrum to calculate the virial coefficients and hence the equation of state of an ideal anyon gas. However, as we have seen in the last chapter, so far, the quantum mechanics of only two-anyon system has been solved 103

104

Fractional Statistics and Quantum Theory

exactly in a few potentials while not even one iV-body (N > 3) problem has been completely solved as yet. As a result, to date only the second virial coefficient of an ideal anyon gas has been analytically computed. As far as the higher virial coefficients are concerned, it has been shown that the third and the higher virial coefficients do not receive any contribution at order a either around the bosonic or the fermionic end. Further, 03(0) to ae(a) have been computed perturbatively upto O(a2) around both the bosonic and the fermionic ends. Besides, by using the numerical and the perturbative methods, 0,3(0) has been computed for any a(0 < a < 1). The plan of this chapter is the following: In Sees. 4.2 and 4.3, we first give the results for the statistical mechanics of an ideal Fermi and Bose gas respectively in two dimensions. In particular, we show that for an ideal Bose gas there is no Bose-Einstein condensation in two dimensions (unlike in three and higher dimensions), and further the specific heat of the Bose and the Fermi gas are identical in two dimensions. Analytical expressions for all the virial coefficients are also derived. In Sec. 4.4, the second virial coefficient of an ideal anyon gas is calculated in three different ways and all the approaches are shown to give an identical answer. In Sec. 4.5, computation of the third virial coefficient of an ideal anyon gas is discussed from several different angles and exact and approximate results about it are discussed at length. Finally, some results about the higher virial coefficients CLN(N > 3) are mentioned in Sec. 4.6. 4.2

Ideal Fermi Gas in Two Dimensions

In this section we briefly discuss some exact results for an ideal Fermi gas in two dimensions. While the general formalism as well as a detailed discussion about the ideal Fermi gas in three dimensions is given in several text books [1,2], not many books have discussed the exact results in two dimensions. Let us consider an ideal Fermi gas in a grand canonical ensemble, in which case the equation of state is given by PA — =ln£(z,A,T) = £ l n ( l + * e - ^ )

(4.1)

from which z is to be eliminated with the help of the equation

N = zlzmc(z,A,T) = Y,TT^

(4.2)

Ch 4-

105

Statistical Mechanics of an Ideal Anyon Gas

where 0 = 1/kT, fugacity z = e^lkT with fi being the chemical potential and k the Boltzmann constant. Here P, A, N, T denote the pressure, area, number of particles and the temperature of the gas respectively. On replacing the summations over e by corresponding integrations, the two equations become (4.3) (4.4)

where A is the mean thermal wave length of the particles of mass m i.e. (4.5)

and , .,

Z100 xn~1dx

1

,

.

On eliminating z between Eqs. (4.3) and (4.4) one then obtains the equation of state of an ideal Fermi gas in two dimensions. The internal energy U is given by

(4.7) Thus, in two dimensions one has quite generally the relationship PA = U.

(4.8)

We shall see that an ideal Bose gas (as well as an ideal classical gas) also satisfies the same relationship in two dimensions. The specific heat Cv of the gas can now be obtained from Eq. (4.7) by making use of the formulae 2^[/,,W] = / „ - ! «

C = {%) \

U 1

/ N,A

(4.9)

(4-10)

and the relationship (dz\ \dTJN,A

zfx{z) T To{z)

(4.11)

106

Fractional Statistics and Quantum Theory

One obtains C^_

h{z) h(z)

Nk ~ 2 7 ^ y - -W) •

^12>

Similarly, one can show that Cp/Cv, Helmholtz free energy AH, and the entropy 5 are given by (4.13)

AH=Nii-PA = NkT\\nz-^\]

«-^=™ [ # - 4

(4.14)

(4 15)

-

It is worth noting that by Cv we mean here (in two dimensions), specific heat at constant area. Let us first discuss the properties of the Fermi gas at low density p(= N/A) and high temperature T. In this case, it follows from Eq. (4.4) that (4.16)

i.e the gas is highly non-degenerate. Let us now consider / i (z) as given by the integral (4.6). It is a standard integral which can be analytically done giving

(4 i7)

uz)=r^ri=Hi+z)-

-

Since at low p and high T, f\{z) « 1, hence z « 1 in that case. In case z is small but not extremely small in comparison to unity then one has to eliminate z between Eqs. (4.3) and (4.4) and obtain the equation of state in the form of virial expansion i.e. PA

_ f2(z)

^

2 |

_ ,

(4.18)

where ai are the (dimension-less) virial coefficients. It is worth pointing out that in most text books on statistical mechanics, a; are usually so defined that they are dimension-full. We now show that for two dimensional Fermi

Ch 4-

107

Statistical Mechanics of an Ideal Anyon Gas

gas all the virial coefficients can be analytically calculated. To that purpose note from Eqs. (4.4) and (4.17) that p\2

= f1(z)=ln(l

(4.19)

+ z).

On using Eq. (4.9), fi{z) can then be obtained by integration i.e.

f2(z) = I £ M dz=

f-z ln(l + z) dz

(4.20)

along with the boundary condition /2(0) = 0. On making the substitution ln(l + z) = y (= p\2) it is easily shown that for y < 2w 2

M^

°°

2/+1

+ T + ^WTiy**-

(4.21)

From Eqs. (4.3), (4.4), (4.19) and (4.21) it then follows that the equation of state is

Mf =

1+

— + ttWTiyB21-

(4 22)

"

On comparing Eqs. (4.16) and (4.23), we then obtain the virial coefficients for an ideal Fermi gas in two dimensions

4 = i, 4 = \, 4n+2 = o, 4n+1 = ~f^ where B^n are the Bernoulli numbers. Some of the low lying i?2n

B

B

B

B

B

* = \' ^-k' « = k> * = -k' ™ = k-

(4.23) are

(424

-^

Since Bin are alternatively positive and negative, it follows that [n = 1,2,...) a 4n _! > 0, ain+i < 0.

(4.25)

Also notice that a^ > 0 and all higher even virial coefficients are zero. This is a special property of the two dimensional Fermi gas (as well as the Bose gas as we shall see in the next section). The specific heat Cv can now be immediately calculated from Eq. (4.12) or more simply from Eqs. (4.18), (4.10) and (4.8). We find (4.26)

108

Fractional Statistics and Quantum Theory

Remarkably enough, only the odd virial coefficients contribute to the specific heat of an ideal Fermi gas in two dimensions. Further, at any finite temperature, the value of the specific heat is smaller than its limiting value which is Nk. We shall see below that the specific heat of an ideal Fermi gas decreases monotonically as the temperature of the gas falls. The other thermodynamic quantities can now be easily calculated (note that fo(z) = z-^fi(z) = z/(l + z)). It is left as an exercise to the reader. In the next section we shall show that in two dimensions, the specific heat of an ideal Bose gas is identical to that of an ideal Fermi gas. Let us now consider the behavior of an ideal Fermi gas at low temperature and high density such that z and hence X2p » 1. As \2p —> oo, the gas is said to be completely degenerate and in this limit the various thermodynamic quantities take a particularly simple form. As is well known, in the limit T —• 0 (which corresponds to X2p —> oo), the mean occupation number of the single particle states e(p) becomes 1(0) for e(p) < no (> Mo) where JJLQ is the chemical potential of the system at T = 0. Thus at T = 0 all single particle states upto e = no are completely filled. The limiting energy /xo is commonly called the Fermi energy and denoted by EF and the corresponding Fermi momentum is denoted by pp. The denning equation for these parameters is

[Fa(e)de = N, Jo

(4.27)

where a(e) is the density of states which in two dimensions is given by

(4.29)

It is then easily shown that (p = N/A)

Similarly, the ground state (or zero-point) energy of the system turns out to be

Eo = [Fsa(e)de

=^

P

%

(4.30)

and hence the ground state energy per particle and the ground state pressure are given by (4.31)

Ch 4-

109

Statistical Mechanics of an Ideal Anyon Gas

Thus PQ OC p (and not p 5 / 3 as in three dimensions). What happens at low but finite temperatures? In this case \2p and hence z is finite and large compared to unity. One then expands fn(z) in powers of (lnz)" 1 . On using the well known Sommerfeld lemma [1], /2,/i,/o are given by / 2 (z) = i ( l n z ) 2 + ^ + O ( l / z )

(4.32)

f1{z) = laz + O(l/z)

(4.33)

/„(*) = l + O(l/z).

(4.34)

Using Eqs. (4.33), (4.29) and (4.4) we have kT\nz = n=^—=eF. (4.35) m Thus for the two dimensional ideal Fermi gas, \i — ep is not only true at zero but even at low temperatures. Similarly using Eqs. (4.3), (4.11), (4.15) and (4.32) to (4.34) one finds that

S-?N(£)V-S-=?K(£)1 <-> ^ - ^ &-! + £( I.}* Nk~ 3 T F ' Cv ~ + 3 [TFJ

^-TT-YUV

J' M~TT>

(437)

{ A1}

(4 38)

-

where Tp is the Fermi temperature (Tp = £F/k). Note that these are all exact expressions at low T (T < < 2>) with no higher order correction terms in powers of (T/Tp). For T ~ TF one has to use the full expressions for /„ and calculate the various thermodynamic quantities numerically. From such a study it can be shown that the specific heat monotonically increases as T increases and reaches its classical value of Nk as T —> oo.

110

4.3

Fractional Statistics and Quantum Theory

Ideal Bose Gas in Two Dimensions

The equation of state for an ideal Bose gas in a grand canonical ensemble in two dimensions is given by PA —

= In C(z, A, T) = - £ ln(l - ze~^)

(4.39)

from where z is to be eliminated with the help of the equation

N = z§-zki£(z,A,T) = J2 z - i j e _ i -

( 44 °)

e

The summands appearing in Eqs. (4.39) and (4.40) diverge as z —> 1 since the term corresponding to p = 0 (and hence e = 0) diverges. On splitting off the terms in Eqs. (4.39) and (4.40) corresponding to e = 0 and replacing the rest of the sum by an integral, one obtains

§f = ^ » W - \ta(l" *>

(4-41)

where i 9n(z) = = ^ r

roc /

T-n-irfr . • \ x

r(n) yxo z-^^ - 1

(4.43)

We have not taken the lower limit of the integral to be 0, since for go and gi, the state e(= x/(3) = 0 does contribute to the integral and this contribution is quite significant (and in fact divergent as z —• 1). However, the last term in Eq. (4.41) is negligible for all values of z and may be dropped altogether in the thermodynamic limit. This is easily seen as follows. Since the quantity z/(l — z) is identically equal to A^o hence the term (—-^ ln(l — z)) is identically equal to (^ ln(7V0 + 1)) and is at most of O{N~1 inN) and hence negligible in the thermodynamic limit. As a result, the internal energy of the system is given by

Comparing Eqs. (4.41) and (4.44) we then find that in two dimensions, a la ideal Fermi gas, an ideal Bose gas also satisfies the relationship PA = U.

Ch 4-

111

Statistical Mechanics of an Ideal Any on Gas

Let us first consider the case of low density and high temperature. In this case the last term in Eq. (4.42) is negligible since this case corresponds to z « 1. Further, the lower limit of integration in gi{z) as denned in Eq. (4.43) can be safely taken to be zero since for z « 1 there is no divergence in gi(z). Once the lower limit is taken to be zero, g\ (z) can be obtained analytically

9i(z) = l

t°°

rlT

p ^ ; — T = -ln(l-2 f ).

(4.45)

We can now obtain z in terms of p\2 by using Eqs. (4.42) and (4.45). We get z = 1 - e~x2p .

(4.46)

We can now obtain 52(2) from g\{z) by integrating the relation gi(z) = z-§-z92{z) i.e.

(4.47)

g2(z) = -J^-Ml-z)

and using the boundary condition, 52(0) = 0. On using — ln(l — z) = y(= X2p), we find that for \y\ < 2ir

(4 48)

^^-T+gdkyfc-

-

On substituting the above expression for z and 52(2) hi Eq. (4.41) we obtain the equation of state in the form of a virial expansion. In particular, on comparing with Eq. (4.23), we find that in two dimensions, the virial coefficients for an ideal Bose gas are identical to those of an ideal Fermi gas except for the second virial coefficient which is equal and opposite in the two cases i.e. R

°1

=

a

l

P

=

1 > a2

P1 =

~a2

R =

R

W

T 1 a2n+2 ~ a2n+2

=

">

where Bin are the Bernoulli numbers. The specific heat Cv of an ideal Bose gas can now be computed by using U = PA and the relation (4.10) and one finds that Cv of the 2-dimensional ideal Bose gas is identical to that of the ideal Fermi gas [3], both being given by Eq. (4.26). It is worth repeating that only the second virial coefficient

112

Fractional Statistics and Quantum Theory

is different in the two cases but a2 does not contribute to the specific heat in two dimensions. One very important consequence of this fact is that unlike in three dimensions, for two dimensional ideal Bose gas, the value of the specific heat at finite temperature is smaller (and not larger) than its limiting (asymptotic) value of Nk (see Eq. (4.26)). Thus the (Cv-T) curve has a positive slope at high temperature. Further, as T —> 0, the specific heat must tend to zero. Thus unlike in the three dimensional case, Cv need not pass through a maximum in the two dimensional case. In fact we will see below that in two dimensions, an ideal Bose gas does not exhibit Bose-Einstein condensation. The other thermodynamic quantities are easily calculated in the limit of z —> 0 and is left as an exercise to the reader. In particular, note that the Helmholtz free energy AH, entropy S and Cp/Cv for an ideal two dimensional Bose gas are given by AH = N/j, - PA = NkT \\azL

~\] 9i{z)\

(4.50)

Here, go{z) = z-§-zgx{z) = z/(l - z) . Let us now consider the interesting but tricky case of the high density and low temperature. We now show that unlike the case of three and higher dimensions, in two (and lower) dimensions, there is no Bose-Einstein condensation [4]. The key to the whole argument is Eq. (4.42) in which the last term is identically equal to No/A, No being the number of particles not only in the ground state e — 0 but in fact the total occupation of particles in the states from e = 0 to e = eo- Thus this equation can also be written as

^o=f_^L_. A

Jxo z - ^ - 1

(4.53)

v

;

Clearly the integral is singular at z = 1. To see this, let us set z — 1. The integral is easily performed and we find that there is a logarithmic

Ch 4.

Statistical Mechanics of an Ideal Anyon Gas

113

divergence i.e.

*LzJ0l~]n(eo/l3).

(4.54)

This divergence can be interpreted in two ways: (i) One can choose any arbitrary £o as the lower cut-off in which case Wo is indeed macroscopic, but then it is not really a condensation into one state (e = 0) and the particles are distributed over a large number of states in the interval 0 < e < EQ. (ii) One can choose e0 -* 0 but in this case, the number of particles in the excited state is as large as possible, and is even larger than NQ. We thus conclude that unlike in three and higher dimensions (in fact unlike in 2 + 5 dimensions, 6 > 0), in two dimensions, there is no accumulation of macroscopically large number of particles in a single quantum state (e = 0), but there is a band of states near e = 0 that gets occupied. Thus there is no Bose-Einstein condensation in two and lower dimensions. This is consistent with the celebrated Mermin-Wegner-Berezinskii theorem [5] which asserts that there is no long range order, i.e. no phase transition in two dimensions for either ideal Bose gas or even for interacting Bose gas so long as the interactions are of short range. To study the thermodynamic behavior of the various quantities in the limit z —> 1, it is useful to get the expansions of 9i(z),g2(z) and go(z) in the powers of 5 = — In z which is a small positive number. An explanation is in order at this stage. Since g\(z) as defined by Eq. (4.43) diverges at z = 1 in the limit xo —* 0, hence while discussing the z —> 1 behavior of 9i(z), it is better not to separate out the contribution from the e = 0 state while defining Eqs. (4.41) and (4.42). We thus drop the last terms in both these equations and take the lower limit of integration xo to be 0 in Eq. (4.43). Prom Eq. (4.43) with XQ = 0, it is easily shown that as z —> 1 (i.e. J-»0)

ffl

(6) = - ln(l - z) = - In 5 + S- - ^ + O{5A)

(4.55)

go(6) - -§g9i(S) = I - \ + ^ + O(S3)

(4.56)

and hence

114

Fractional Statistics and Quantum Theory

while <72(S) can be obtained by integrating gi(S) i.e. 92{5) = - f gi(S)d5 + g2{6 = 0).

(4.57)

Using Eq. (4.55) and the fact that (4.58)

^L=^/6 e

—i

we find that 2

z2

D

4

92(S) = ^+Slnd-5---+O(S3).

(4.59)

Let us now consider Eq. (4.42) which for z ~ 1 takes the form pX2=gi(z^l)

= -\n5+^ + ....

(4.60)

On using this in Eq. (4.41) and using the fact that U = PA, we have

so that as in the ideal Fermi gas case, as T —> 0, the specific heat of an ideal Bose gas in two dimensions also varies as T. Similarly the entropy is given by

On the other hand Cp/Cv diverges as T —> 0 since CP

a

=

2g2(z)g0(z)

^!w

z

^ 7T2 e ^ 2

r

^o

' Y(^F "" °°-

.. ...

(464)

It is easily shown that whereas Cp/Cv — 2 as T —> oo, it is finite and > 2 at finite T and diverges as T —> 0. The adiabatic of an ideal Bose gas is also easily studied. Now an adiabatic process implies the constancy of 5 and N and hence of z and in turn of p\2. Thus we have the relationship a oc 1/T where a = A/N. Similarly, the constancy of z and p\2 implies the relationship P oc T2. Eliminating T from these two relations we then have Pa2 = constant

(4.65)

Ch 4-

Statistical Mechanics of an Ideal Anyon Gas

115

as the desired equation for the adiabatic of an ideal Bose gas in two dimensions.

4.4

Ideal Anyon Gas: Second Virial Coefficient

In the last two sections we have discussed at some length the statistical mechanics of a two dimensional ideal Bose as well as Fermi gas and obtained the temperature and density dependence of the various thermodynamic quantities. We were able to do that since the distribution functions for Bose-Einstein and Fermi-Dirac statistics are already well known. This in turn is related to the fact that the wave function for the -/V-bosons (Nfermions) is simply a symmetrized (anti-symmetrized) product of the single particle wave functions. Unfortunately, as seen in the last chapter, even the two-anyon spectrum (wave function) is not related to the corresponding single particle spectrum (wave function). Thus the path taken to study an ideal Bose or Fermi gas is not going to be useful for studying the statistical mechanics of an ideal anyon gas. However, we can look upon an ideal anyon gas as an interacting Bose or Fermi gas and use the techniques which are usually used to study the interacting Bose or Fermi systems. For example, one of the most popular method to study an interacting Bose or Fermi gas is to use the method of quantum cluster expansions as developed by Kahn [6] as well as Uhlenbeck and Beth [7]. Since this method is extensively discussed in almost all the books on statistical mechanics [1] hence we shall not discuss all the details of the method but merely summarize the key formulae required for our purpose. The cluster expansion is specially suitable for systems such as dilute gases. We shall study the low density, high temperature limit of an ideal (noninteracting) anyon gas (i.e. interacting Bose/Fermi gas) using the virial expansion i.e. the equation of state can be expressed as J^

= l+a2(p\2)

+ a3(p\2)2 + ...

(4.66)

where 02,03,... are the (dimension-less) virial coefficients while p is the density and A the thermal wave length. How does one evaluate 02,03,... for an interacting Bose/Fermi gas? One usually expresses them in terms of the cluster coefficients 62, &3, ••• • In particular, in the low density, high temperature approximation, the grand canonical partition function can be

116

Fractional Statistics and Quantum Theory

written as a cluster expansion -ln£ =^ 6 ^

(4.67)

i=i

where b\ are the cluster integrals, z is the fugacity and A is the area of the system. On the other hand, as we have seen in the last two sections, the pressure P and density p are given by kT P=— ln£

(4.68)

(469)

P

-4X^C)A,T

On inserting the cluster expansion (4.67) in Eqs. (4.68) and (4.69), we get oo

oo

P = kT^blzl,

p = ^2lhzl.

(=i

(4.70)

i=i

As a result, the equation of state (4.66) takes the form OO

OO

i

r

zl

j2biz = C£tti )U+ ^ ( 6=1

1=1

L

OO

i

->

J

a 2

/

4

OO

5>i* )+a 3 A ( J2 i )

^ 1=1

'

\

lb zl

^ (=1

/

2

-i

+••• • (4-71) J

On expanding both sides of this equation and equating the coefficients of equal powers of z, we obtain the expressions for the virial coefficients in terms of the cluster integrals ; the first few of which are a2A2 = - | , a

3

A

4

=4a^-^,....

(4.72)

On using the fact that the grand canonical function can be written as an infinite sum of canonical partition functions ZN i-e. oo

C = J2 *NZN

(4.73)

JV=O

one can express the cluster integrals 6; in terms of the canonical partition functions up to Zj. In particular, on using the fact that for small x, ln(l + x)=xx2/2 + x3/3 + ... we get from Eqs. (4.67) and (4.73) 2Z Z h 7 IA h i~ l 3Z3-3Z2Z1+Z13 h 3= 6i = Z1/A, b2 = — ^ 4 — ' " §]4 '

t4'74)

Ch 4.

117

Statistical Mechanics of an Ideal Any on Gas

It must be emphasized that all these expansions are meaningful only in the infinite area limit, so that particular care must be taken in their evaluation. Using this formalism we shall now calculate the second virial coefficient of an ideal anyon gas by treating it as an interacting Bose gas. On using Eqs. (4.72) and (4.74) we have 2

-A(2Z2-Zf) 2Z2

'

^

'

Now, since we are treating an ideal anyon gas as an interacting Bose gas hence we shall break up a2 (a) into two pieces; one from an ideal Bose gas and the rest from the interaction. In particular, we write O2 (a)

= ffl2(0) - lim -Aj[Z 2 (a) - Z2(0)}.

(4.76)

Here we have used the fact that Z\ is the same for fermions, anyons and bosons since no statistical effect is present in the one body problem. We have explicitly mentioned the limit A —> 00 to emphasize the point that all these expressions are meaningful only in the infinite area limit. Here 0 < a < 1 and with our notations, a — 0 corresponds to bosons. It may be noted that in two spatial dimensions, one has h = I/A2

(4.77)

Zx = biA = A/X2 .

(4.78)

and hence

Note that we have defined the equation of state (4.18) such that a2,O3,... are all dimension-less. We now compute a2(a) for an ideal anyon gas by different methods and show that all the approaches give the same answer which is finite. Conceptually, the fact that a2(a) is finite is a nontrivial statement since the anyonic interaction is a long range interaction and not a short range interaction as is usually assumed while deriving the cluster expansion. In particular, if the particles interact by a two-body potential, then a necessary condition for the existence of the virial expansion in the thermodynamic limit of infinite volume and constant density, is that if the potential decreases as r~n at a large distance r, then [8,9] n>d,

(4.79)

118

Fractional Statistics and Quantum Theory

where d is the configuration space dimension. Thus, naively, the virial expansion should not hold good for the non-interacting anyon gas since the statistical interaction between the anyons may be represented by a vector potential proportional to 1/r. However, it seems that the above criterion does not apply in the case of the vector potentials which is the case of anyons. Put another way, if one looks at the anyon Hamiltonian, then one finds that one has not only the two-body but also the three-body interactions coming from the non-trivial braiding effects and hence those theorems of statistical mechanics which are derived by assuming only the two-body interaction among the particles may not apply in the case of anyons. In fact there is a widespread belief that the virial expansion exists for an ideal anyon gas. The fact that a2(a) is indeed finite provides strong support to this belief. a2(a) using oscillator as a regulator In order to calculate a2(a) by using the harmonic oscillator as a regulator, we must first figure out the equivalent of ^4 —> oo limit in this approach. This is easily done by computing Z\, the one particle partition function for a particle of mass m in two dimensions in an oscillator potential of frequency w. In this case, it is well known that the energy eigenvalues are (4.80)

En = (nx + ny + l)hu> =(n + l)huj , n = nx + ny

with degeneracy being n+1. Here n, nx, ny = 0,1, 2,.... Hence the canonical partition function Z\ is given by

4sinh2(M^)

h

l

'

Taking the high temperatures limit and using Eq. (4.78), one is led to the identification A A2

\ h2f32uj2

2

2TT

.

m(3A

Thus the harmonic frequency w2 can be thought of as proportional to the inverse of the area A and hence the infinite area limit [A —• oo) becomes equivalent to the zero harmonic frequency limit (i.e. u> —> 0). This is physically understandable since putting the system in a harmonic potential confines the particles to move in a finite region; removing the harmonic force (u> —> 0) amounts to removing any restriction on the space (i.e. A —> oo).

Ch 4.

Statistical Mechanics of an Ideal Anyon Gas

119

Before we proceed with computing a2 (a) using the harmonic regulator, we must dispose off one more subtle point in connection with the harmonic regularization. For two particles, the harmonic potential is w2(r2 +

r2) =

2a;2R2 +

^

r

2

(4-g3)

where R = (ri + r 2 )/2 is the center of mass coordinate while r = i^ — r2 is the relative coordinate. Thus, in the two body problem a;2 m — 2LO2. Generalizing to the Z-body case, it is easily shown that in that case uJc.m. = ^ 2 - Since it is LJc.m. which is actually related to the area (see Eq. (4.82)), hence it follows that the two-dimensional cluster integrals h must be multiplied by I. In view of the fact that a2A2 = — 62/&1 ( s e e Eq. (4.72)) it then follows from Eqs. (4.76) and (4.78) that with harmonic regularization, 02(0) can be written as

«(.)~i-»a[*"-V (l 'i

(484)

where we have used the fact that a2(0) = af = —1/4 (in units of A2). It is convenient to separate out the contribution to Z2 due to center of mass by using Z2 = Z%m-Z2 = ZXZ2

(4.85)

so that a2 (a) takes the simple form 1 a2(a) = - - - 2 lim [Z2(a,w) - Z2(0,u)] •

(4.86)

Let us now compute the canonical two-body relative partition function Z2(a,ui) which is given by Z2(a, w) = Tre-0H^a'^

(4.87)

where H2 is the two-body relative Hamiltonian which has been extensively discussed in the last chapter. As has been shown in the last chapter, the energy eigenvalues (including degeneracy) of the two anyon relative Hamiltonian in an oscillator potential are given by En(a,uj) = (2n+l + a)froj, deg. = (n + l) = (2n + 1 - a)fno,

deg. = n

(4.88)

120

Fractional Statistics and Quantum Theory

where n — 0,1,2,.... Hence J?2(a, w) is given by oc

Z2(a,uj) = J2[(n+l)e-(2n+1+a'>hl3"

+ ne-(2n+1-a)hf3"}.

(4.89)

n=0

These sums are easily done and we obtain 2smh (hptJ) Note that both ^2(a,w) and Z2(0,u;) diverge like w~2 as w —» 0 i.e. they diverge linearly with the area A as A —> oo. However, the divergence cancels in their difference and hence 02(0) turns out to be finite and given by [10] a2(a)

= _ I ( l - 4 a + 2a 2 ).

(4.91)

Not surprisingly, for a = 0 one gets back the correct Bose value of 02 (0) = a^ = —1/4. However, what is really remarkable is the fact that even for a = 1 one gets back the correct Fermi value i.e. 02(1) = af = +1/4 thereby providing a nontrivial check on the calculations. Further a 2 (a) interpolates continuously between the bosonic value (—1/4) and the fermionic value (+1/4). We remark that 02(0) must clearly be periodic in a with period Aa = 2. In particular, if a = 2j + S (i.e. quasi bosonic case) with |<5| < 1 and j an integer, then the allowed values of \l — a\ are |<5|, 2 ± 5,4 ± 6,... and hence the entire derivation goes through as before with the obvious replacement of a by S. Thus one can say that a2(a)=

L

- i ( l - 4 a + 2a2) *

J periodic

(4.92)

where the subscript indicates that we are to extend these results for \a\ > 1 in a periodic fashion. A plot of 02(0;) as a function of a is given in Fig. 4.1 [11]. Notice that 0.2(0:) has cusps at the Bose values of \a\ = 2n, (n = 0,l,2,...). Thus there is a purely statistical attraction between anyons so long as 0 < a < l/-\/2 while there is repulsion between them in case 1/V2 < a < 1. At the special point a = 1/A/2, the statistical interaction is absent. Instead of computing a 2 (a) in the bosonic basis, one could also compute it in the fermionic basis in which case, instead of Eq. (4.86) one has the

Ch 4-

Statistical Mechanics of an Ideal Anyon Gas

121

Fig. 4.1 Second Virial Coefficient (in units of A2/4) of an ideal anyon gas as a function of a.

equation (note v = 1 — a) a2(u) = a2(u = 0) - 2 lim [Z2{u,u) - Z2(0,u)} w—>-0

(4.93)

where a2{y = 0) = a2 = j . In the last chapter, we have seen that the spectrum of two-anyons in an oscillator potential is given by Enrti(v, u) = (2nr + \l-v\

+ \)hu

(4.94)

where in the fermionic basis I = ±1,±3,.... Thus the spectrum and the degeneracy in a compact form are En(i/,u) = (2n ± v)hjj , deg. = n where n{=nr + ^±i) == 1,2,3,.... Hence ^J

2sinh 2 (^o;)

(4.95)

122

Fractional Statistics and Quantum Theory

From Eqs. (4.93) and (4.96) it then follows that a2(v) = \(l-2v2)

(4.97)

which is the same as given in Eq. (4.91) since u = I — a. Notice that unlike the bosonic end, there are no cusps at the fermionic end (the linear term in v is absent). It is worth emphasizing that the harmonic oscillator potential was merely used as a regulator to discretize the two-anyon spectrum and the final answer for ai{o) is in fact independent of the regulator used. For example, instead of an oscillator, one could instead consider the two-anyons in a circular box of radius R so that the spectrum is again discrete and then take the limit R —> oo at the end of the calculation. Historically, in fact, this is how a2(a) was first calculated [11,12] and only later it was calculated by using the harmonic confinement [10] and it was shown that both the approaches give the same answer for 02(0:)- In fact 02(0;) for an ideal anyon gas has also been calculated by using the path integral approach [11] and once again one obtains the same answer. In a way this approach is more satisfactory, since, even though one is still presented with the delicacy involved with extracting the finite difference of two divergent expressions, there is no necessity to impose a finite area constraint (which is necessary in the other two approaches in order to perform the mode counting). It is highly satisfying that all the approaches give the same answer for 02(0:) thereby establishing the regularization independence of the result. We thus have obtained the remarkable result that 02(0) for a noninteracting gas is finite even though the anyons do experience a long ranged, statistical vector interaction. As mentioned before, a priori it is not at all obvious whether a virial expansion should exist in the case of noninteracting anyon gas since the statistical interaction is long ranged. What about the higher virial coefficients? Would they also be finite? While it is believed that all the higher virial coefficients would be finite, so far, there is no rigorous proof of it. It is thus clearly very important to calculate the higher virial coefficients. However, it is very difficult to obtain the full spectrum of multi-anyons (N > 2) in any external potential and hence till today no one has been able to compute the third and the higher virial coefficients exactly. We therefore discuss an alternative method for evaluating the second virial coefficient due to Uhlenbeck and Beth [7] and Kahn [6]. The advantage of this method, called the semi-classical approximation, is that it does not require the knowledge of the eigenspectrum. Thus, hope-

Ch 4-

Statistical Mechanics of an Ideal Anyon Gas

123

fully, one may be able to extend this method to evaluate the higher virial coefficients. 02(0?) Using Semi-classical Approximation We now show that the second virial coefficient of an ideal anyon gas can be analytically evaluated by using the semi-classical approximation and it in fact yields the exact quantum result [13]. As seen in the last chapter, the relative classical Hamiltonian for twoanyons is given by

H$ = ^ + {Ve ~ hf 2

(4.98)

m mr where m is the mass of an anyon and pg and pr are the canonical momenta corresponding to the coordinates 6 and r respectively. Hence the canonical classical, partition function is given by iPe h Z* = -\~ f }} drd d0dPe . 2 2 f exp f - pi £ + (2nti) J I [m mr2 J J Pr

(4.99)

The limits of integration are 0 < r < 00, — 00 < pr < 00, 0 < 0 < 2?r, — 00 < P0 < 00. On shifting the variable of integration from pg to pg — ha we see that the integral becomes independent of a thereby confirming once again that the anyonic statistics is a genuine quantum mechanical effect. Let us recall that the classical equations of motion derived from the anyon Lagrangian are in fact independent of a and hence same as for the bosonic case. To unravel the quantum effect, one recognizes that pg = HI where I = 0,±2, ±4 ... in the bosonic basis. The semi-classical partition function is thus obtained from Eq. (4.99) by replacing J dpg by ^X^> with the sum over appropriate I according to the basis used. The above integral is divergent and must be regularized so as to extract the interacting part of the partition function Z 2 (a) — Z2{0). One could regularize it either by adding an external harmonic oscillator potential or by putting the particles in a circular box of radius R so that each particle moves in an area of TTR2. Needless to say that both the methods give the same answer in the limit of R —> 00 or vanishing harmonic potential. We therefore present only the R-cutoff method [13,14] and leave the other method as an exercise to the interested reader. It is worth pointing out that even in the i?-cutoff method, 02 (a) is still given by the same expression as in the oscillator basis, i.e. by Eq. (4.86). This is because, whereas in the oscillator case one had to multiply the cluster integral 6; by I, in the i?-cutoff basis one has to take care of the fact that each particle moves in an area of nR2.

124

Fractional Statistics and Quantum Theory

On performing the trivial pr and 9 integrals and replacing / dp$ by H £]; we find that for a given partial wave

(41 0)

^a) = W2[dreA^l-a?\

°

a

In order to compute a2(a), we now have to calculate J2i(^2( ) ~ ^2(0)), where 1 = 0, ±2, ±4,... in the bosonic basis. On using the Poisson summation formula 00

J-

I—

2 2 e-« (p+*)

1-

00

-1

1 + 2 *£ cos(nnb)e~^2^2

= ^

p= — 00

L

n=l

(4.101) -I

where p is an even integer, we obtain

J2 \Zl2{a) - z£(0)] = - £ £ [ 1 - C0S(n7ra)J / rdre- 3 " 2 '- 2 /^ . A

(even

-70

n=l

(4.102) On letting R —> CXD we thus have

£ [^(oO - 3(0)] = - ^ f; / even

[1 C

- °l ( n 7 r a ) ] + O(E-).

(4.103)

n=l

This is immediately evaluated by using the summation formula (0 < x < 2TT)

E

COS KX

7TZ

7TX

X

,

A

.

We then find on using Eq. (4.86) that 02(01) is again as given by Eq. (4.91) i.e. the semi-classical approximation gives the exact quantum result for 02(0:). This is a remarkable result. One could also calculate a2(a) in the fermionic basis and show that even in that case the semi-classical approximation gives the exact result as given by Eq. (4.97). We leave it as an exercise to the interested reader. The fact that the semi-classical approximation is exact for 02(0), even though remarkable, is not that surprising! In fact, we had anticipated this result in 1989 itself when we came across an interesting paper by Comtet and Ouvry [15] in which they showed that 122(0:) is related to the axial anomaly of the (1 +1 )-dimensional fermionic field theory in the classical flux tube background. Now it is well known [16] that only one loop contributes to the axial anomaly while two and higher loops do not contribute to it, i.e. the one loop or the semi-classical approximation is exact for the axial

Ch 4'

Statistical Mechanics of an Ideal Anyon Gas

125

anomaly. Note that this result is valid in any even space-time dimensions. The fact that the semi-classical approximation is exact for the (1 + 1)dimensional axial anomaly and further that this axial anomaly is related to a2(a) then strongly suggests that the semi-classical approximation is also exact for 0,2(0:). Why is the semi-classical approximation exact for 02(0;)? While the reason is not completely clear, one possible answer could be the following. We have seen that the noninteracting anyon gas can be regarded as a noninteracting Bose/Fermi gas plus interaction which is a dimension-less scale invariant interaction. Thus the only scale in the problem is the thermal wave length A and it is known that dimensionally 02 (a) = CX2 where C is a dimension-less constant (depending only on a but not on A). Now, since the semi-classical approximation by construction must be exact as T —> 00, and since C is merely a dimension-less constant (which is T-independent), hence it follows that the semi-classical approximation must be exact for 0.2(0;). By extending this logic, we also conjecture that the semi-classical approximation may even be exact for all the higher virial coefficients. Of course it is not clear whether the higher virial coefficients can be computed within the semi-classical approximation. As an illustration of these arguments, we now discuss an example of an interacting anyon gas with a scale invariant interaction and show that in this case also the semi-classical approximation for 0.2(0:) is exact. Let us consider an interacting anyon gas when the relative interaction between the two anyons is repulsive and given by

*% = £

(4-105)

where g is a dimension-less constant. Thus the total relative Hamiltonian for the two anyons (after taking out the center of mass) is given by (see Eq. (4.98)) Hrel =

PJ+ ( P ^ M ! + _l_

(4 106)

We shall first calculate the exact second virial coefficient and then the same by using the semi-classical approximation. The exact second virial coefficient 0-2(0:, g) for this system can be easily obtained by either using the harmonic confinement or by putting the particles in a circular box or even by using path integral approach [17] using directly the relative Hamiltonian as given by Eq. (4.106). All the approaches give the same answer. As an illustration let us compute it by using the harmonic confinement.

126

Fractional Statistics and Quantum Theory

The relative spectrum of two-anyons experiencing both g2/mr2 and the harmonic interaction (\muj2r2) is easily written down by following the discussion of the last chapter

EH:l= [2n + l + y/(l-a)2+g2}tKj

(4.107)

where, n = 0,1,2,... while 1 = 0, ±2,... in the bosonic basis. In the bosonic basis, a2{a) as given by Eq. (4.86) takes the form OO

OO

"^

r

^ Ue-Q«+W»

a2(a,g)-a2(0,g)=limJ2

n=0l=-oo'-

} - exp [ - ^ 2 +
(exp [-y/{l-a)2+g2

(4.108)

Notice that both 0,2(0, g) as well as 02(0, g) are logarithmically divergent as w —> 0. However, their difference is finite since the divergence is independent of a. On using the Poisson summation formula to display explicitly the periodic nature of Eq. (4.108) we obtain OO

a2(a,g) - a2(0,g) = lim V [ 1 - cas(irna)]In OJ—>0 ^ — '

(4.109)

n=l

where /„ - lim .

-„•• . / <

= nir Few comments are in order here.

dxe2innxe-Vn2+92

^-Kl{mtg).

(4.110)

(1) The second virial coefficient is a smooth function of the statistical parameter a for g > 0. In particular, the cusps at the Bose values of a = ±2n (n = 0,1,2,...) are removed in the presence of the hard-core g2/r2 interaction. (2) In the limit g —> 0 , the Bessel function K\(x) ~ 1/x and one recovers the non-analytic result as given in Eq. (4.91). (3) In the large g limit, iGXa;) ~ \f^e~x ( n °te x = rmg) and hence a2(a, g) - 03(0,0) =s ^

7T

exp {-ng)[l - cos(na)}

(4.111)

which shows that with increasing interaction strength g, the influence of the statistics becomes negligible.

Ch 4-

127

Statistical Mechanics of an Ideal Anyon Gas

(4) This calculation also demonstrates that a long ranged interaction of the form g2jr2 leads to divergent a,2(a,g) in the thermodynamic limit. This is in conformity with the theorem quoted earlier in Eq. (4.79) that the virial expansion exists only if n > d in case the two-body potential fall off like r~n at long distance and if d is the configuration space dimension. On the other hand, we also have the long ranged statistical interaction which however leads to a finite 02(0:) thereby showing that the finiteness of the second virial coefficient of an ideal anyon gas is a highly nontrivial result. (5) The potential g2/r2 behaves like a hard-core potential in two dimensions since the probability amplitude for finding two anyons at the same point is zero for any a (including a — 0) provided g2 > 0. We now show that the semi-classical approximation gives the exact quantum result for [o2(a, g) - 02(0,5)]. Following Eq. (4.99), the classical canonical partition function for the relative motion is given, in this case, by

a

?-^/«'[-''{g+(w"^+''}]**^-'"")

On doing the trivial pr and 6 integrals and replacing J dpg by h ]T^ we find that for a given partial wave

*" = Wx I* ***»[-£*{{l-afl

+ )

*l

(4113)

'

On using the Poisson summation formula as given by Eq. (4.101), we then find that 00

l

l

00

J2 [Z 2(a,g) - Z 2(0,g)} = - J £ [ 1 - cos(mra)] 71=1

/ = — OO

x

y0 r d r e x p [ - ^ - ^ H -

(4 114)

-

On letting R —> 00 , using the integral / dxe-(ax+b'^ =2(-)1'2Kl(2Vab), a Jo

(4.115)

128

Fractional Statistics and Quantum Theory

and Eq. (4.86) we then find that a2{a,g) - a 2 (0, 5 ) = -£• ^

" I 1 ~ coa^ira^K^nirg)

(4.116)

n—l

which is the exact quantum result as given by Eqs. (4.109) and (4.110). We thus have shown that the semi-classical approximation is not only exact for the second virial coefficient of a non-interacting anyon gas but also for an interacting anyon gas so long as the interaction is scale invariant. As explained above, this is because in both the cases, apart from the thermal wave length A, there is no extra dimensional parameter in the theory. As a further support to this argument, we now consider an interacting anyon gas with a dimension-full interaction and show that in this case the semi-classical approximation does not reproduce the exact quantum result for a2(a). Second Virial Coefficient for Anyons with Hard-disk Repulsion In the last chapter, we had consider the scattering of two identical anyons by the hard disk potential [18] V(r) = oo, = 0,

r < a, r>a,

(4.117)

where r is the relative distance between the two anyons. It is worth pointing out that the virial coefficients of a hard-disc boson or fermion gas have been discussed before [19,20,21]. The Hamiltonian for the relative motion is given, in polar coordinates by Hrel =

Pl+(Pl^f+v{ry

(4 . n8)

We shall first calculate the exact 02(0;, a) and then the same by using the semi-classical approximation and show that in this case the semi-classical approximation is not exact since apart from the thermal wave length A, the disc radius a represents the second scale in the problem. As shown in the last chapter, the phase shift for the I'th partial wave is given by (see Eq. (3.69)) tan^_a|(fc)=^-|(fca)

(4.119)

where Jv and Nv are cylindrical Bessel and Neumann functions respectively.

Ch 4-

129

Statistical Mechanics of an Ideal Anyon Gas

On using the well known Beth-Uhlenbeck formula [7,2]

Za(a, a) - 2k{a, 0) = ± £

f° & % ^ e ^

(4.120)

and Eq. (4.86) we obtain

a2ia,a) - a2(a,0) = 4, j ^ ^ ° ° *

M

^

a

) ^_ Q|(jfea)] (4-121)

where a 2 (a, 0) is the second virial coefficient of the noninteracting anyon gas as given by Eq. (4.91) and / in the above equations is even, since we are working in the bosonic basis. Detailed numerical results for a2(a,a) have been given in [18] which we refer to the interested reader. Let us now compute the same virial coefficient 02(0;, a) by using the semi-classical approximation. Following Eq. (4.99), the canonical partition function for the relative motion is given by

where V(r) is as given by Eq. (4.117). On doing pr and 6 integrals and replacing J dp$ by h J^, we find that 00

1

l

Yl[Z!i(<*,a)-Z 2{a,0)] = -^Jo

fa

^

e x

f

A2

1

P [ - ^ - a | 2 J - (4.123)

On using Eq. (4.86) and the integral fR I exp(-b/r2) dr = Rexp(-b/R2) - Vbn[l - erf{Vb/R)} (4.124) Jo where erf(x) is the error function of the argument, we then find that 00

a2(a,a)-a2(a,0)

= V2\ Y, {ae- (Ci/o)2 - v^C/fl -erf(Cj/a)]} . (4.125) ;=-oo

Here / is even and & = \l — a\\/\/%n. On comparing it with the exact quantum expression as given by Eq. (4.121) we find that for this problem, the semi-classical approximation is not exact. This is only to be expected since there are two dimensional parameters A and a in the problem and hence except as T —• 00, the semi-classical approximation is not expected to be exact.

130

4.5

Fractional Statistics and Quantum Theory

Ideal Anyon Gas: Third Virial Coefficient

Unlike the second virial coefficient, the higher virial coefficients are not known as yet. This is because the full quantum spectrum and hence the canonical partition function for ./V-anyons (N > 3) is not known as yet. In a way this is a pity since only the third and the higher virial coefficients contain information about the nontrivial braiding effects. Several people have, however, obtained partial results which give us some qualitative idea about these higher virial coefficients. In this section we shall discuss the known results about the third virial coefficient and in the next section we shall discuss the results about the higher virial coefficients. In order to calculate the third virial coefficient, one has to calculate the canonical three-body partition function Z^[a) for anyons. Now, for free anyons, each energy level and hence Z% is clearly divergent. To get finite Zz{a), let us as before introduce the confining harmonic interaction and finally at the end of the calculation, we shall switch off the harmonic interaction. Before we come to the computation of Zz{a), let us see how to compute Zz (or in general ZN) for bosons and fermions. As is well known, in this case, one first solves the single particle problem and then one forms all TV-body wave functions by constructing all Slater permanents for bosons or determinants for fermions with single-particle wave functions. The energy eigenvalues and their degeneracy follow immediately from here and hence ZM can be immediately obtained. For example, as seen in the last chapter, for particles of mass m interacting in the plane with a harmonic potential of frequency u, the energy eigenvalues are given by En,i = (n + I + 1)OJ

(4.126)

where n = 0,1,2,... and I = 0,1,2,.... From here Z\ as well as Z^ and Z% are easily calculated and are given by Eqs. (4.81), (4.85), (4.90) and (4.96). Let us now calculate Z^. We remark that the bosonic three-body wave functions are totally symmetric products of the three single particle wave functions and hence can be labeled by (ni, li), (n 2 , h), (ri3, ^3)- Further, on taking into account the symmetry imposed by the Bose statistics and the fact that S

n1,Jlln2,i2,n3,J3 = K + n 2 + «3 + h + h + h + 3)HiO

(4.127)

Ch 4-

131

Statistical Mechanics of an Ideal Anyon Gas

it is easy to show that ZB

=

3

coSh(3W + 2cosh2(^) 32sinh2(^)sinh2(n/3w)sinh2(5Mi£) •

In the case of fermions, one must exclude all states in which two or more quantum numbers are the same. We thus find F_

l + 2cosh2(^)

" ~ 3 2 s i n h 2 ( ^ ) sinh2(ft/3o;) s i n h 2 ( ^ ) '

Z

(4 129)

'

Let us now address the question of computing Zz («)• Unfortunately, the above procedure cannot be applied to the anyons since in this case, the iV-body wave function cannot be constructed from the products of the single-particle wave functions. Put differently, in the case of the anyons, the relevant group is the braid group and not the permutation group. However, there is another way which can be of some help. In the last chapter, we have obtained a class of exact solutions for the problem of three-anyons in an external harmonic oscillator potential and found that for all of them the energy varies linearly with a. We could therefore calculate the partition function Z3n(a) by including the contribution from these linear states and see if for a = 0 and 1 it agrees with Z3 ' as given above. That would be another way of knowing whether there are any missing states in the three-anyon spectrum as obtained in the last chapter. In fact not only Zl3in(a) but even Z%n(a) have been calculated [22,23] by using the exact linear solutions for the ./V-anyon problem as obtained in the last chapter and it has been shown that

On comparing with Eqs. (4.128) and (4.129) we see that Z\in (a = 0) ^ Zf and also Z3n {a = 1) =£ Z3 . Note however that Z2(a) (which can be easily obtained by using Eqs. (4.81), (4.85) and (4.90)) exactly agree with Zl2in. This then provides another evidence that unlike the two-anyon case, there are missing (nonlinear) states in the three-anyon spectra. Same can also be proved in the case of ./V-anyon spectra (N > 3). It is however, amusing to note that even though Zl3m(0) and Z| m (l) do not agree with exact Z3 and Z[ respectively, the difference with respect to the exact result is the

132

Fractional Statistics and Quantum Theory

same i.e. ZB

3

_

Zlin(a

3

=

n

l

,

CQ S h 2 (^)

'

2

16smh (^)Smh2(hf3uj)smh2(^) = Zl - Zl3in(a = 1).

(4.131)

Let us write the total canonical partition function for three-anyons as Z3{a) = Zl3in{a) + Zf(a)

(4.132)

l

where Z3 (a) is not known as yet, since so far not a single nonlinear state is analytically known. As a first step, therefore, let us estimate a3(a) as coming from the linear states alone. On using Eqs. (4.72) and (4.74) and the fact that with the harmonic regularization, one is required to multiply bi by a normalization factor of I in two dimensions we have (4.133) a3{a) = Aa22{a) - 2 lim x2 \3Z3(a) - 3Z2(a)Z1 + Z\ . x^O [ J Here x = hfiw and use has been made of Eqs. (4.82) and (4.78). On using Eqs. (4.81), (4.90) and (4.130) we find that as x -> 0 Z ^ ^ - ^

+

+ Oix2)

U - l 2 + 48 + 8 + C l J + -

(4.134)

<4135)

*•(-)=« iSS + ^ - ^ + i ) On using these expressions for the partition function, we find that in the x —> 0 limit, al3n(a) diverges like I/a;4. However, remarkably enough, the coefficients of the divergent terms (i.e. 1/x4 and 1/x2 terms) turn out to be independent of a so that even though al3 ™ (a) is divergent (when only Zl3m(a) is used to compute it), al3m(a) — al3n(a') turns out to be finite. In fact, one finds quite remarkably alin(a)-a3in(a')

=0

(4.137)

Ch 4-

Statistical Mechanics of an Ideal Any on Gas

133

which is due to some highly nontrivial cancelations. Thus, if one only includes the linear states, then 03(0) — 0,3=0,3 = 1/36. In other words, only the nonlinear states contribute to Aa.3 = 0,3(01) — 03(0). In fact using Eqs. (4.133) to (4.136) and the expression for 02(0;), it is easily shown that Aa3(a) = - 6 lim AZf(o,u>).

(4.138)

UJ—>0

It is worth pointing out that within the linear approximation, even 0,3 and af are divergent even though, as seen in Sees. 4.1 and 4.2, the exact ones are indeed non-zero and finite. This divergence is due to the non-inclusion of several states within the linear approximation. The divergence in a3(a) is in fact due to the missing nonlinear states. Since none of the nonlinear states is analytically known as yet, hence, till today, we do not know the exact expression for the third virial coefficient of an ideal gas of anyons. Some Exact Results for 113(0) Even though the exact third virial coefficient of the ideal anyon gas is unknown so far, using the perturbative, the semi-classical and the numerical approaches, a fair amount of information about 0,3(0) is available by now. We shall merely quote these results and we refer the interested reader to the original literature. It has been shown that unlike 0.2(0), a3(a) does not receive any contribution at order a either around the bosonic or the fermionic end and hence has no cusps in it [24,25]. Subsequently, a3(a) has been calculated upto second order in a [26,27], and it has been shown that around both the fermionic and the bosonic ends a3(a) = ± + ^+O(a3).

(4.139)

Around the same time, it was shown [28] that a3(a) satisfies the remarkable relation a3(a) = a3(l - a)

(4.140)

i.e. 0,3(a) is symmetric under a —> 1 — a. This is interesting because it means that it suffices to calculate a3(a) for say 0 < o < 0.5. The key ingredient in the proof is the fact that the nonlinear states in the three-anyon spectrum in an external harmonic oscillator potential are mirror symmetric (on reflecting around a = 1/2) i.e. for every regular nonlinear wave function ip(a), there is a corresponding regular nonlinear wave function

134

Fractional Statistics and Quantum Theory

•0(l — o:), a fact first observed in the three-anyon spectrum obtained numerically, and later on proved rigorously [28]. This fact also means that the two nonlinear states must cross at a = 1/2. Now consider the full three-body partition function as given by Eq. (4.132). Taking into account the mirror symmetry of the nonlinear states, it then follows that Z3(a) - Z3(l - a) = Zl3in(a) - Z\in(\ - a). This implies that only the exactly known linear states contribute to as(a) — 03(1 — a). On using Eq. (4.137) and the fact that a$ = af\ we then immediately recover relation (4.140). Even though this relation does not determine 0,3(a), nevertheless, it puts strong restrictions on the possible forms of 0,3(0). In fact there is a strong belief that a^(a) is not a polynomial function of a. Finally, 0,3(0) has been calculated by numerically computing the threeanyon spectrum in an oscillator potential [29,30] as well as by using a path integral representation of the partition function for the anyons in an oscillator potential [31]. These calculations clearly show that 0,3 (a) is finite for any a. Secondly, it has been shown that 03(0) can be well approximated by only a few terms from the Fourier series . . 1 sin2 Tra 4 fi a3(a) = —-\———g—hcism wet + c2 sin ira + ....

. (4.141)

Note that this series is consistent with periodicity as well as with the exact relations as given by Eqs. (4.139) and (4.140). The initial hope was that ci, C2, and all higher coefficients are perhaps zero so that 0,3(0) has a nice, remarkably simple, form. However, a recent, very accurate numerical computation [32] shows that whereas cn = 0, n > 2, C\ is non-zero even though very small. In particular, the numerical calculations show that C! = -(1.652 ± 0.012) x 10" 5 .

(4.142)

If true, this calculation suggests that the task of calculating 03(0) analytically may be quite difficult.

0,3(0) within Semi-classical Approximation There is another approach i.e. the semi-classical approximation which might help in obtaining the third virial coefficient. The main advantage of this approach is that one need not know the three-anyon spectrum. Besides, as we have seen in the last section, the semi-classical approximation gives the exact a2(a) for not only an ideal gas of anyons, but also for an interacting (but scale invariant) gas of anyons. Since there is no other scale in the problem of non-interacting anyons except thermal wave length A,

Ch 4-

Statistical Mechanics of an Ideal Anyon Gas

135

hence we suspect that if one can compute 03 (a) within the semi-classical approximation, then it will be exact. So the question is whether one can really compute 03(0) using the semi-classical approximation. The Lagrangian for the noninteracting three anyon problem is given by [13]

L=^(r21+rl + vl) + Mj2eiJ

(4-143)

i<j

where 0y = tan~1[(yi — yj)/(xi — Xj)\. The center of mass motion may be separated by transforming to Jacobi coordinates R = - ( r i + r 2 + r3), p=—(n-r2),

7?=-^(ri+r2-2r3).

(4.144)

The corresponding classical Hamiltonian for the relative motion can now be worked out. Some tedious but straight forward algebra yields

Hf = ^ [(p, + ahhf + {pn - ahf2)2 + + ^(P8P - ah(l + / 3 )) 2 + i ( p e , - aft/4)2]

(4.145)

where / i , / 2 , / 3 and 7*4 are functions of p, rj and the relative angle ip = 9p — 6rj- I n particular, / i , . . . , / i are given by _ 6p772sin2V' , __ 6p277sin2^ 7i = Y) ' /2 = D ' 7*3 = ^ ( P

2

- 3r72 cos2V), h = ^ ( 3 7 ? 2 - P2 c o s 2 ^ )

(4-146)

where D = r\zr%z = p4+r]4-6p2r]2 cos 2ip. It may be noted that / 3 + / 4 = 2. Here the generalized momenta pi are as usual given by pi = dL/dqi, where Qi are p, f] and the corresponding polar angles 6P, 6n. The classical canonical relative partition function for three-anyons is given by Zf = 7 ^ ^ J dpdppdr1dpr,depdpepd9ridpen exp(-/3ff 3 d ).

(4.147)

The pp and pv integrations are performed trivially. Further, to obtain the semi-classical partition function, we replace pep by hlp and pgn by hln in

.136

Fractional Statistics and Quantum Theory

the phase space integrals. In this way one obtains -I

yoo

rCG

I"

riit

- ^ (j% - "(1 + A)]2 + £ [ ! , - «/«]2)] • (4-148) This integral is divergent and must be regularized to extract the nontrivial and interacting part of the partition function AZ3(a)(= Zs(a) — Z3(0)). As in the second virial coefficient case, we apply a finite cut off R i.e. consider three-anyons in a circular box of radius R and finally take R —> oo. For a finite cut off R, integration limits on p and r\ are rather complicated. However, for large values of the cut off R, such that R% = 2nR2/X2 »

1,

(4.149)

p and 7] may vary independently of each other, with the limits 0 < p < R/^/2,

0 < r\ < y f-R- In this notation, the cut off R has again the

same interpretation as in the two-body case, i.e., A = wR2 is the area in which a particle is confined, and the finite size effect in the thermodynamic quantities is negligible when Eq. (4.149) is satisfied. We would like to calculate the interaction part of the relative three-body partition function by summing over the partial waves i.e.

AZ3(a) = J2 E ^ 3 P ' S a ) " ^''"(0)].

(4-150)

Here, in essence, lies the main difficulty with the semi-classical formalism. Recall that in the two particle problem, summing over the even (odd) partial waves ensured the bosonic (fermionic) basis. In the three particle problem, however, specifying lp and ln does not uniquely determine the symmetry of the state. This is because, in the three particle case, apart from the fermionic and the bosonic states, one also has the so called mixed-symmetry states. For example, summing lp over only even (odd) integer valves and lv over all integer values would ensure that the fermionic (bosonic) contribution is excluded, but the basis still contains the mixed symmetry admixture. Thus, no further progress is possible in estimating 03(0:) within the semi-classical approximation unless one can work out a way for eliminating the contribution of the mixed symmetry states. What has been done so far, is to sum over all integral values of lp and l\, as would be done in a Boltzmann basis. Of course this is no substitute for

Ch 4-

137

Statistical Mechanics of an Ideal Anyon Gas

a genuine quantum calculation. It turns out, though, that such an 03(0), though approximate, is consistent with the many exact constraints available in the literature. Within the Boltzmann basis, AZ 3 (a) may be written as (4.151)

AZ3 = AZ+ + AZ~ + AZ%

where the superscript m denotes the mixed-symmetry contributions. Historically, AZ3 was first evaluated numerically [13] by taking the cutoff RB = 50 (see Eq. (4.149)) and summing over all lp and lv such that \lP\, \ITI\ < 1200. Extensive numerical checks were performed to make sure that the sum is convergent. An accuracy of one part in 105 was maintained in the numerical work. It was found that within this accuracy AZ3 and AZ2 satisfy the remarkable relation AZ3 = ^AZ2

(4.152)

or AZ3 = ^AZ2

where AZ2 denotes the two-anyon interaction partition function in the Boltzmann basis i.e. AZ2 = AZ} + AZz

(4.153)

which is obtained by summing over all integer I. Notice that if each particle moves in an area A = nR2, then the total and the relative two (and three) anyon partition functions are related by 2A AZ 2 = ^ - A Z 2 ,

^A AZ3 = ^AZ3.

(4.154)

What is the significance of the relation (4.152)? This relation shows that even though there is vector interaction between all the particles, the third anyon effectively decouples (in the area A) and does not contribute directly to the interacting part of the partition function. The relationship (4.152) is quite nontrivial and it has an interesting consequence as far as the virial coefficient (in the Boltzmann basis) is concerned. As has been shown by Baumgartl [33], in the Boltzmann basis, the second and the third virial coefficients are given by Aaf o i (a) = - ^ A 2 2 ( a )

Aai°\a)=4(Aa*°l(a))2-^-\AZ3-j^AZ2\

(4.155)

.

(4.156)

138

Fractional Statistics and Quantum Theory

Recall that our virial coefficients are dimension-less. On using Eqs. (4.91), (4.97), (4.153) and (4.155), we can easily obtain Aaf °l(a) within the semiclassical approximation (recall that the semi-classical approximation is exact both for a£(a) and a^"(a)) \2

Aaf o i (a) = —-(AZf + AZz) = Aa $ (a) + Aa^ (a) = a ( l - a ) . (4.157) In view of the remarkable relation (4.152), we then find that Aaf °'(a) = 4a 2 (l - a) 2

(4.158)

where Aaf ol (a) = 03(0:)-a 3 (0), with a3(0) = ^ . The connection between this Aaf o l (a) and the quantum one is not clear and only a full quantum calculation can clarify the situation. Even though approximate, it turns out that this Aaf (Q) is consistent with the many known exact results. For example, Aa^ol(a) goes to zero at a = 0 and a = 1 as any correct quantum calculation should (note that 03(0) = 03(1) = 1/36). Further, we also find that there is no O(a) contribution to Aafol(a), a result which has subsequently been rigorously derived for the full quantum a3(a) [24,25]. Thirdly, Aaf 0 '(a) satisfies the remarkable relation [28] Aaf°'(a) = A a f ^ ( l - a ) .

(4.159)

As expected Aaf o '(a) overestimates the true answer i.e. whereas Aaf°'(a) = 4a 2 + O(a3),

(4.160)

the true quantum coefficient satisfies [26] Aa3(a) = ^+O(a3).

(4.161)

It must be emphasized here that the result (4.158) crucially depends on the equality (4.152) which was obtained numerically. Can one prove it analytically? We now present an unpublished work by Singh [14] where he has been able to derive it rigorously. Let us consider A^3(a) as given by Eq. (4.150) where ^3"'" (a) is as given by Eq. (4.148). On using the Poisson summation formula 00

£ p=-OO

1— 1-

e-a>{p+bf

=

VJL N + "-

00

2

2

21

J- cos(2n7r6)e- 1 ^ 71=1

(4.162) J

Ch 4-

139

Statistical Mechanics of an Ideal Anyon Gas

where p is any integer, one can write AZ^(a) as (4.163)

AZ3(a) = h + I2 + I3 where h = ^l A

pdp •'O

4TT /"A

h = TJ

vdr] Jo

Jo

JO

fVlk

dil;\[coB(2nira(l ~J

f2v

pdp / Jo

77^77 / io

+

f3))-l]e—&— (4.164)

^

_4»w

d^ / . [cos(2rn7ra/4) — l]e

^

m=1

(4.165) ^3 = T T / A -^

P^P / ^

^

/ Jo

dip V V ) cos(2n7ra(l m=ln=i L

+ / 3 )) cos(2m7ra/4) - l] e ^ I ' V ^ V ) / * 1 _

(4 _ 166)

We shall evaluate these integrals in the limit R —•> oo and show that to the leading order, only / i is nonzero while I2 and 73 vanish. Consider the integral Ji first. Let us first re-scale the variables p and 77, i.e. define >=2^'"=#*-

(4 16?)

-

Then, in the limit i£ —> oo, I\ can be written as c\ D 2

/

/*OO

i = ^2A2_/

/»1

^^y

/>2TT

°°

#5>os(2n7rQ(l + / 3 ) ) - l ] e - n V 2 .

^ M /

(4.168) Now in the limit R —> 00, / 3 as given by Eq. (4.146) tends to 0 and hence we have 7

R2 ^ [cos(2mra) - 1] i = 3^2 2 ^ •

(4-169)

n=l

Proceeding in the same way, it is easy to see that to the leading order, h, h —> 0 as R —• 00. For example, as R —* 00, J 2 can be written as nn2

I2 =

. / ^ 2 A2 j

/*OO

0

/»1

/*27T

crdcr / /xd/i / 70 Jo

°°

di/' 2\.[cos('^rn7rafi) ^-^

2 2 2 3

~ l]e

^

• (4.170)

140

a

3

a

*

Fractional Statistics and Quantum Theory

a0

a

a2

M

°

h.

°

°

±^e + (iefos ^ ^ I + ^ T I 1 1 1 ^ ! I 8 i V5+1 _ 25 i 3+V6 , 9 i 4+%/lQ^ 4 8 V 6 l n 3 - V l "^ 32V1O 1 I I 4-Vloy " 1 '27y^ i I i V-5-1

On using the fact that as R —> oo, f\ —> 2, we then see that there is no O(J22) contribution from 72- Similarly one can show that there is no O(R2) contribution from /3, so that /\Zz{ot) is equal to 7i which is as given by Eq. (4.169). We must now similarly compute £\Z-2,{a) within the semi-classical approximation. In Eq. (4.103), we have already evaluated it using the bosonic basis i.e. we only summed over the even values of I and hence used the identity as given by Eq. (4.101). We now must sum over all integer values of I instead, and hence on using the summation formula (4.162) in Eq. (4.100) we immediately find that

n=l

On comparing with Eq. (4.169) we then obtain the remarkable relation (4.152). A note of caution is in order here. The derivation given here has only shown the validity of relation (4.152) to leading order in R as R —> oo. However, it is not clear if there are sub-leading corrections to AZ3(a) or not. This is an important point, since as is clear from Eq. (4.156), if there are sub-leading correction terms, they could contribute to Aaf ol(a). 4.6

Ideal Anyon Gas: Higher Virial Coefficients

In the last section, we have seen that the task of calculating even the third virial coefficient is quite difficult and till today not much is known about as(a) except its value upto the second order in a and the mirror symmetry relation (4.140). Not surprisingly, even less is known about the higher

Ch 4.

Statistical Mechanics of an Ideal Anyon Gas

141

virial coefficients aN(a)(N > 3). What has been shown so far is that unlike 02(a), none of the higher virial coefficients have any cusp i.e. none of them have any term at order a in their expansion [34,25] either around the fermionic or the bosonic ends. Further, exact expressions for 04(a), 05 (a) and a§(a) have been obtained to second order in a around both the bosonic and the fermionic ends [26,27]. In Table 4.1 [26], we have given expressions for 0,2(0) to ae(a) upto second order in a in both the fermionic and the bosonic basis. Note that the upper (lower) sign in the Table corresponds to expression in the bosonic (fermionic) basis. From the Table we find that whereas the coefficient of the a2 term in a2(a) and 03(0) is the same at the bosonic and the fermionic ends, same is not true for the higher virial coefficients 04,05,06- One immediate consequence of this is that unlike 03(0), the higher virial coefficients (TV > 3) do not satisfy the mirror symmetry relation Ojv(a) = ajv(l — a). From the Table one also notices a proliferation of logarithms in the higher virial coefficients, the argument of the latter being the roots of simple second-degree equations with integer coefficients. It is also clear from the Table that the complete summation of the thermodynamic potential even to second order in a is very difficult. Thus even the problem of a noninteracting anyon gas is rather nontrivial. Note however that at least to second order in a, one has a well-defined virial expansion. This is a nontrivial result, since because of the long range vector interaction, there could have been a breakdown of the virial expansion, as happens for example, in the case of g2/r2 interaction. One immediate consequence of these results is that to first order in a, the specific heat of an ideal anyon gas is the same as that of a Bose or a Fermi gas. This is because only 02 (a) receives contribution at order a, but in two dimensions, a 2 (a) does not contribute to Cv. Finally, using Eqs. (4.26) and Table 4.1, we can write down a perturbative expression for the specific heat of an ideal anyon gas which is valid to second order in a i.e. | | = 1 - a3(a)(pA2)2 + 2o4(a)(p\2)3 + 3o5(a)(pX2)4 + 4a6(a)(pX2)5 + .... (4.172) Much more progress is required before we can know the various thermodynamic properties of an ideal anyon gas. We strongly believe that unless one can understand the properties of an ideal anyon gas, any calculation including interactions will always remain unreliable.

142

Fractional Statistics and Quantum Theory

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] [27] [28] [29] [30]

R.K. Pathria, Statistical Mechanics (Pergamon Press, Oxford, 1972). K. Huang, Statistical Mechanics (John Wiley and Sons, Inc., 1963). R.M. May, Phys. Rev. 135A (1964) 1515. I am grateful to Somen Bhattacharjee for a discussion on this issue. N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; V.L. Berezinskii, Sov. Phys. JETP 32 (1971) 493. B. Kahn, Ph.D. Thesis (Utretch) 1938, Reprinted in Studies in Statistical Mechanics, eds. J. de Boer and G.E. Uhlenbeck (North-Holland, Amsterdam 1965). G.E. Uhlenbeck and E. Beth, Physica 3 (1936) 729 ; 4 (1937) 915. G.H. Wannier, Statistical Physics (Dover, New York, 1987). J. Myrheim, Dr. Philos. Thesis, Univ. Trondehim, Unpublished (1993). A. Comtet, Y. Georgelin and S. Ouvry, J. Phys. A22 (1989) 3917. D.P. Arovas, R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B251 (1985) 117. J.S. Dowker, J. Phys. A18 (1985) 3521. R.K. Bhaduri, R.S. Bhalerao, A. Khare, J. Law and M.V.N. Murthy, Phys. Rev. Lett. 66 (1991) 523. Virendra Singh, Unpublished (1992). A. Comtet and S. Ouvry, Phys. Lett. B225 (1989) 272. S.L. Adler and W.A. Bardeen, Phys. Rev. 182 (1969) 1517 ; For a Pedagogical introduction to chiral anomalies see, S.B. Treiman, R. Jackiw and D. Gross, Lectures on Current Algebra and its Applications (Princeton University Press, Princeton 1972). D. Loss and Y. Fu, Phys. Rev. Lett. 67 (1991) 294. A. Suzuki, M.K. Srivastava, R.K. Bhaduri and J. Law, Phys. Rev. B44 (1991) 10 731. R.L. Siddon and M. Schick, Phys. Rev. A9 (1974) 907. W.G. Gibson, Mol. Phys. 49 (1983) 103. S.K. Sinha and Y. Singh, Mol. Phys. 44 (1981) 877. For detailed discussion see, A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics, Lect. Notes in Phys. m 14, (Springer-Verlag, Berlin, 1992). J. Dunne, A. Lerda, S. Sciuto and C. Trugenberger, Phys. Lett. B277 (1992) 474. J. McCabe and S. Ouvry, Phys. Lett. B260 (1991) 113. D. Sen, Nucl. Phys. B360 (1991) 397. A. Dasnieres de Veigy and S. Ouvry, Phys. Lett. B291 (1992) 130 ; Nucl. Phys. B388 (1992) 715. R. Emparan and M.A. Valle Basagoiti, Mod. Phys. Lett. A8 (1993) 3291. D. Sen, Phys. Rev. Lett. 68 (1992) 2977 ; Phys. Rev. D46 (1992) 1846. J. Law, A. Suzuki and R.K. Bhaduri, Phys. Rev. A46 (1992) 4693. J. Law, A. Khare, R.K. Bhaduri and A. Suzuki, Phys. Rev. E49 (1994) 1753.

Ch 4-

Statistical Mechanics of an Ideal Anyon Gas

143

[31] J. Myrheim and K. Olaussen, Phys. Lett. B299 (1993) 267; B305 (1993) E428. [32] S. Mashkevich, J. Myrheim and K. Olaussen, Phys. Lett. B382 (1996) 124. [33] B. Baumgartl, Z. Phys. 19 (1967) 148. [34] A. Comtet, J. McCabe and S. Ouvry, Phys. Lett. B60 (1991) 372.

Chapter 5

Fractional Exclusion Statistics

Fool, said he, "Space is Length. Interrupt me again, and I have done" — E.A. Abbott in Flatland

5.1

Introduction

So far we have discussed the anyonic fractional statistics. We have seen that, stated simply, anyons are particles whose many particle wave function picks up a phase of e ±M7r , (0 < a < 1) under the exchange of the position of any two anyons. We have also seen that the concept of anyonic statistics is only valid in two dimensions. However, this is not the only way to define quantum statistics. One could instead think of defining it through the concepts of statistical mechanics. One such attempt was made by Haldane [1] who introduced the concept of exclusion statistics. This approach generalizes the notion of the Pauli exclusion principle which in a way is the corner-stone of quantum statistical mechanics. The nice thing about this approach is that a la Fermi-Dirac and Bose-Einstein statistics, the notion of exclusion statistics can be introduced without specific reference to spatial dimension. In the last few years several remarkable results have been obtained through this approach. Special mention may be made of the work of Wu [2] and Isakov [3] which made a general formulation of quantum statistical mechanics for an ideal gas of particles obeying fractional exclusion statistics and obtained occupation number distribution functions for them. Needless to say that the Bose-Einstein and the Fermi-Dirac distribution functions are contained here as special cases. In the specific case of particles occupying the lowest Landau level, the same distribution has also been 145

146

Fractional Statistics and Quantum Theory

derived by Dasnieres de Veigy and Ouvry [4]. Here the term ideal means that the only interaction between the particles is statistical and the interactions are independent of the momentum (or the energy) scale. The idea of exclusion statistics has recently been generalized from systems with a single species of indistinguishable particles to systems with several species. In that case, the idea of exclusion statistics has been generalized to that of mutual exclusion statistics [2,3,5,6] and the distribution functions have also been obtained [2]. Further, the many body states and the operator algebra for exclusion statistics has also been written down [7,8]. Several people have shown that the excitations in exactly solvable many-body problems in one dimension like Calogero-Sutherland model [9,10], quasi-particles in Lutinger Model [11] and <5-function gas [12,13] obey this statistics [5,14,15]. Further, excitations in the Haldane-Shastri spin models [16] have been shown to obey [17] the fractional exclusion statistics. In fact it has rigorously been shown that quasi-particles with nontrivial exclusion statistics exist in a class of models that are solved by the Bethe ansatz [3,5]. Application of this statistics to other condensed matter systems has also been considered [18]. A crucial property of exclusion statistical interactions is that they cause shifts in single particle energies at all scales [14,19]. A remarkable feature of this statistics is that the exclusion acts across a set of levels unlike the Fermi or Bose statistics where the exclusion principle is stated with a single level in mind. It is this crucial difference that results in the occurrence of the so called negative probabilities [20,21]. In this chapter, we shall first introduce the basic idea of fractional exclusion statistics and then derive the occupation number distribution function for an ideal gas of particles obeying such statistics. Using this distribution function, we shall then study several thermodynamic properties of such a gas of particles. Finally, we shall also briefly discuss the resolution of the paradox regarding the occurrence of negative probabilities [22,19].

5.2

Distribution Function

Statistics is the distinctive property of a particle (or of an elementary excitation) that plays a fundamental role in determining the macroscopic or the thermodynamic properties of a quantum many-body system. Some time ago, Haldane [1] introduced the concept of fractional exclusion statistics which generalizes the notion of Pauli exclusion principle. For exam-

Ch 5.

Fractional Exclusion Statistics

147

pie, whereas a single-particle quantum state can accommodate an arbitrary number of identical bosons, no two identical fermions can occupy one and the same quantum state. This gives rise to profound differences between the properties of systems obeying the Fermi-Dirac and the Bose-Einstein statistics. Let us define the statistics g of a particle by 9 = -dN+^~dN

(5.1)

where N is the number of particles and d^ is the dimension of the oneparticle Hilbert space obtained by holding the coordinates of iV— 1 particles fixed. Obviously g — 0forbosons since any number of particles can occupy a given state (so that djv+Aiv = d/v)- On the contrary, g = 1 for fermions as implied by the Pauli exclusion principle since it excludes a single state for the remaining particles, the one they occupy. It should be noted that there is no notion of periodicity in the way we have defined the statistics here and it makes sense to consider the cases with g > 1. Clearly, however, g > 0. On the basis of this formula, Haldane proposed the following combinatorial formula for the number of many-body states (W) of N identical particles occupying a group of p states \p+(N-l)(l-g))l N\\p-gN-(l-g)]\-

{

'

For g = 0 and g = 1, this reproduces the well known formulas in the case of bosons and fermions respectively i.e. WB

=

M(p-l)! '

WF=

N\{P-N)\-

(5 3)

'

Using these ideas let us now derive an expression for the occupation number distribution functionforparticles obeying the fractional exclusion statistics. Perhaps it is worthwhile to first spell out the framework of the exclusion statistics. One is here considering situations in which say, p states (p >> 1) in each cell are confined to a finite region of matter i.e. it is finite and extensive, being proportional to the size of the matter region in which the particle exists. Let us imagine dividing the one-particle states into a large number of cells with p > > 1 states in each cell and then count the number of configurations with rii particles in the i'th cell. We are (as is usual in quantum statistical mechanics) considering here an ideal situation in which the total energy and the total number of particles are fixed and given by the simple

148

Fractional Statistics and Quantum Theory

sum

E = J2ni£i> N = Y,mi

(5.4)

i

On generalizing Eq. (5.2), we then have

One underlying assumption in all this is that the definition of the fractional exclusion statistics as given by Eq. (5.1), is applicable locally in phase space. We shall critically examine this assumption later. Let us now minimize the quantity S as given by S = Y)P + (l - g)(ni - l)] ln[p + (l - g){rn - I)] -

mhim

i

-\p-l-

g(rii - 1)]ln[p - 1 - g(m - 1)] - fie^i + /3/j.rii (5.6)

with respect to the occupation numbers n,. Here (3 and /3/x are the Lagrangian multipliers which enforce the constraints of fixed energy and particle number. On differentiating Eq. (5.6) with respect to rii and neglecting terms of 0(1) which are negligible compared to lnp, we obtain dS — = 0 = (1 - g) ln[p + (1 - g)m) -Inm+g

In(p - gm) - /3(e4 - /x) (5.7)

so that the average occupation number n, satisfies [1 - gni]s[l + (1 - g)^}1'9

= nt e«e<-">.

(5.8)

Here Hi = n^/p. Thus the occupation distribution function is given by n = —^

(5.9)

where the function u){^) satisfies the functional equation w(Os[l + OJ(£)]1-9 = £ = e^-rt

(5.10)

with j3 = 1/kT (k being the Boltzmann constant), and \x is the chemical potential. For the special cases of g = 0 and 1 we have w(£) = £ — 1 and w(£) — £ and we immediately recover the familiar Bose and the Fermi distributions respectively. For other values of g, n has to be obtained

Ch 5.

Fractional Exclusion Statistics

149

numerically. However, for special cases including g = | , 2, | , 3, \, 4, it can be obtained analytically. For example, for g — 1/2, Eq. (5.8) becomes a quadratic equation which can be easily solved yielding

fa)9=* =

[1 + 4e2(et-ri/kT]i/2

•

(5-U)

For the intermediate statistics (0 < g < 1), it is not hard to select the solution ui of Eq. (5.10) that interpolates between the bosonic and the fermionic distributions. In particular, when Xi = exp[(ei — fi)/kT] is large, we have u> ~ 1/XJ and hence we recover the Boltzmann distribution function rii « e-^'^/kT

(5.12)

which is valid at sufficiently low densities and/or high temperatures for any statistics. It is also worth noting that Xj is always non-negative and so is ui and in fact u> lies between 0 and 1. It then follows from Eq. (5.9) that for any g rii < 1/5

(5.13)

thereby expressing the generalized exclusion principle for the fractional statistics. In particular, at T = 0, £ = 0 and hence w = 0 if £$ < /i while if e > /i then £ = oo and hence u> = oo. Thus, for any g ^ 0, we see that the average occupation numbers for single particle states with continuous energy spectrum obey

^0

tor £j > no

thereby showing that at T = 0, particles obeying exclusion statistics exhibit a Fermi surface and that below the Fermi surface, the average occupation number is 1/g for each single-particle state which is in agreement with the concept of the fractional exclusion statistics. This is the so called generalized exclusion principle locally in phase space and the particles obeying such a principle have been termed as FES particles. They have also been termed as g-ons [20].

150

5.3

Fractional Statistics and Quantum Theory

Thermodynamics of an Ideal Gas

Let us consider an ideal gas of particles in D dimensions obeying the fractional exclusion statistics with dispersion e(p) = atf .

(5.15)

On using the fact that in D dimensions the volume is given by

it is easy to show that [25] n{e)de sD/a

P =A

(5.17)

Jo where

A

W T O |

+

(518)

I>-

Further, one can also show that

rn{e)dee(Dl°-1\

^ =^

(5.19)

Here n(e) is the distribution function as given by Eqs. (5.9) and (5.10). It is now easy to show that the system satisfies the relationship (5.20)

PV = ?-U where U is the internal energy.

Constant Density of States Let us first specialize to the case D =
/• w(0)

Jo

dtu (

\

1 - u\

J

\n + kT[gln(l-uj)-lnu)\)

^v = ^L rh-

(5.21)

(522)

Ch 5.

151

Fractional Exclusion Statistics

On integration this yields 0,(0) = 1 - e~p/AkT

(5.23)

where w(0) satisfies relation (5.10) with e = 0. On eliminating w(0) we then obtain the chemical potential /x in terms of the density p [23] M(p,

T) = ^ +fcTln(l - e - " / A f c T ) .

(5.24)

Further, the first two integrals in Eq. (5.21) can be easily performed, yielding P=^-

2

2A

+ kTpln[l - e -"/

AfcT

] - A(kT)

2

/

/"u(0)

JO

J

- ^ - lnw . 1— W

(5.25)

On making the change of variable w

= 1 _ e-p'/AfeT

(526)

in the last integral, one finally obtains

p

'€ + i£d"'w^^-y

<527)

Low Temperature and High Density In this case p » AkT, and hence, the upper limit of integration in Eq. (5.27) can be taken to be oo. In that case the integral can be done analytically yielding

P 4 = §£ + ^ W

(5.28)

and hence Cv/V = |TT2AA;2T which is valid to all orders in T. High Temperature and Low Density In this case, p' « AkT and hence exp(p'/AkT) can be expanded and a term by term integration can be done yielding the equation of state

^

= 1+ i ^ -

1

^

) +

|(2iT^(A^

(5 29)

-

where B21 are the Bernoulli numbers. We thus have the remarkable result that for an ideal gas of fractional exclusion statistics (FES) particles with constant density of states (D = a), only the second virial coefficient a
152

Fractional Statistics and Quantum Theory

identical to those of an ideal Bose or Fermi gas in two dimensions (see for example Eqs. (4.22) and (4.23)). Thus, in two dimensions, the case D = a corresponds to an ideal gas of non-relativistic FES particles. This is because, in that case AkT = mkT/2irh2 = X~2 (A being the thermal wavelength), and we find that the equation of state for an ideal fractional exclusion statistics gas in two dimensions is the same as that of an ideal Bose or Fermi gas except for the second virial coefficient. However as seen in the last chapter, in two dimensions, the second virial coefficient does not contribute to the specific heat and hence we have the remarkable result that the specific heat of an ideal fractional exclusion statistics gas in two dimensions (with D = a = 2) is independent of g. From Eq. (5.29) as well as from the results of the last chapter on the third (and the higher) virial coefficients of an ideal anyon gas, it follows that an ideal anyon gas is not an ideal fractional exclusion statistics gas since unlike the FES case, for an ideal anyon gas, 03(0) (and even aN^a), N > 3) are very different from those of an ideal Bose or Fermi gas. Ideal Gas with D ^ a Let us now consider the more general case when D ^ a which is relevant, for example, for an ideal gas of non-relativistic FES particles in one space dimension. Let us first take out the explicit T-dependence of P and N/V by substituting e = xkT in Eqs. (5.17) and (5.19). We obtain

•j^ = Cf§+1(z),y=Cf§(z)

(5.30)

where C = AF(— + l)(kT)D/a

(5.31)

1 f°° fm(z) = ^r^ n(x,z)xm-1dx r(m) J o

(5.32)

while fm(z) is given by

with n(x,z) being as given by Eq. (5.8).

Ch 5.

Fractional Exclusion Statistics

153

Low Density and High Temperature This corresponds to the situation when fn(z) = ^ < < 1. This implies that z itself is smaller than unity. In this case n(x, z) can be expanded as oo

n{x,z) = ^2h(ze-x)1

(5.33)

/=i

where b\ = 1 while 62,63,... can be immediately obtained by substituting this expansion in Eq. (5.8). In this way, we find that 6l

= jj(l-^).

(5.34)

p

P=I

For example, 62 = (1 — 2g). On using the expansion of n(x, z) as given by Eq. (5.33), we then find that for small z, fm(z) is given by

/-(.>- + £ + " + £ •

<0-»»

An elimination of z between the two equations in (5.30) can now be carried out by first inverting the series appearing in the equation for N/V and then substituting this expansion into the series appearing in the equation for P/kT. In this way, one can show that the equation of state has the form of a virial expansion, i.e.

Here the coefficients ai are the virial coefficients of the system. Some of the low lying ones are n -

3

h

-

(g

_ (2g-l)2 3 / + 22Dla 263 D CT 1

b\ 2 /

2D CT

a

-

^364_,36263

~ 1/2) (3fl-l)(3fl-2) ZD/a+l

(W7\

\.d-J°)

56|

Notice that the second virial coefficient is positive for g > 1/2 while it is negative for g < 1/2 so that the purely statistical interaction for an ideal fractional exclusion statistics gas is attractive (repulsive) for g < 1/2 (> 1/2). Not surprisingly, in the special case of D = a, these virial coeffi-

154

Fractional Statistics and Quantum Theory

cients agree with those given by Eq. (5.29). In particular a^ is independent of g while a\ = 0 thereby providing a nontrivial check on the calculations. Hence, the specific heat of an ideal fractional exclusion statistics gas is given by

-?hW)0-!H- <-> Thus, at finite temperature, the value of the specific heat is smaller (larger) than its limiting value of DNk/a provided (g - 1/2)(1 - D/cr) is < (>) 0. High density and low temperature In the case when N/VC » 1, the functions involved can be expressed as asymptotic expansions in powers of (In z)" 1 and this corresponds to the case of degenerate gas. As N/VC —> oo, the functions assume a closed simple form and this is the case of a completely degenerate fractional exchange statistics gas. Since the average occupation number for an ideal fractional exchange statistics gas is 1/g for each single particle state (see Eq. (5.14)), hence it is easily shown that in D dimensions with dispersion e(p) = apa, the Fermi momentum and the zero-point energy per particle of the system are given by /ffAT(f + l)\1/Dh

E0

(D/a\

Thus an ideal fractional exclusion statistics gas at T = 0 behaves similarly to an ideal Fermi gas and in fact EQ/N is identical in the two cases. Let us now extend this discussion to that of low but non-zero temperatures in which case we must consider asymptotic expansion in powers of (lnz)" 1 - the so called Sommerfeld expansion. If we look at the expressions forP and N/V as given by (5.17) and (5.19) respectively, we see that the typical integral is of the form r°°

I[f] = / def(e)n(e). Jo

(5.42)

Ch 5.

Fractional Exclusion Statistics

155

Using Eq. (5.10) this can be re-expressed as an integral over the variable w, i.e. r(0)

du

f[n + gkT\n(l-u>)-kT\mv}. Jo 1— w At low T this can be written as I[f] = kT

I[f] = 9-1 /

f(e)de + J2

JkT lnw(O)

J^Q

{ l

-^Ci(9)f(ri

(5.43)

(5-44)

-

where Cj(fl)=/

^-{[slna-^-ln^'-tfflna-o;)]'}.

(5.45)

Note that here w(0) = 1 — 0{e~tL/gkT) and can be set to unity upto the non-perturbative corrections. We note that C\ (g) is a polynomial of order (I - 1) in g, i-i

Cl(g) = Y/Ci,k9k

(5.46)

fc=0

where

Q,k = (-1)'"* ([) f - lnfc(W) ln;"fc(l - W ).

(5.47)

Thus the first few coefficients are [20] C0(g) = 0, C1(fl) = y , C2(5) - 2(1 - g)((3), C3(g) = ^ ( l - 9~ + g2^

(5.48)

where £(3) is Riemann zeta function (defined by YAL\ js ~ 1-202). Notice also the duality relation Q,k = (-1)1-1 Ci,i-k-i

(5.49)

which is satisfied by C;^. Using the Sommerfeld expansions for the particle number and the energy, the chemical potential and the specific heat can be easily shown to

156

Fractional Statistics and Quantum Theory

satisfy

"•4-£(!-0(£)' + "]

«"•*

where fi0 = (gp/A)a/D is the T = 0 chemical potential and p — N/V is the particle density. Thus Cv oc T at low T(T «TF). General Remarks Let us close this discussion about the ideal fractional exclusion statistics gas by mentioning some of its remarkable properties. (1) Duality: It is not difficult to verify that the distribution functions at g and 1/g are related by the following formula [20,24] 1 - gng(x) = (l/g)n1/g(-x/g)

(5.52)

where x = (3(s — /z). For example, from the basic Eq. (5.8), it is easy to show that the distribution function for g = 2 is given by 1 1/2 ( - ) P - = 2 - (i+4e-)i/»

(5 53)

"

and one then immediately verifies the duality relation by using Eq. (5.11). (2) Sub-Poissonian Statistics: One can check that for any g (> 0), the ideal FES particles, like the fermions, obey sub-Poissonian statistics. For example, using the formula [25]

it is easily shown that for g = 1/2 and g = 2 one has (A^)Vi/2 = n(l-n2/4),

(5.55)

(An) 2 | s = 2 = n ( l - n ) ( l - 2 n ) .

(5.56)

Ch 5.

157

Fractional Exclusion Statistics

Note that in contrast, the bosons (g = 0) are super-Poissonian satisfying (5.57)

{Anj*\g=o = n(l + n).

5.4

Negative Probabilities

Finally we would like to point out the problem of negative probabilities with the fractional exclusion statistics [20,21,26], its origin and its resolution. Consider the expression (5.2) which can also be rewritten as W

_ r\p+l + (N-l)(l-g)] - T(N + l)T\p-g(N-l)] •

(5 58)

-

Note that the dimension W of the many particle Hilbert space for N particles, when the cell includes p states, vanishes at g = m/(N — 1) where m = p,p + l,...,p + N — l. This gives the correct result for fermions namely W = 0 if iV > p, since g = 1 is then one of the zeros of W. However, for the other values of g (> 0), W does not vanish if gN > p and indeed can be negative when gN > p. This is easily seen, for example, when g = 1/2 provided N > 2p and is an even integer. There is in fact a more serious problem. But before we discuss it, we would like to show that the dimension formula (5.2) of Haldane while true for p » 1 is not correct for arbitrary P [21,19]. It is amusing to notice that the solution to this question comes from the analysis of Ramanujan [27], the celebrated Indian mathematician. Ramanujan considered the equation aqXr - X" + 1 = 0,

(5.59)

where a may be complex and q, r are positive numbers. The general solution for Xr is Xr=Y,CN(r,q,p)aN,

(5.60)

JV=O

where Co(r, q,p) = 1 and Ci(r, q,p) = p and JV-l

CN(r,q,p) = j^l[(p

+ Nr-jq),

N>2.

(5.61)

158

Fractional Statistics and Quantum Theory

If we now choose r = 1 — g and q = 1 then this CN reduces to r n n 1 n\ CN(l-g,l,p) =

P(p+[i-g}[N-i}-iy.

m(p-gN)[

'

(5 62)

-

which is clearly different from the dimension formula (5.2) as given by Haldane. Note, however, that both (5.2) as well as (5.62) reproduce the correct bosonic and ferniionic dimension formulas as given by Eq. (5.3) in case g = 0 and g = 1 respectively. It is amusing to notice that the formula (5.62) was also independently obtained by Polychronokos [21] with the restriction that any two particles are at least g sites apart when placed on a periodic lattice. If instead, one requires that any two particles are g sites apart but without the restriction of periodicity then one would derive the Haldane's dimension formula (5.2). The important point is that if we now put X = (1 + u~1) and a = (5.59) then we immediately obtain Eq. (5.10), derived e-0(s-/j.) m gq earlier by Wu. Thus, it is the dimension formula (5.62) which precisely leads to the distribution function derived earlier and not the Haldane formula (5.2). Note, however, that in the limit of p » 1, CN(l-g,l,p) = WN(g,p)+o(^\.

(5.63)

Having obtained the correct dimension formula, let us now explain the more serious problem with the exclusion statistics. The grand canonical partition function of the system may be written as oo

Z(X) = (1 + u-ly = J2 CN(1 - g, l,p)e-NKe-ri ,

(5.64)

N=0

with w satisfying Eq. (5.10). Here one is assuming that all the energy levels are degenerate with energy e. Note that this is an exact expression for Z and no assumption about the single particle dimension p is required. The negative weights arise when one insists on expanding 1 + ui~1 in powers of e'0^-^. Using Eq. (5.64) it follows that OO

1

1 + a," = J2 C "(! - 9,1,1) e-"« e -">.

(5.65)

71=0

O n e c a n n o w s h o w [21] t h a t t h e w e i g h t s Cn(l — g,l,l)

0,(1-2,1,1)= f[ ( l - - ) , m=2

^

'

are

(5.66)

Ch 5.

Fractional Exclusion Statistics

159

which are always negative in case gn > m. This is then the problem of negative weights associated with the exclusion statistics and is claimed to be inherent in this statistics. However, as emphasized in [22,19], the negative probabilities arise because of our insistence on the factorization implied by Eq. (5.64). For example, on combining Eqs. (5.64) and (5.65) we obtain

( ) = E ( Q C ^ (! - 9,1, l))e-*e-"> £, n> ,

Z X

"3

(5.67)

3

where the sum is an unconstrained one over all sets of occupation numbers. The over-counting resulting from this unconstrained sum is compensated by the occurrence of negative weights. As has been shown [22,19], the particles obeying the fractional exclusion statistics may be characterized by constraints on the sets of occupation numbers. There are no negative probabilities if these constraints are obeyed. If we relax these constraints, then negative weights arise in order to compensate for the resulting over counting. This is analogous to other situations in physics where negative probabilities arise. For example, in gauge theories the negative probabilities arise in the ghost sectors. Ghosts come from the Jacobian associated with nonlinear gauges which essentially ensure the correct counting of states. Another example is that of Wigner distribution function in quantum mechanics which is not positive definite precisely because some constraints are relaxed. Summarizing, a unique feature of the exclusion statistics is that the exclusion acts across a set of levels unlike in the case of Fermi or Bose statistics where the exclusion principle is stated with a single level in mind. It is this property which is responsible for the occurrence of negative probabilities. 5.5

Gentile Statistics

In 1940, Gentile [28] generalized the Fermi-Dirac and the Bose-Einstein statistics by allowing for the possibility of (no more than) p (p = 1,2,...) particles occupying the same quantum state and derived expressions for the distribution function and other thermodynamic quantities. It has recently been shown [29] that these results are also valid for a g-fermion [30,31,32] provided q is complex and takes values on a unit circle. In this section, we shall briefly summarize the results for the thermodynamics of an ideal gas of particles obeying Gentile statistics.

160

Fractional Statistics and Quantum Theory

Following the discussion of an ideal fermion gas, it is easily seen that the grand partition function of an ideal Gentile gas of order p is given by

(5 68)

-

c=n O^Pj e 1=0

e

l

\

x

J

where x = ze~l3e, z = e^M and the product is over all possible single particle states with energy e. The mean occupation of a single particle state with energy e is given by

(5-69)

^)-{iiiA/{iiA-

As expected, this formula reduces to the familiar expression for the bosons and the fermions in the limit p —> oo and p = 1 respectively. If the particles obey the dispersion relation e = apa then it can be shown that in D space dimensions one has K1

Jo

V 2^,i=ox /

Elimination of z between these equations leads to the equation of state for this system. One can study this equation in the limit of high density, low temperature as well as low density, high temperature.

High Temperature, Low Density In order to evaluate the virial coefficients, we define

F {z p) =

*>

i

fM

f°°

r

p

yr> ldy

~

i

p

lzl& vl

e

i

T, ~ £ ^ ~ n

(5-72)

where y = (5e. In this case we can write ^=CFz+1(z,p);

y=CFu(z,p)

(5.73)

where C is as given by Eq. (5.31). For small values of N/CV, one can express z in terms of N/CV and hence obtain the equation of state

PV

^ / Ny-i

Ch 5.

161

Fractional Exclusion Statistics

It is lengthy but straightforward to show that Fv(z,p) is identical to the corresponding bosonic value up to terms of O(zp). The first term that is different from the corresponding bosonic value is of O(zp+1). In particular one finds Fv{z,p) = F*{z,p) -

+ O(z^2).

(p+1)v_1

(5.75)

As a result, the first p cluster coefficients as well as the virial coefficients are the same as those of the corresponding ideal Bose gas while the higher ones are different. In particular, it is easily shown that if one writes

z

= bicv)+bAcV)

+

+ +b

- Acv)

-

(5 76)

"

then bn = b% foi n
(5.77)

As a result, one can easily show that an = a% , n < p « P + 1 = «p+l +

P {p+ 1)D/a

(5-78)

•

One can also show that as in the previous case, the internal energy U is again given by U = j^PV and hence from Eq. (5.73) we have

U NkT

DF§+1(z:P)

a Fo(z,p)

{

'

'

a

The specific heat is immediately obtained by differentiating this equation with respect to T, keeping ./V and V constant. We obtain Nk~\Nk)B~

(p + l)D/° \CV)

+U

\CVj

•

(5 80)

-

Thus for large T, the specific heat of an ideal Gentile ideal gas approaches the asymptotic value of D/a from above.

High Density, Low Temperature For T —> 0, (n) is a step function with a step at the Fermi energy e = ep i.e. at zero temperature all single particle states up to SF are completely filled with p particles per state while all states with e > SF are empty.

162

Fractional Statistics and Quantum Theory

Thus for any finite (integral) p, one has a Fermi surface. It is amusing to note that an ideal Gentile gas, specially for reasonably large p, behaves like an ideal Bose gas at high temperatures and like an ideal Fermi gas at low temperatures. For example, at low T, the specific heat is linear in T. Further, for any finite value of p, the specific heat vs temperature curve does not show any discontinuity. Note that in the bosonic case, there is a kink at the critical temperature Tc due to the Bose condensation. The fact that the Gentile statistics is very different from the anyonic statistics clearly comes out from the expressions for the virial coefficients as given by Eq. (5.78). We see that for any p > 2, a2 = af ^ 0,2(01), while for p = 1 one has an ideal Fermi gas which clearly is very different from an ideal anyon gas. 5.6

New Fractional Exclusion Statistics

Recently Polychronakos [21] has suggested another form of the fractional exclusion statistics. It is based on the following combinatorial formula for putting N particles in p states w

=

p(p-g)(p-2g)...(p-(N-l)g) ^

^ ^

This can be thought off as a different realization of the fractional exclusion statistics idea: the first particle put in the system has p states to choose, the next one has p — g due to the presence of the previous one and so on. Further, one divides by AH to avoid over-counting. Such a form is an improvement over the form of fractional exclusion statistics from several angles. Firstly, unlike the fractional exclusion statistics case, here the mean occupation of a single particle state has a nice simple, closed form as given by [33] (n) =

\

(5.82)

so that analytical expressions for all the thermodynamic quantities can be immediately obtained. Secondly, at least for g = 1/p with p being an integer (i.e. a fraction of a fermion), the probabilities as given by Eq. (5.81) are all positive for TV up to p and vanish beyond that. Besides, for g < 0 all probabilities are positive and non-zero. Thirdly, there is a maximum single-level occupancy in accordance with the fraction of a fermion that g represents.

Ch 5.

163

Fractional Exclusion Statistics

Clearly the fermions and the bosons correspond to g — 1 and g = — 1 respectively while g = 0 corresponds to quantum Boltzmann statistics since the distribution is valid for all temperatures and densities. Thus one can say that such a system has a bosonic (g < 0) and a fermionic (g > 0) sector, with the Boltzmann statistics as the separator. If the particles obey the dispersion relation e = apa, then it can be shown that in D space dimensions one has — =A /

kT

-5

ePe+gz

Jo

S =A / ° ° ^

(5.83)

e^de

y

'

(5.84)

s^-'de.

Elimination of z in these equations leads to the equation of state for the system. Let us study this equation in the two extreme limits.

High Temperature, Low Density In order to obtain the virial coefficients, we define

(585

'•"•''-fHjjr*^

»

where y = 0e. In this case we can write ^=CF±+1(z,g),

^ = CFz(z,g).

(5.86)

It is easily shown that the cluster coefficients are the same as in the case of the ideal Bose (Fermi) gas if g < 0 (> 0) and the virial coefficients are given by [33] an+1(g)

= \g\na*+1 n

= g a^+1

if g<0 if g > 0

(5.87)

where a^'F are the virial coefficients of an ideal Bose/Fermi gas in D dimensions with dispersion e = ap". Finally, let us consider the case of the quantum Boltzmann statistics, i.e. g — 0. In this case the integral as given in Eq. (5.85) takes a simpler form 1 f°° Fv{z,g = Q) = -— ze-*yi-Hy 1 WJ Jo

= z.

(5.88)

164

Fractional Statistics and Quantum Theory

As a result, one finds that the ideal quantum Boltzmann gas satisfies the ideal gas equation -^pp = 1 and hence a\ = 1 while all the higher virial coefficients are zero (an = 0,n > 1). From Eq. (5.87) it is clear that this fractional exchange statistics gas is very different from an ideal anyon gas, since, for example, the second virial coefficient is very different in the two cases. Even if we identify \g\ with 1 - 4a + 2a 2 so that a2(|£/|) = 0.2(0), we will find that the corresponding third virial coefficients do not agree. For example, with this identification, we have a3(\g\) = (l-4a

+ 2a2)ai.

(5.89)

However this as does not satisfy the exact relation o 3 (a) = o 3 (l - a)

(5.90)

which is satisfied by the anyons [34]. The case of high density, low temperature can also be considered in a straightforward way and we leave it as an exercise to the reader. Summarizing, we have seen that several other forms of the fractional statistics have been proposed in the literature but none of them is the same as the anyonic fractional statistics.

References [1] F. Haldane, Phys. Rev. Lett. 67 (1991) 937. [2] Y.-S. Wu, Phys. Rev. Lett. 73 (1994) 922. [3] S.B. Isakov, Phys. Rev. Lett. 73 (1994) 2150 ; Int. J. Mod. Phys. A9 (1994) 2563 ; Mod. Phys. Lett. B8 (1994) 319. [4] A. Dasnieres de Veigy and S. Ouvry, Phys. Rev. Lett. 72 (1994) 600; Mod. Phys. Lett. A10 (1995) 1. [5] D. Bernard and Y.-S. Wu, cond-mat/9404025; 6th Nankai Workshop on New Developments of Integrable Systems and Long-ranged Interaction Models, eds. M.L. Ge and Y.-S. Wu (World Scientific, Singapore, 1995). [6] T. Fukui and N. Kawakami, Phys. Rev. B51 (1995) 5239 ; J. Phys. A28 (1995) 6027. [7] D. Karabali and V.P. Nair, Nucl. Phys. B438 [FS] (1995) 551. [8] P. Mitra, hep-th/9411236. [9] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197 ; 12 (1971) 419. [10] B. Sutherland, J. Math. Phys. 12 (1971) 246, 251 ; Phys. Rev. A4 (1971) 2019.

Ch 5.

Fractional Exclusion Statistics

165

[11] F.D.M. Haldane, J. Phys. C14 (1981) 2585; J. Bagger, D. Nemeschansky, N. Seiberg and S. Yankielowicz, Nucl. Phys. B289 (1987) 53 and references therein. [12] C.N. Yang and C.P. Yang, J. Math. Phys. 10 (1969) 1115. [13] E. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605. [14] M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 72 (1994) 3629 ; 73 (1994) 3331. [15] A.P. Polychronakos, Nucl. Phys. B324 (1989) 597 ; Phys. Rev. Lett. 69 (1992) 703. [16] F.D.M. Haldane, Phys. Rev. Lett. 60 (1988) 635 ; B.S. Shastri, 60 (1988) 639. [17] F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, d. Bernard and V. Pasquier, Phys. Rev. Lett. 69 (1992) 2021. [18] R.K. Bhaduri, R.S. Bhalerao and M.V.N. Murthy, J. Stat. Phys. 82 (1996) 1659 ; D. Sen and R.K. Bhaduri, Phys. Rev. Lett. 74 (1995) 3912 ; R.K. Bhaduri, M.V.N. Murthy and M.K. Srivastava, 76 (1996) 165. [19] M.V.N. Murthy and R. Shankar, IMSC/99/04/12. [20] C. Nayak and F. Wilczek, Phys. Rev. Lett. 73 (1994) 2740. [21] A.P. Polychronakos, Phys. Lett. B365 (1996) 202. [22] S. Chaturvedi and V. Srinivasan, Phys. Rev. Lett. 78 (1997) 4316. [23] S.B. Isakov, D.P. Arovas, J. Myrheim and A.P. Polychronakos, Phys. Lett. A212 (1996) 299. [24] A.K. Rajagopal, Phys. Rev. Lett. 74 (1994) 1048. [25] R.K. Pathria, Statistical Mechanics (Pergamon Press, Oxford, 1972). [26] S.B. Isakov, Phys. Rev. B53 (1996) 6585. [27] B.C. Berndt, R.J. Emery and B.M. Wilson, Adv. Math. 49 (1983) 123. [28] G. Gentile, Nuovo Cim. 17 (1940) 493; 19 (1942) 106. [29] R. Dutt, A. Gangopadhyaya, A. Khare and U.P. Sukhatme, Int. J. Mod. Phys. A9 (1994) 2687. [30] L.C. Biedenharn, J.Phys. A22 (1989) L873. [31] A.J. Macfarlane, J.Phys. A22 (1989) 4581. [32] R. Parthasarathy and K.S. Vishwanathan, J. Phys. A24 (1991) 613. [33] R. Acharya and P. Narayana Swami, J. Phys. A27 (1994) 7247. [34] D. Sen, Phys. Rev. Lett. 68 (1992) 2977.

Chapter 6

Introduction to the Chern-Simons Term One touch of Nature makes all worlds akin — E.A. Abbott in Flatland 6.1

Introduction

In the last five chapters, we have studied the quantum and the statistical mechanics of non-relativistic anyons. No knowledge of field theory was required. In the next two chapters we shall study relativistic and nonrelativistic quantum field theories where anyons occur either as solitonic excitations (Chapter 7) or as elementary field quanta (Chapter 8). We shall see that all this is possible provided these field theories contain the topological action, i.e. either the Chern-Simons term or its incarnation, the Hopf term. It may therefore be worthwhile to first introduce the ChernSimons (CS) term (in 2+1 dimensions) and discuss its various properties in some detail [1]. This is what we propose to do in this chapter. The most important property of the Chern-Simons term in 2+1 dimensions is that when it is added to a theory containing the gauge kinetic energy term, it makes the gauge field quanta massive and yet the action is gauge invariant! Further, in the non-abelian case, the coefficient of the Chern-Simons term has to be quantized as otherwise, the theory is not well defined. The Chern-Simons term violates the discrete symmetries of parity (P) and time inversion (T), even though the combined symmetry PT is still preserved. Because of this term, the vacuum polarization tensor, in 2 + 1 dimensions, has an extra, unusual, parity (P) and time inversion (T) violating piece. As a result, the vacuum of such a theory exhibits an unusual effect called the magneto-electric effect. Further, just like the gauge field 167

168

Fractional Statistics and Quantum Theory

mass term, the Chern-Simons term, at least in the abelian case, can also be generated by spontaneous symmetry breaking. Unlike the Maxwell term, for the Chern-Simons term, the gauge invariance automatically ensures the Lorenz invariance. Besides, a gauge theory with pure Chern-Simons action is an example of a topological field theory since this term has the same form in the flat and the curved space-time without any additional metric insertions. Finally, note that the Chern-Simons term is not a luxury, at least in the non-abelian case, since, even if one does not add it to the action at the classical level, it is automatically induced by the quantum corrections. It is worth pointing out that in recent years, the Chern-Simons term has found diverse applications, from condensed matter physics to supergravity and from string theory to mathematics. As we shall see in the next two chapters, the possibility of anyonic statistics can be elegantly formulated by using the Chern-Simons term. Further, this term is known to play an important role in the fractional quantum Hall effect (Chapter 10) and in anyonic theories of superconductivity (Chapter 9). As mentioned above, a gauge theory with pure Chern-Simons action results in a topological field theory. The only observables of this theory are Wilson loops, whose expectation value gives rise to knot invariants. Considering such a theory with a non-compact gauge group, for example the 2+1 dimensional Poincare group, gives a consistent quantum theory of 2+1 dimensional gravity. Finally, theories in 2+1 dimensions with the Chern-Simons term may have physical significance in case these models in their Euclidean version emerge as the high-temperature limits of the realistic four-dimensional theories. 6.2

What is the Chern-Simons Term?

Consider the Lagrangian density for classical electrodynamics in 3+1 dimensions as given by C = —F^F^

+^(i 7 M D" -

TO)V>

(6.1)

where F^ = d^Av — d^A^ and D^ = d^ — ieA^ is the covariant derivative. Here <9M = gf-, while A^ and ip denote the gauge potential and the Fermi field respectively. This Lagrangian is invariant under the local gauge transformation V>(z) -> eiea^iP(x),

A^x) -> A^x) + d^aix).

(6.2)

Ch 6.

Introduction to the Chem-Simons Term

169

Similarly, for massless fermions (TO = 0), this Lagrangian is also invariant under the (global) chiral transformation ^ x)

^

e

^ ( z ) , A^x)

-> A^x).

(6.3)

The naive expectation was that, these two symmetries i.e. the gauge and the chiral symmetries, which are valid at the classical level, will continue to hold good even in the quantum theory. As a consequence, one expected that the vector and the axial vector currents j ^ — tpj^ip and j ^ = ip'y^751/' which are conserved at the classical level, will continue to remain conserved even in the quantum theory. It has however, been shown [2] that this is not so. There is no regularization which can simultaneously preserve both these symmetries at the quantum level. Because of the unexpected result, it was called an anomaly at that time (and unfortunately even today it is called so), even though the correct name should have been quantum mechanical symmetry breaking. Remarkably, the entire effect comes only from the one loop diagrams and two and higher loops do not contribute to the anomaly. In view of our strong faith in the gauge symmetry, one therefore says that it is the chiral symmetry which is broken by the one loop quantum corrections. By now it is well known that this feature is in fact universal, and valid in any even dimension 2n, and also in both abelian and non-abelian gauge theories. In particular, there is a gauge singlet (axial) anomaly in any even dimension, (2n) so that the divergence of the gauge singlet axial current, even for massless fermions, is non-zero and proportional to the corresponding Chern-Pontryagin density P^n in that (even) dimension 2n i.e. &>jl(x) oc P2n.

(6.4)

It is also well known that the Chern-Pontryagin density can always be written as a total divergence P2n = 9MA" , n = 0,1,2,..., 2n - 1.

(6.5)

The object AM, for a particular value of fi (say /1 = 2n-l) naturally lives in odd (2n-l) dimensions and is known as the Chern-Simons density in that dimension. Thus, whereas the Chern-Pontryagin density lives in even space-time dimensions, the Chern-Simons density lives in odd space-time dimensions. For example, the gauge singlet anomaly in 3+1 dimensional

170

Fractional Statistics and Quantum Theory

quantum electrodynamics is given by d^jl = ^-e^F^F^

=

Z7T

TT

e

ld»{e^XaAvFx°)

(6.6)

so that the abelian Chern-Simons term in 2+1 dimensions is given by

Jcs=

I CCSd3x

<x /'cPx£vXcrAl'FXa .

(6.7)

Throughout this book we shall mainly be concerned with this Chern-Simons term or its non-abelian generalization. Let us therefore discuss in some detail the various properties of this term.

6.3

Gauge Invariant Mass Term

Let us consider pure electrodynamics in the presence of the Chern-Simons term in 2+1 dimensions [3,4] C = - ^F^F^ + ^e^F^Ax .

(6.8)

Since in 2 + 1 dimensions the mass dimension of A^ is 1/2, hence it follows that the mass dimension of the parameter /x is 1. The field equation following from this Lagrangian is W™

+ lfa0FecfS = 0

(6.9)

which is invariant under the gauge transformation A^ —> A^ + d^a. On the other hand, the Chern-Simons Lagrangian density changes by a total derivative so that the corresponding action is indeed gauge invariant. We thus find that, unlike the Chern-Pontryagin term which has only a nontrivial topology but no dynamics (being a total divergence), the Chern-Simons term (at least the non-abelian one) has nontrivial topology as well as nontrivial dynamics in it. The field Eq. (6.9) can also be written as

Wv + -e^ada J *FV = 0

(6.10)

where *FV is the dual field strength which is a vector in 2+1 dimensions i.e. *FV = ^euaf}FaP

• F^ = SfiUa*Fa

.

(6.11)

Ch 6.

Introduction to the Chern-Simons Term

On operating by (g^ - j^s^sd6)

171

to Eq. (6.10), we get

(n + fj,2)*Fp=0

(6.12)

which clearly shows that the gauge field excitations are massive with the gauge field mass /x being the coefficient of the Chern-Simons term. It is gratifying that /i 2 occurs in Eq. (6.12) with the correct sign for a propagating particle. Note that we have no a priori control over this sign since the Chern-Simons Lagrangian is linear in /z (this // can be both positive or negative). We have thus shown that the Chern-Simons term when added to the Maxwell kinetic energy term for the gauge field, acts as the gauge invariant gauge field mass term. It is worth adding that this remarkable property of having a gauge invariant mass term for the gauge field in the action itself is very special to 2+1 dimensions. It may be added here that the Chern-Simons action alone does not have real dynamics of its own since the corresponding field equation is F^ = 0. Thus the Chern-Simons gauge field is a non-propagating field whose dynamics comes from the fields to which it is minimally coupled. In particular, consider C = ^e^xF^Ax

+ A^J* .

(6.13)

This gives rise to the equations of motion p=J°^fj,B,

(6.14)

r = fiefiEj .

(6.15)

We shall see in the subsequent chapters that these two are the most basic relations in the Chern-Simons field theories. The relation (6.14) is a Gauss law constraint which on integrating over the whole space gives a remarkable relation between the Noether charge Q and the total flux $ i.e. Q=

/ pd2x = fj, / Bd2x = n^.

(6.16)

We shall see in the next chapter that in view of this relation, the abelian Higgs model with the Chern-Simons term admits charged vortices which in fact have been shown to be charged (extended) anyons. Note that these objects have non-zero and quantized magnetic flux as well as Noether charge. Yet another unusual property of the Chern-Simons term is that the Chern-Simons action is only first order in the time derivative. As a result,

172

Fractional Statistics and Quantum Theory

A\ and A2 appear as canonically conjugate fields. On the other hand, the Maxwell kinetic energy term is second order in the time derivative and hence A\ and A2 can both be regarded as coordinate fields which are canonically conjugate to the electric fields E\ and E2 respectively. 6.4

Behavior Under C, P, and T

Let us consider the behaviour of the Chern-Simons term as well as the Dirac Lagrangian CD = i ^ . f l " - m)ip

(6.17)

under the discrete transformations of charge conjugation (C), parity (P), and time inversion (T). Here, ip is a two component spinor with mass m (> 0) and the mass dimension of ip is 1. We use the following twodimensional realization of the Dirac algebra 7 ° = (T

3

, f1=UT1,

y 7 " = g^ - ie^ala

I2 = i
(6.18)

; cT = diag.Q., - 1 , -1)

(6.19)

where az are the usual Pauli matrices. It is easily shown that under charge conjugation CTV+

CA^C-1 = -Ap , CipC-1 =

(6.20)

so that the action is invariant under charge conjugation C. On the other hand, under parity transformation, the gauge and the Fermi fields transform as follows PA0'2(t,T)p-1=A°'2(t,rf),

PA1{t,r)p-1

= -A1{t,r'),

Pipit, r)P~l = orV(*,r') •

(6.21) (6-22)

Note that in 2+1 dimensions, the parity transformation is somewhat unusual i.e. r = (x, y), r' = (—ar, y) (or (x, — y)). On the other hand, (—x, —y) corresponds to rotation (and not space reflection). As a result, we find that the mass terms for both the Fermi and the gauge fields (i.e. mtyty and the Chern-Simons term) are not invariant under parity (P). Similarly, time inversion changes the signs of both the mass terms since TA*{t,v)T-x = A°(-t,r),

TA^iOT"1 = -A(-t,r),

(6.23)

Ch 6.

Introduction to the Chern-Simons Term

TWrfT-1

= o*1>(-t,T).

173

(6.24)

Thus, both the Chern-Simons term as well as the fermion mass term, mipi/j are non-invariant under parity (P) as well as under time inversion (T). However, they are invariant under the combined operation PT and hence the CPT symmetry is still valid. Note that in 3 + 1 dimensions though, mipip is separately invariant under parity (P), charge conjugation (C) and time inversion (T) . Finally, let us talk about the photon spin. One can show that the Chern-Simons photon spin is 1(—1) if Chern-Simons mass \x > 0 (< 0) while the spin of the massless photon is zero. Further, in either case, the photon has only one degree of freedom. One simple way to see this result is to note that the angular momentum M as well as the Pauli-Lubanski invariant £tivaMilvPa are odd under parity transformation. Hence, in a parity conserving theory with one degree of freedom (this is the case when fi — 0), the spin must be zero since additional degrees of freedom are unavailable to form the parity doubled, non-zero spin states [5]. By the same reasoning, a single degree of freedom must carry non-vanishing spin in case parity is violated (// ^ 0). If we consider a theory containing both topological masses ±/x, then both spins ±1 are present and hence parity is conserved and the theory has two degrees of freedom. A similar thing also happens in a conventional massive (gauge non-invariant) vector theory which also possesses two degrees of freedom with spin ±1 and the theory is parity conserving. 6.5

Coleman-Hill Theorem

It turns out that because of the parity (P) and the time inversion (T) violating but gauge invariant Chern-Simons term, the most general form for the vacuum polarization tensor consistent with Lorentz and gauge invariance is more general than in other dimensions i.e.

IWfc) = (fc 2 ^ - fcMMni(fc2) - ie^xk^ik2).

(6.25)

Note that the second term on the right hand side is odd under parity (P) as well as under time inversion (T). Clearly, any parity (P) and time inversion (T) violating interaction will contribute to U2(k2). For example, the fermion mass term, which violates both parity (P) and time inversion

174

Fractional Statistics and Quantum Theory

(T), does contribute to n2(fc2) at one loop. Remarkably enough, it was discovered that at two loops, however, there is no contribution to n 2 (0) and hence to Chern-Simons mass [6,7]. Inspired by this result, Coleman and Hill [8] have in fact proved under very general conditions that 112(0) receives no contribution from two and higher loops in any gauge and Lorentz invariant theory including particles of spin 1 or less (An open question is whether this is also valid for higher spin theories, specially spin-3/2). Their proof only requires that the matter fields be massive so that one does not have to worry about the infrared problems. Further, they also assume that no part of the free electro-magnetic Lagrangian density is hiding in the matter part of the Lagrangian. It may be noted that their result is valid even for non-renormalizable interactions in the presence of the gauge and Lorentz invariant regularization. Coleman and Hill also claimed that at one loop, the only contribution to n2(0) can come from the fermion loop. This is, however, incorrect. In particular, there is no reason why parity (P) and time-inversion (T) violating interactions involving spin-0 or spin-l particles should not contribute to 112(0) at one loop. In fact, it has been shown that the parity violating spin-0 [9] as well as spin-l interactions [10] do contribute to ^ ( 0 ) at one loop. It may be noted here that apart from these situations which were overlooked by Coleman and Hill, there are other situations where the initial assumptions of the theorem are not satisfied, and not surprisingly, ^ ( 0 ) does get further radiative corrections. One such situation is if there are massless particles present, in which case, infrared divergences spoil the proof of the theorem [8,11,12]. Another case is if Lorentz or gauge invariance is not satisfied, a situation found in the non-abelian case. A third case is that of spontaneously broken scalar electrodynamics [13,14], where the term quadratic in the gauge field explicitly violates one of the assumptions of Coleman and Hill. However, recently it has been shown [9] that if the theorem is restated in terms of the effective action, rather than n2 (0), then at one loop, there are no radiative corrections to the Chern-Simons term in the effective action even though there are corrections to 112(0) at one loop. Summarizing, one finds that under very general conditions, the ChernSimons mass does not get any contribution from two and higher loops. This is a very important result but a deeper understanding of it is still lacking.

Ch 6.

6.6

Introduction to the Chern-Simons Term

175

Magneto-Electric Effect

There are many crystals in nature like chromium oxide, which show the magneto-electric effect i.e., they also get magnetically polarized in an electric field and electrically polarized in a magnetic field [15,16]. It is well known that this effect depends upon having a CP-asymmetric medium. Mathematically, the signal for the magneto-electric effect in 2+1 dimensions is that the relation between the excitation fields D and H and E and B is modified to A - x^Ej

+x^B;

H = X{m)B + x|me)£?i.

(6.26)

It has been shown [17] that the vacuum of the 2 + 1 dimensional quantum electrodynamics with Chern-Simons term as given by Eq. (6.8) along with the minimal interaction (say as given in Eq. (6.13)) also shows the magneto-electric effect. In particular, it has been shown that both x\ are and Xi non-zero and proportional to fcjll^fc2). Of course this is not really surprising if one remembers that the Chern-Simons term violates the discrete symmetries of parity (P) and time inversion (T). 6.7

Chern-Simons Term by SSB

We have seen above that the Chern-Simons term provides mass to the gauge field. Now, usually the gauge field mass is generated by spontaneous symmetry breaking; hence it is worth enquiring whether the Chern-Simons term can also be generated by spontaneous symmetry breaking. The answer to the question is yes [18]. This is because, unlike other dimensions, in the 2+1 case, one can have a more general definition of the covariant derivative. In particular, it is easily seen that V^

= (0M - ieA^ - ige^xF"^

(6.27)

also transforms as a covariant derivative, since the field strength FvX by itself is gauge invariant. Obviously, the same thing is also true for a spin-0 charged scalar field. Now consider the following generalized abelian Higgs model in 2 + 1 dimensions

£ = ~\F^F^

+ i(2V)*(D'V) - a(|0|2 - a2)2

(6.28)

where the generalized covariant derivative is as given by Eq. (6.27) while 4> denotes the charged scalar field. On expanding the term |(X>;i<£)*(T>'1),

176

Fractional Statistics and Quantum Theory

we have

+ ig*F^l((P*d'i4>-4>d^cl)*) + ese MI/A (dM")A A |0| 2

(6.29)

so that if 4> acquires a nonzero vacuum expectation value then the abelian Chern-Simons term is generated from the last term of this equation. Clearly a similar mechanism should also work for the non-abelian case, but technically it is a tougher problem since one also has to generate the non-linear term. Hopefully, it will be solved in the near future. It is worth noting that the generalized covariant derivative gives rise to the Pauli type non-minimal interaction Cint

= ig *F^

, J" = - (f>d>*(f>*

(6.30)

for even the charged scalar fields. Since the Pauli type non-minimal term gives rise to the (anomalous) magnetic moment, this implies that in 2 + 1 dimensions, even scalar particles can have non-zero magnetic moment. Further, even point particles can have non-zero magnetic moment. It should be noted here that such a non-minimal interaction is not really a luxury; it has been shown [19] that even if such a non-minimal interaction is absent at the tree level, it is automatically generated at the one loop by the radiative corrections. The tree level anomalous magnetic moment can produce some very unusual effects. For example, it can lead to an attractive interaction between the like charged particles [20] so that charged scalar pairs can condensate into bound states. Further, even though the tree-level non-minimal interaction is a non-renormalizable interaction, it has been shown [21] that at one loop it still results in a renormalizable interaction. For a special value of the non-minimal coupling constant g [22], it has been shown that the physical photon mass does not receive radiative correction at the one loop level [23].

6.8

Lorentz Invariance from Gauge Invariance

One of the remarkable properties of the abelian Chern-Simons term is that in this case the Lorentz invariance of the action automatically follows from the gauge invariance. In contrast, notice that the most general form of the

Ch 6.

Introduction to the Chern-Simons Term,

177

gauge invariant Maxwell Lagrangian in classical electrodynamics in 3 + 1 dimensions is C = E 2 + aB 2 .

(6.31)

It is only the demand of the Lorentz invariance which tells us that a = — 1 (In the 2 + 1 case, one has aB 2 since B is a pseudo scalar but the same argument is still valid). On the other hand, if one writes the Chern-Simons action as

Ics=[£csd3x

(6.32)

where CCS = eijEiAj

+ aBA° ,

(6.33)

then the demand of the invariance of Ics under the gauge transformation Ap —> Afj, + d^a fixes a and uniquely gives us the Chern-Simons action which is automatically also Lorentz invariant.

6.9

Quantization of Chern-Simons Mass

Let us now discuss the Chern-Simons term in the non-abelian gauge theories. We shall mention only those properties which are special to the nonabelian Chern-Simons term. To begin with, notice that the non-abelian Chern-Simons term has an extra term compared to the abelian case i.e.

jics) = ^ l ^

x £

^

t r

^

A x

_ 1A»AVA^)

(6.34)

where A^ and F^ are matrices A^ = gTaA1;

F^ = gTaF«v = d»Av - d ^ + [A^, Av].

(6.35)

Here, Ta are the representation matrices of the gauge group G satisfying [Ta,Tb] =fabcTc

(6.36)

where fabc are the structure constants of the group. In the case of SU(2), Ta = ra/2i. Here r a (a = 1,2,3) are the Pauli matrices.

178

Fractional Statistics and Quantum Theory

Let us now consider a non-abelian gauge theory with the Chern-Simons term as given by A» = ^tr(F^F^)

- ^e^xtr

(F^AX

-

\A^A

X

\

(6.37)

The field equation which follows from here is

V "

+ f £uaPFafi = 0

(6.38)

where D^ = d^ + [Ap, ]

(6.39)

is gauge covariant. As in the abelian case, it immediately follows that the Chern-Simons term provides a gauge invariant gauge field mass /x. In particular, the field Eq. (6.38), in terms of the dual field, takes the form Dli*Fv - DV*F^ - fiF^ = 0

(6.40)

where the dual field *FV is as defined by Eq. (6.11), which in the nonabelian case satisfies the Bianchi identity Dll*F'i = 0.

(6.41)

On applying another covariant derivative on Eq. (6.40) and using the identify [D^Dv] = Fpv, we have (£>„£>" + n2)*Fx = eXSr,[*F5, *F"]

(6.42)

which is the non-abelian analogue of Eq. (6.12). As in the abelian case, the non-abelian Chern-Simons Lagrangian density changes by a total derivative under an infinitesimal local gauge transformation so that the corresponding action is invariant under such a gauge transformation. However, the Chern-Simons action is not invariant under finite (also called homotopically non-trivial, or those which are not continuously deformable to the identity) gauge transformations as given by

AM —» U^A^U

+ U-'O^U.

(6.43)

As a result, one finds that the action corresponding to the Lagrangian (6.37) transforms as follows

Ch 6.

Introduction to the Chern-Simons Term

Ina —+ Ina + V J

179

(PxE^trfdviA^dxUp-1])

+ | /'d 3 xe^ A ir[(^C/)[/- 1 (^t/)C/- 1 (aAC/)t/- 1 j . (6.44) Let us consider those gauge transformations which tend to the identity at temporal and spatial infinity so as to avoid a convergence problem in Eq. (6.44) i.e. U{X)X^? I.

(6.45)

It is now easily seen that the gauge field dependent surface integral in Eq. (6.44) vanishes. However, the last term in the integral is non-zero. It can be converted to a surface integral once the integrand is rewritten as a total derivative. This can be made manifest by using an explicit parameterization for U. For example, in the case of 5/7(2) (more generally, we choose SU{2) sub-group of the gauge group G; for reasons that will be clear soon), one can make use of the exponential parameterization U(X) = exp(iaa9a(x)).

(6.46)

In this way one can show that under large gauge transformations, Ina is not invariant but transforms as Ina - Ina + ^ W )

(6-47)

where

LO(U) = —jL f^xe^tr^d^U-^d^U-^dx^U-1]

(6.48)

is the winding number of the gauge transformation U [24]. In particular, if the gauge group G is such that the third homotopy group of G is non-trivial i.e. TT3(G) = Z

(6.49)

where Z is the additive group of integers, then under these so called large gauge transformations, the action transforms as Ina ~* Ina + ^J^m

(6.50)

where m is an integer. Note in particular, that Eq. (6.49) is true for any gauge group G of which SU(2) is a sub-group. We thus conclude

180

Fractional Statistics and Quantum Theory

that the action corresponding to the non-abelian gauge theory with the Chern-Simons term is not invariant under the large gauge transformations but changes by 87r2fj.m/g2. However, in the path integral formulation, the action itself may or may not be gauge invariant but, it is the exponential of the action (exp(i/ na )) which should be gauge invariant. In this way we conclude that the non-abelian gauge theory with the Chern-Simons term does not make sense in 2 + 1 dimensions unless the Chern-Simons mass fi is quantized [4] in units of g2/4n i.e. - ^ g2

= 2nn or n = —n,

(6.51)

4TT

where n = 0, ± 1 , ± 2 , . . . . It is worth noting that the gauge coupling constant g (as well as the electro-magnetic coupling constant e) have mass dimension of 1/2 in 2 + 1 dimensions. This mass quantization is reminiscent of the famous Dirac quantization in the case of magnetic monopole [24]. An important question to address is whether the quantization condition (6.51) is respected by the quantum corrections. This issue was considered by Pisarski and Rao [25] for the case of a pure gauge theory (i.e. without any matter field). They found that the quantization is indeed preserved at one loop; however, the integer on the right hand side of Eq. (6.51) is shifted by N in case the gauge group G = SU(N). Subsequently, it has been shown that there are no further corrections from two and higher loops in the limit of the pure Chern-Simons gauge theory [26]. How does the quantization condition modify in the presence of the matter fields? It has been shown that so long as the scalar field does not break the non-abelian gauge symmetry, then the quantization condition remains unaltered. The massive fermions, of course, modify the quantization condition [25,27] ; the right hand side of Eq. (6.51) being shifted by -^f-fTft, where TR is the Casimir generator for the gauge group G (i.e. tr{TaTb) = —5abTR), in case the fermions are in the fundamental representation of the gauge group G. Thus the quantization is preserved so long as TR is an integer. Much more interesting is the case of partial (spontaneous symmetry) breaking of a non-abelian gauge symmetry. In this case it has been shown that if the non-abelian gauge symmetry SU(N) is spontaneously broken to say SU(M) ® 17(1) (or even several U(l)'s), then the one-loop radiative correction to the right hand side of the quantization condition (6.51) [28,29] arises purely from the unbroken non-abelian sector in question, the orthog-

Ch 6.

Introduction to the Chern-Simons Term

181

onal U(l) sector makes no contribution. This implies that the coefficient of the Chern-Simons term is a discontinuous function over the phase diagram of the theory. Finally, if the SU(N) symmetry is completely (spontaneously) broken, then it turns out that the one loop correction is a complicated function of the three mass scales in the problem (i.e. Chern-Simons mass //, symmetry breaking mass scale and the physical Higgs mass scale) and hence the Chern-Simons mass is now no more quantized [30]. However, it is not clear whether in this case the calculation is that of the coefficient of the Chern-Simons term alone. 6.10

Parity Anomaly

Is our entire discussion about the Chern-Simons term merely of academic interest? Put differently, some one might argue that since the Chern-Simons term violates both the parity (P) and the time inversion (T) symmetries, why should one, in the first place, add such a term to the action? The answer to this question, at least in the non-abelian gauge theories, is that even if one does not add the Chern-Simons term to the action at the tree level, it is automatically generated by the one loop radiative corrections due to the so called parity anomaly [27,31]. In particular, consider the action

I[A», 1>] = Jd3x [^L*r(fMI/f"") + *^yM(0* - ieA^J

(6.52)

for an odd number of massless doublet of fermions in the fundamental representation coupled to SU(2) gauge fields (more generally any gauge group G of which SU(2) is a sub-group so that Eq. (6.49) is satisfied; and the fermions are required only to be in the fundamental representation). This action is invariant under the gauge transformations (both large and small) as well as the discrete transformations of parity (P) and time inversion (T). However, the effective action Ieff [A], obtained by integrating out the fermionic degrees of freedom, must violate one of the two symmetries. In other words, there is no regularization which can simultaneously maintain the invariance of / e // [A] under the large gauge transformations as well as parity P and time inversion T. In view of the tremendous success of the gauge principle, one usually maintains the gauge invariance at the cost of the parity (P) and the time inversion (T) by simply adding the Chern-Simons term to the action (alternately one can also regulate it by

182

Fractional Statistics and Quantum Theory

using the P and T violating Pauli-Villars regularization). In this way, one finds that the Chern-Simons term is induced by the radiative corrections even if it is absent at the tree level. This is very similar to the way the Chern-Pontryagin term is induced in even dimensions due to the gauge singlet (chiral) anomaly (the parity anomaly can similarly be shown to exist in higher odd dimensions). Note however, that whereas the anomaly in even dimensions is in the divergence of the current, in odd dimensions there is no anomaly in the divergence of the current. Instead the anomaly is in the current itself. In particular, there is a piece with the wrong parity in the current. Further, whereas the even dimensional anomaly is for both the abelian and the non-abelian gauge theories, the parity (P) and the time inversion (T) anomaly in odd dimensions is only in the non-abelian gauge theories; after all, there are no large gauge transformations in the abelian case in the first place!

6.11

Topological Field Theory

One of the most remarkable property of the Chern-Simons action is that it depends only on the antisymmetric tensor Eiiv\ and not on the metric tensor g^v. As a result, the Chern-Simons action in the flat and the curved space is the same. Hence, the Chern-Simons action, in both the abelian and the non-abelian cases, is an example of the topological field theory [32,33,34]. It might be mentioned here, that the topological field theories give a natural framework for understanding the Jones polynomials of the Knot theory in terms of three dimensional terms. Further, these theories have shed new light on conformal field theories in two space-time dimensions. Finally, the gravitational Chern-Simons term has also been considered [4,35] and shown to have some remarkable properties. In particular, whereas the massless Einstein theory in 2 + 1 dimensions is trivial, it acquires a propagating, massive, spin-2 degree of freedom when the Chern-Simons term is present. Further, even though this topological term has third time derivative dependence, yet the theory is ghost-free and unitary and one has a consistent quantum theory. The contribution of the topological mass term to thefieldequations also has a natural geometric significance: it is the three dimensional analogue of the Weyl tensor. As in the gauge theory, the gravitational Chern-Simons action also responds non-trivially to large gauge transformations and a possible quantization of the dimension-less quantity k2/j,, is formally the same as in the vector case. Here, /J, is the gravitational

Ch 6. Introduction to the Chern-Simons Term

183

Chern-Simons mass while k~2 is the coefficient of the Einstein part of the action with the dimension of mass. However, in the gravitational case, the internal group is non-compact SU(2,1), rather than the compact 50(3) or SU(2) and hence quantization will not occur since the maximal compact subgroup, S0(2), is homotopically trivial.

References [1] For a detailed review of the various properties see for example, A. Khare, Fort. der. Physik 38 (1990) 507 ; Proc. Indian Natn. Sc. Acad. A61 (1995) 161. [2] S.L. Adler and W.A. Bardeen, Phys. Rev. 182 (1969) 1517 ; For a detailed discussion of the chiral anomalies see, S.B. Treiman, R. Jackiw and D. Gross, Lectures on Current Algebra and its Applications (Princeton Univ. Press, Princeton, 1972). [3] W. Siegel, Nucl. Phys. B156 (1979) 135; J. Schonfeld, Nucl. Phys. B185 (1981) 157; R. Jackiw and S. Templeton, Phys. Rev. D23 (1981) 2291; C.R. Hagen, Ann. Phys.N.Y.) 157 (1984) 342. [4] S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48 (1982) 975; Ann. of Phys. 140 (1982) 372. [5] B. Binegar, J. Math. Phys. 23 (1982) 1511. [6] Y. Kao and M. Suzuki, Phys. Rev. D31 (1985) 2137. [7] M. Bernstein and T. Lee, Phys. Rev. D32 (1985) 1020. [8] S. Coleman and B. Hill, Phys. Lett. B159 (1985) 184. [9] A. Khare, R.B. MacKenzie and M.B. Paranjape, Phys. Lett. B343 (1995) 239. [10] C. R. Hagen, P.K. Panigrahi and S. Ramaswami, Phys. Rev. Lett. 61 (1988) 389. [11] G.W. Semenoff, P. Sodano and Y.-S. Wu, Phys. Rev. Lett. 62 (1988) 715. [12] V.P. Spiridonov, JETP Lett. 52 (1990) 513 ; V.P. Spiridonov and F.V. Tkachov, Phys. Lett. B260 (1991) 109. [13] S.Yu. Khlebnikov, JETP Lett. 51 (1990) 81. [14] V.P. Spiridonov, Phys. Lett. B247 (1990) 337. [15] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, Second Edition (Pergamon Press, Oxford 1963). [16] T.H. O'Dell, The Electrodynamics of Magneto-Electric Media (NorthHolland, Amsterdam, 1970). [17] A. Khare and T. Pradhan, Phys. Lett. B231 (1989) 178. [18] S.K. Paul and A. Khare, Phys. Lett. B193 (1987) 253, B196 (1987) E571. [19] I.I. Kogan, Phys. Lett. B262 (1991) 83. [20] Y. Georgelin and J.C. Wallet, Phys. Rev. D50 (1994) 6610. [21] M.E. Carrington and G. Kunstatter, Phys. Lett. B321 (1994) 223. [22] J. Stern, Phys. Lett. B265 (1991) 119.

184

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

Fractional Statistics and Quantum Theory

Y. Georgelin and J.C. Wallet, Int. J. Mod. Phys. A7 (1992) 1149. R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). R.D. Pisarski and S. Rao, Phys. Rev. D32 (1985) 2081. G. Giavarini, C.P. Martin and F. Ruiz Ruiz, Nucl. Phys. B381 (1992) 222. A.N. Redlich, Phys. Rev. Lett. 52 (1984) 18 ; Phys. Rev. D29 (1984) 2366. A. Khare, R.B. MacKenzie, P.K. Panigrahi and M.B. Paranjape, Phys. Lett. B355 (1995) 236. L. Chen, G. Dunne, K. Haller and E. Lim-Lombridas, Phys. Lett. B348 (1995) 468. S.Yu. Khlebnikov and M. Shaposhnikov, Phys. Lett. B254 (1991) 148. A. J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 51 (1983) 2077. A.S. Schwarz, Lett. Math. Phys. 2 (1978) 247. E. Witten, Comm. Math. Phys. 121 (1989) 351 ; For the abelian case see, M. Bos and V.P. Nair, Phys. Lett. B223 (1989) 61. D. Birmingham, M. Blau, M. Rakowsky and G. Thompson, Phys. Rep. 209 (1991) 129. E. Witten, IAS Preprint, HEP-88/32.

Chapter 7

Soliton as Anyon in Field Theories

O day and night, but this is wondrous strange — E.A. Abbott in Flatland

7.1

Introduction

In the last chapter, we have discussed in detail the various properties of the Chern-Simons term. In this chapter, we demonstrate the most dramatic effect of this term. In particular, we discuss a variety of relativistic as well as non-relativistic models with Chern-Simons term and show that they admit charged vortex solutions of finite energy in 2+1 dimensions [1,2]. It has been shown that these charged vortices are nothing but charged, extended, anyons thereby providing us with a relativistic model for the charged (extended) anyons. Remarkably, unlike in any other known soliton example, several relativistic models admit both the topological as well as the non-topological self-dual (and hence non-interacting) charged vortex solutions in case the parameters of the theory acquire some special value. In the non-relativistic abelian Higgs model with pure Chern-Simons term, one has non-topological charged vortex solutions, all of which can be written down in a closed form. Apart from these, we shall also discuss the soliton solutions of a CP1 model with the Hopf term (which is an incarnation of the Chern-Simons term) and show that the model admits electrically neutral, extended, soliton solutions of finite energy in 2+1 dimensions and that these solitons are examples of relativistic neutral (extended) anyons. The plan of the chapter is as follows. In Sec. 7.2, we shall consider an abelian Higgs model with Chern-Simons term in 2 + 1 dimensions and discuss in some detail the construction and the various properties of the 185

186

Fractional Statistics and Quantum Theory

charged vortex solutions of finite energy. In Sec. 7.3, we shall focus on the special case of the abelian Higgs model with pure Chern-Simons term and show that this model admits both topological as well as non-topological selfdual charged vortex solutions of finite energy. In Sec. 7.4, we shall consider the non-relativistic limit of this model and show that such a model admits (non-topological) self-dual charged vortex solutions. Further, in this case, all the solutions can be written down in a closed form. In Sec. 7.5, we discuss the CP1 model with the Hopf term in 2 + 1 dimensions and show that it admits extended neutral soliton solutions of finite energy.

7.2

Charged Vortex Solutions

Before we discuss the charged vortex solutions, it might be worthwhile to mention how such solutions were historically discovered. A long time ago, Abrikosov [3] wrote down the electrically neutral vortex solutions in the Ginzburg-Landau theory which is a mean-field theory of superconductivity. Subsequently, these vortices were experimentally observed in the type-II superconductors. Nielsen and Olesen [4] rediscovered these solutions in the context of the abelian Higgs model which is essentially a relativistic generalization of the Ginzburg-Landau theory. These people were looking for string-like objects in relativistic field theory. It turns out that these vortices have finite energy per unit length in 3 + 1 dimensions (i.e. finite energy in 2 + 1 dimensions as the vortex dynamics is essentially confined to the x-y plane), quantized flux, but are electrically neutral and have zero angular momentum. Subsequently, Julia and Zee [5] showed that the SO(Z) Gerogi- Glashow model which admits t'Hooft-Polyakov monopole solution, also admits its charged generalization i.e. the dyon solution with finite energy and finite, non-zero, electric charge. It was then natural for them to enquire whether the abelian Higgs model, which admits neutral vortex solutions with finite energy (in 2+1 dimensions), also admits its charged generalization or not. In the appendix of the same paper [5], Julia and Zee discussed this question and showed that the answer is no i.e. unlike the monopole case, the abelian Higgs model does not admit charged vortices with finite energy and finite and non-zero electric charge. More than ten years later, Samir Paul and I [6] showed that the Julia-Zee negative result can be overcome if one adds the Chern-Simons term to the abelian Higgs model. In particular, we showed that the abelian Higgs model with the Chern-Simons term in 2+1 dimensions admits charged vortex solutions of

Ch 7.

Soliton as Anyon in Field Theories

187

finite energy and quantized, finite, Noether charge as well as flux. As an extra bonus, it was found that these vortices also have non-zero, finite angular momentum which is in general fractional. This strongly suggested that these charged vortices could in fact be charged anyons which was subsequently rigorously shown by Prohlich and Marchetti [7]. Strictly speaking, what one has obtained are the charged soliton solutions and not the vortex solutions, but because of the close connection with the neutral vortex solutions, one has continued to call them as charged vortices rather than charged solitons. This fact should be remembered in this entire discussion. Consider an abelian Higgs model with Chern-Simons term as given by the Lagrangian density

C = --AFllI/F^+l-{D^)*{D^)-CA[\^-^

+t±s^xF^Ax (7.1)

where /n is the Chern-Simons mass, <j> denotes complex scalar field and D^fi is the covariant derivative i.e. D^

= {d^-ieA^)4>.

(7.2)

Here <j>, A^ as well as the gauge coupling constant e have mass dimension of 1/2 while C4 and C^ have mass dimensions of 1 and 2 respectively. Note that the corresponding action (/ = f £d3x) is invariant under the local gauge transformations <j>{x) —> eiea<*V(a0, Mx)

— • A^x) + d^a(x).

(7.3)

In order to obtain the charged vortex solutions, let us consider the following ansatz A(x,t) = - e e C 0 ( g ( r ) ~ n ) , 0(x,t) = C0f(r)ein9, A0(x,t) = C0h(r) r (7.4) where g(r),h(r), f(r) are the dimension-less fields, r is the dimension-less length, while Co has mass dimension of 1/2 i.e. r = eCop, Co = y/C2/2C4.

(7.5)

Note that p and 9 are related to x and ybyp— ^Jx1 + y2 and tan# = y/x. It turns out that even though the Lagrangian (7.1) has so many parameters, the dynamics essentially depends on two dimension-less variables, S and A

188

Fractional Statistics and Quantum Theory

defined by A = V^CVe 2 , S = fi/eCo .

(7.6)

The field equations which follow from here are g"{r)-^g\r)-gf

(7.7)

= 5rti{r)

h"(r) + h'(r)-hf2=5-g'(r)

fir) + I/'(r) _ If where g'(r) = dg(r)/dr. be

+

(7.8)

^ / ( 1 _ / 2 ) = _fh2

(7g)

The corresponding field energy can be shown to

(7.10) Several remarks are in order at this stage. (1) In the neutral vortex case too, the same ansatz (as in Eq. (7.4)) has been used except that in that case fi = 0 and one assumes AQ and hence h(r) = 0. (2) As expected, in the limit h = 0 (i.e. Ao = 0) and S = 0 (i.e. fx = 0) the field equations reduce to those of the neutral vortex case. Prom the Gauss law Eq. (7.8) it also follows that if S (i.e. fx) is non-zero, then Ao must also be non-zero thereby justifying the ansatz (7.4). (3) The boundary conditions for finite energy solutions are lim f(r) = l,h(r) = 0 = g(r)

(7.11)

lim f(r) = 0, g(r) = n, h(r) = (5

(7.12)

r—>co

i—>0

where j3 is an arbitrary number while n = 0,±l,±2.... (4) Prom these boundary conditions it immediately follows that the magnetic flux is quantized in units of 2vr/e i.e.

^[Ba*X J

= -^rrdr(1-d-l) e Jo

\rdrj

= 2^. e

(7.13)

Ch 7.

Soliton as Anyon in Field Theories

189

It may be noted that even for the neutral vortices, the flux is quantized in units of ^ . The underlying reason for the flux quantization is same in both the cases i.e. both are topological objects with the underlying boundary conditions being such that there is a non-trivial mapping from the space time to the group manifold i.e. 7Ti(f/(l)) = Z, with Z being the set of integers, forming a group under addition. (5) Prom the Gauss law Eq. (7.8), it then follows that these vortices also have a non-zero and finite Noether charge which is quantized in units of 2irfj,/e. This is easily seen by noting that in terms of the electric and the magnetic fields, the Gauss law equation can be written as V - E + /xB = /3

(7.14)

where p is the Noether charge density. On integrating both sides of this equation, it then follows that

Q = I pd2x = n f Bd2x = ^ n .

(7.15)

Note that J V • Ed2x = 0, since, because of the Higgs mechanism, both E and B fall off exponentially at long distances. This is probably for the first time that the quantization of the Noether charge has followed from purely topological considerations. In a sense, relation (7.15) can be looked upon as the (2+l)-analogue of the Witten effect [8]. Let us recall the work of Witten who had shown that in the presence of the CP and T violating Chern-Pontyagin term, the t'Hooft-Polyakov monopole acquires electric charge whose fractional part is proportional to the coefficient of the Chern-Pontryagin term. It must however be remembered that whereas the Witten effect is purely a quantum mechanical effect, in our case, the vortices acquire a non-zero charge at the classical level itself due to the presence of the Chern-Simons term. (6) It is also clear from here that in the abelian Higgs model (without the Chern-Simons term), one cannot have vortices having simultaneously the finite energy as well as the finite, non-zero Noether charge. The point is, in the absence of the Chern-Simons term, the Gauss law Eq. (7.14) gives on integration

Q= I pd2x= fv •Ed2x.

(7.16)

The only way Q can be non-zero and finite is if there is a non-zero contribution to the integral around r —> 0 i.e. if E —> 1/r as r —> 0. But

190

Fractional Statistics and Quantum Theory

in that case, the electrical field energy fE2d2x diverges logarithmically [5]. (7) The energy-momentum tensor T^u for this model can be obtained by varying the curved space form of the action with respect to the metric

T^=\{D^r{Du4>) +

^{Dv4>yD^-glxv{c-^eaMF^AA

(7.17) where the Lagrangian L is as given by Eq. (7.1). Note that the ChernSimons term, being independent of the metric tensor g^v, does not contribute to the energy momentum tensor T^. Using this T^ and the field equations, the angular momentum carried by the charged vortices can be shown to be J-Jdxe

xllOJ

- -

2g

-~e2n

~ ~~^T-

I 7 - 18 )

Thus, unlike the neutral vortices, the angular momentum of the charged vortices is non-zero and is solely determined by their charge and flux. Besides, the angular momentum of n superimposed charged vortices is n2 and not n times the angular momentum of a single vortex. Further, since the Chern-Simons mass /x is not quantized in the abelian case, hence this angular momentum J can in general take any fractional value. This strongly suggests that these charged vortices are charged anyons. In fact, since the charged vortex has non-zero charge and flux, hence, following the arguments given in Chapter 2, it follows that the charged vortex is indeed a charged anyon with statistical parameter 9 = 2nJ where J is as given by Eq. (7.18). Thus the solitons of the abelian Higgs model with the Chern-Simons term provide us with a relativistic field theory model for the extended charged anyons. Prohlich and Marchetti [7] have in fact rigorously proved that these charged vortices are charged anyons. They also show that the charged vortices cannot be localized in bounded regions but can be localized in space-like cones in three-dimensional Minkowski space-time [9]. (8) It is amusing to note that the local charge [10] and angular momentum [11] induced on a neutral (Nielsen-Olesen) vortex by fermions is precisely the same [12] as the local charge and the angular momentum of the charged vortex as obtained here provided the Chern-Simons mass \x is chosen to have that value which is obtained by integrating out the massive (and hence parity and time reversal invariance violating) fermions [13,14],

Ch 7.

Soliton as Anyon in Field Theories

191

(9) The magnetic moment of these vortices can be computed by using the field equations and one can show that, whereas for the neutral vortices it is equal to the flux $ ( = 2im/e), the charged ones acquire an extra contribution

Kz = f{v x J)z d2x= — + — [°° rh(r)dr . J e e Jo

(7.19)

Unusual Higgs Mechanism One must now solve the field Eqs. (7.7) to (7.9) and show the existence of the charged vortex solutions. To date, no analytic solution has been obtained of these field equations. However, it is easily seen that for large r, the asymptotic values of the gauge and the Higgs fields are reached exponentially fast g(r) = a±y/re~'n±r

+ ...

(7.20)

ft(r) = T^pe-" ± r + ...

(7.21)

f{r) = l+pe-Xr

(7.22)

+ ...

where a± and j3 are dimension-less constants while the dimension-less vector meson mass rj± is given by

V± = Jl^±l-

(7.23)

However, it has subsequently been shown [15,16] that the solution with rj+ does not exist for all r. On noting that the field Eqs. (7.7) to (7.9) are invariant under r —• - r , it is easily shown that the behavior of the gauge and the Higgs fields around r = 0 is given by g{r)=n + a1r2 +O(r4) h(r) = S3 + ai5r-

+ 0{rA)

f(r) = a2T .M + O( r l"l+ 2 ).

(7.24)

(7.25) (7.26)

192

Fractional Statistics and Quantum Theory

Detailed numerical work has subsequently confirmed the existence of the radially symmetric charged vortex solutions with these boundary conditions [17]. These correspond to n superimposed vortices. The qualitative behaviour of the charged vortex solutions which follows from here is as follows : the magnetic field B decreases monotonically from its non-zero value at the core of the vortex (r = 0) to reach zero as r —• oo with the penetration length l/?7_, while the Higgs field increases from zero at the origin to its vacuum value at infinity with coherence length I/A. Finally, the electric field Ep which is radial, vanishes both at r = 0 and r = oo reaching the maximum in between at some finite r. It is worth pointing out that as in the quantum Hall effect, for the charged vortex solutions too, E(= Ep) is at right angles to J(= jg) and both in turn are at right angles to B. Why does one obtain two asymptotic solutions for g and h, i.e. for the gauge fields Ag and Ao? As we shall now show, this is because of the unusual nature of the Higgs mechanism in 2 + 1 dimensions in the presence of the Chern-Simons term. Normally, one cannot have a gauge invariant gauge field mass term in the action and in fact that is why the Higgs mechanism was invented in the first place to give mass to the gauge field by spontaneous symmetry breaking. However, 2 + 1 dimension is rather unusual and it turns out that the gauge field can in fact acquire gauge invariant mass in the following four different ways. (1) Only the Maxwell term is present and further there is also the Higgs mechanism; this is what happens in the abelian Higgs model. (2) Both the Maxwell and the Chern-Simons terms are present in the action but there is no Higgs mechanism; this has been discussed in detail in the last chapter. (3) Both the Maxwell and the Chern-Simons terms are present and in addition there is also the Higgs mechanism in operation; this is exactly what is happening in the case of the charged vortex solutions discussed above. (4) Only Chern-Simons term is present and there is also the Higgs mechanism; we shall discuss this case in the next section. In order to understand how the gauge field becomes massive in the four cases, let us consider the quadratic part of the gauge field Lagrangian that follows after the spontaneous symmetry breaking from the Lagrangian (7.1)

Cguad = ~F^F^

+ ^XF^AX

+ -f-A^

.

(7.27)

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Soliton as Anyon in Field Theories

193

The case (1) follows when /z = 0. This is the case of the standard Higgs mechanism by which the gauge particle acquires mass. Note that for /i = 0, it is a parity conserving theory; the gauge particle has two degrees of freedom with both spins +1 and —1 being present. The case (2) corresponds to Co = 0 i.e. there is no spontaneous symmetry breaking. As seen in detail in the last chapter, the Chern-Simons term provides mass to the gauge field. It is a parity violating theory with the massive gauge particle having only one degree of freedom with spin +1 or —1 depending on if the Chern-Simons mass /i > 0 or < 0. Let us now discuss the case (3) when both the Maxwell and the Chern-Simons terms are present and further there is also the spontaneous symmetry breaking. Clearly such a theory must still propagate only two massive modes. However, since the Chern-Simons term violates both the parity (P)and the time inversion (T) symmetries, hence one must have two massive, parity violating propagating modes. The question is : in what way are the two massive modes propagated by Eq. (7.27) so as to reflect the parity violation in the theory? This question has been analyzed in detail [18] and it has been shown that in this case Cquad corresponds to Proca equation with the Chern-Simons term. It propagates a self-dual field with two distinct Chern-Simons type masses and that corresponding to each mass there is one propagating mode. Further, the two masses (in dimension-less form ) are precisely r}± as given by Eq. (7.23) thereby explaining the reason for the occurrence of two asymptotic solutions rj±. Finally, let us discuss the case (4) when only the Chern-Simons term is present and there is also the Higgs mechanism [19]. In order to consider the case in which the Maxwell term is decoupled from Eq. (7.27), consider £quad/e2 as given by £, — £guad _ ^quad — e2 —

1 jp p^v JJ^ -pnv 4e2 r M " f + 4e2E'"'A

A \ , ^ 0 A Afi 2 '

(7 ?K\

Now the limit in which the Maxwell term is decoupled from tquad may be achieved by taking e 2 —> oo, fi —> oo, k = -^ = fixed.

(7.29)

Note that k which is kept fixed is dimension-less. Now consider the two masses m± as given by Eq. (7.23),

m ± = yU7 0 2 + ^ ± ^ .

(7.30)

194

Fractional Statistics and Quantum Theory

Note that in dimensional form, r]±r = eCoJj±p = m±p (p = \Jx2 + y2). Prom here it follows that in the limit as given by Eq. (7.29), C2 m_ -> -—-, m+ -> oo

(7.31)

so that the mass m+ decouples from the theory. It is interesting to note that the charged vortex solutions exist for all r only in the case of the mass m_ (and not m+). Thus in the case of the pure Chern-Simons term, one finds that after the Higgs mechanism, the gauge field is massive and of Chern-Simons type, propagating one mode. We shall discuss this case in detail in the next section when we discuss the charged vortex solutions in an abelian Higgs model with pure Chern-Simons term. If one is only interested in the masses of the gauge field excitations, then it is in fact sufficient to make a zeroth-order spatial derivative expansion (i.e. neglecting all spatial derivatives). In this limit the quadratic Lagrangian density becomes L = \k2t + £ e " A4 • A,- + e2CgAi • At

(7.32)

where A* = A$(£) is only a function of time. This is the Lagrangian for a charged particle of unit mass moving in a uniform magnetic field of strength H and a harmonic potential well with frequency eCo- Such a quantum mechanical model is exactly solvable and is well known to decouple into two separate harmonic oscillator systems of characteristic frequencies [20,2]

where uc ( = (j.) is the cyclotron frequency corresponding to the magnetic field and u> (=eC0) is the harmonic frequency. Note that UJ± are identical to m± as given by Eq. (7.30). In this language, the pure Chern-Simons limit (7.31) corresponds to the physical limit when the magnetic field strength UJC dominates so that 2

f2

u- -» — = -p- = m_, w+ -> oo. uc k

(7.34)

Thus, this limit corresponds to a truncation of the physical Hilbert space in which the dynamics is projected onto the lowest level. This limit is of relevance in the context of theories of the fractional quantum Hall effect.

Ch 7.

Soliton as Anyon in Field Theories

195

A comment is in order regarding the degrees of freedom count in the abelian Higgs model with the Chern-Simons term as given by Eq. (7.1). In case the symmetry is unbroken, the complex scalar field has two real massive degrees of freedom and the gauge field has one massive excitation (coming from the Chern-Simons term). When the symmetry is spontaneously broken, one component of the scalar field (the "Goldstone boson") is eaten up by the longitudinal part of the gauge field to produce a new massive gauge degree of freedom. Thus in the broken case, there are two massive gauge degrees of freedom and one real massive scalar degrees of freedom (the "Higgs boson"). Vortex-Vortex Interaction One of the most interesting question is whether these charged vortices can be observed experimentally in some planar system. In this context recall that the neutral (Abrikosov) vortices have been experimentally seen in type/ / superconductors. This can be understood from the fact that whereas the vortex-vortex interaction is repulsive in the type-// region (A > 1), it is attractive in the type-/ region of superconductivity [21]. It is thus of great interest to study the charged vortex-vortex interaction and to see when is it repulsive. This has been done both in the perturbation theory (in the Chern-Simons mass) and by the variational calculation [17], and in both the cases one finds that the charged vortex-vortex interaction is more repulsive than the corresponding neutral case with the extra repulsion coming from the electric field of the charged vortex. For example, for 5 = 0.5, one finds that the charged vortex-vortex interaction is repulsive even for A > 0.45 (note that in the neutral case the interaction is repulsive only if A > 1). Note that A and 5 are as given by Eq. (7.6). In order to see that the charged vortex-vortex interaction is more repulsive than the corresponding neutral case, let us consider the case when the Chern-Simons mass is small and expand the charged n-vortex fields in terms of the corresponding neutral vortex fields plus a small correction proportional to 8. In particular, it can be shown that upto O(52), the charged n-vortex fields are given by [17] g(r, A, 5) = g(r, A, 0) + ^4~g(r, A, 0) + O(<54) o

(7.35)

h(r,\,5) = ^g(r,\,0)+O(53)

(7.36)

f(r,\,5) = f(r,\,0) + O(64).

(7.37)

196

Fractional Statistics and Quantum Theory

Here g(r, A, 0) and f(r, A, 0) are the solutions to the corresponding neutral vortex case (i.e. in the pure Maxwell case). On substituting these perturbative solutions in the expression for the field energy as given by Eq. (7.10), one can show that T? 2 (S 2

En{X,5) = En{\,0) + —

+ O(6i)

(7.38)

where En (A, 0) is the energy of the corresponding neutral vortex with vorticity n. It is worth noting that the O(52) correction is positive, proportional to n 2 and independent of A. Prom here it also follows that En(X, S) - nE^X, 5) = En(X, 0) - nE^X, 0) + ^

~ " V + O(54) (7.39)

so that the charged vortex-vortex interaction is always more repulsive than the corresponding neutral case. A word of caution is in order here. The entire analysis here about the charged vortex-vortex interaction is only valid in the case of the superimposed vortices. The problem of the charged vortex-vortex interaction, when the vortices are separated by distance d, is still an unsolved problem.

Fermion-Charged Vortex Interaction In the last few years, the question of the zero energy eigenmodes of the Dirac equation in a solitonic background has been extensively studied [14]. In most cases one finds that because of these zero modes, the soliton acquires fractional charge. Such effects have found physical application in the context of polymers like polyaceteylene as well as in the monopole catalysis of the proton decay. It is thus of interest to study the interaction of the charged vortices with fermions. In this context it is worth recalling that in the case of the neutral vortices, it has been shown that there are precisely n zero modes of the Dirac equation in the background of the neutral nvortex [22,23]. No analogous index theorem has so far been obtained in the case of the fermion- charged vortex interaction. However, one zero mode has explicitly been obtained [24,25] in case n is odd (while none has been obtained when n is even). Based on this work, it has been speculated that in this case there might be a mod-2 index theorem.

Non-abelian Charged Vortex Solutions It is clearly of considerable interest to enquire whether the charged vortex solutions obtained above can be embedded in non-abelian gauge theories

Ch 7.

Soliton as Anyon in Field Theories

197

with the (non-abelian) Chern-Simons term. The first obvious question is whether such vortices could be topologically stable or not. It is easily seen that if G is the gauge group of the non-abelian gauge theory and H is the sub-group under which the vacuum remains invariant after spontaneous symmetry breaking, then the topologically non-trivial vortices are possible only if ni(G/H)^0.

(7.40)

In the case of SU(N) gauge theories, it turns out that no Z-vortices are possible. However, Zjv-vortices are possible in case H is Z^r since Tri(SU(N)/Zff) = %N- It turns out that at least N Higgs multiplets are required so that the vacuum is invariant under ZJV [26,27]. As a result, only one non-trivial charged vortex is possible in the case of SU(2) gauge theory with flux $ = 2ir/g, charge Q — ^ $ = 2nfi/g, and angular momentum J — —QQ/in = —Tr/j,/g2 where g is the gauge coupling constant. However, as discussed in the last chapter, the Chern-Simons mass /J, is quantized in non-abelian gauge theories having SU(2) as its sub-group i.e. q2

fx = j - n , n = 0, ±1, ±2,... 4TT

(7.41)

and hence the vortex charge is gn/2 i.e. it is quantized in units ofg/2 while the angular momentum is quantized in units of 1/4 i.e. J=~-

(7-42)

This is remarkable as it strongly suggests that if the usual spin-statistics connection is valid then whereas the abelian charged vortex is an anyon with any phase factor, the non-abelian (SU(2)) charged vortex can only be a semion, a fermion or a boson. 7.3

Relativistic Chern-Simons Vortices

In the last section we have obtained the charged vortex solutions in case the gauge part of the abelian Higgs model consists of both the Maxwell and the Chern-Simons term. It may be of some interest to enquire whether the abelian Higgs model with pure Chern-Simons term can also admit charged vortex solutions. This question is specially relevant in the context of the condensed matter systems since in the long wave length limit, the ChernSimons term having one derivative dominates over the Maxwell term which

198

Fractional Statistics and Quantum Theory

has two derivatives. It turns out that the answer to the question is yes [28]. One finds that in fact the self-dual charged vortex solutions are possible in this case provided the Higgs potential is not of 4-type but of ^ 6 -type [29,30]. It is worth pointing out here that whereas a Higgs potential of the type X^i^l^P with 0 < i < 4 is renormalizable in 3 + 1 dimensions, Y^i Ci\
= Srh'(r)

-hf

=

(7.43)

5

(7.44)

-g'(r).

However, Eq. (7.9), for the Higgs fields, is still a coupled second order equation. We now show that in case one replaces the standard double well <^4-type potential by the following 6-type potential

V(\4>\) = ^\\2M2-C20f

(7.45)

then even the Higgs field satisfies a first order equation. When the Maxwell term is absent and the Higgs potential is as given by (7.45), the vortex energy (7.10) has the simpler form

£n=< rfr

f

2+

[© T

H2/2+

/2

(? 46)

>- 4 -

This can be rewritten as

3 2 ^p*[(/vi/ ) ?/? 4*^] ±^(w')] En 9 ± 9+ (7.47) which, in view of field Eq. (7.44), takes the form

(7.48)

Gh 7.

199

Soliton as Anyon in Field Theories

This gives a rigorous lower bound on the energy in terms of the flux En >

±TTC20

\g(0) - g(oo)} = ±\eC2 $

(7.49)

since the finite energy consideration requires that f2g vanish at both the ends. This bound is saturated when the following self-dual first order equations are satisfied f(r) = ±-rfg

-Vw = ^ - ± ^ / 2 ( l - / 2 ) -

(7.50)

(7-5D

It is easily checked that these first order equations are consistent with the second order field Eq. (7.9). One can in fact decouple these coupled first order equations and show that the Higgs field / must satisfy the following un-coupled second order equation /"(r) + \f'{r) ~ ~ ^ + ^ / 3 ( 1 - / 2 ) = 0.

(7.52)

Several comments are in order at this stage. (1) The self-dual equations (7.50) and (7.51) are similar to those of the Nielsen-Olesen (neutral) self-dual vortices [31,32] (which are valid only if A = 1) which are given by

f' = ±ll,B=--gf(r) = ±hl-f2).

(7.53)

In that case too, the first order equations can be decoupled and the un-coupled second order equation is

f'{r) + l-f(r) - ¥Q + l-f(l -f)=0.

(7.54)

It is worth pointing out that no analytic solutions are known to either of the Eqs. (7.52) or (7.54). (2) Whereas the Lagrangian for the self-dual neutral vortex case (i.e. Lagrangian (7.1) with [i = 0 and A = 1) is the bosonic part of a N = 1 supersymmetric theory [33], the Lagrangian for the self-dual charged vortex case (i.e. the Lagrangian (7.1) with the Maxwell term being absent and the Higgs potential being as given by Eq. (7.45)) is the bosonic part of a N = 2 supersymmetric theory [34].

200

Fractional Statistics and Quantum Theory

(3) The 4-potential as given in Eq. (7.1) and the <jf>6-potential as given by Eq. (7.45) represent very different physical situations. For example, whereas the ^-potential corresponds to the case of the second order phase transition with T < Tc7/, the 6-potential as given in Eq. (7.45) corresponds to the case of the first order phase transition with T = T* [35]. Hence, whether the charged vortex solutions discussed in this or the previous section would be of relevance would depend on the physics of the problem. (4) One way to understand why the 06-type potential is required for the self dual charged vortices while the 4-type potential is required for the neutral self-dual vortices is that whereas in four space-time dimensions the coefficient of the 4-term is dimension-less, it is the coefficient of the 06-term which is dimension-less in three space-time dimensions. Let us now discuss the most remarkable property of the self-dual Eqs. (7.50) and (7.51). In particular, since the Higgs potential (7.45) has degenerate minima at \<j>\ = 0 and \\ = Co, hence, it turns out that at the self-dual point, one can simultaneously have both the topological and the non-topological charged vortex solutions. It is worth pointing out that at the time of this discovery, no other self-dual system was known which exhibited this remarkable property. Topological Self-dual Solutions

The topological, self dual charged vortex solutions satisfy the same boundary conditions as given by Eqs. (7.11) and (7.12) with /? = h(r = 0) = ±l/25 2 . Note that the upper (lower) sign corresponds to n > 0 (< 0). As a result, the flux $, the Noether charge Q, and the angular momentum J of these charged vortices are again as given by Eqs. (7.13), (7.15) and (7.18) respectively while the energy of these charged vortices is TTCQ \n\. Connection between the self-dual charged vortices and anyons has been discussed in detail in [36]. From now onwards, we shall confine our discussion to the case of n > 0 i.e. those corresponding to the upper choice of sign. Solution with n < 0 are related to these by the transformation g —> —g, f —> / . A countable infinite number of sum rules have been derived [37] and using the first two, it has been proved that the magnetic moment of the topological, self-dual charged n-vortex is given by [38] Kz = 2irn(n+1)—. e Note that for the neutral n-vortex, Kz = $ = 2nn/e.

(7.55)

Ch 7.

Soliton as Anyon in Field Theories

201

No analytic topological self-dual charged vortex solution has been obtained as yet. However, one can show that all the fields approach their asymptotic values exponentially fast i.e. lim

g(r)=a(r/5)Ki(r/6)

(7.56)

lim

f(r) = l-aK0(r/5)

(7.57)

r—»oo

r—>oo

where the constant a is not determined by the behavior at infinity while Kn(x) is the Bessel function. In particular note that at the self-dual point, the vector and the scalar meson masses are equal. On the other hand, near the origin, a power series solution gives

« r ) = "MS)" - stsfr^w*'3"*2 + 8p^hp W i ) S " + 2 + - (7-59) where the constant bn is not determined by the behavior of the fields near the origin. We thus see that the properties of the Chern-Simons topological vortices are almost the same as those of the Maxwell-Chern-Simons vortices. However, there is one major difference between the two, as is clear from Eqs. (7.58) and (7.24). In particular, whereas for the Maxwell-Chern-Simons case, the magnetic field is maximum at the core of the vortex (r —> 0), for the Chern-Simons vortices, the magnetic field is zero at the core of the vortex and is concentrated in a ring surrounding the vortex core. Non-topological Self-dual Solutions Since \<j>\ = 0 as well as \<j>\ = Co are degenerate minima of the Higgs potential (7.45), hence it turns out that one could also have non-topological self-dual charged vortex solutions [38,39]. In this case, the finite energy considerations demand the following boundary conditions f(r) = 0 , g(r) = T « , a > 0

(7.60)

lim f(r) = 0 , g(r) =n for n ^ 0

(7.61)

lim f(r) = rj, g(r) = 0 for n = 0

(7.62)

lim

r—>oo

202

Fractional Statistics and Quantum Theory

where r\ is an arbitrary number while —a (+a) is for n > 0 (< 0). As a result, the flux, the charge, the energy and the angular momentum of these vortices for (n > 0) are 2-K , 2nu, . $ = — ( n + a), Q = n$ = — - ( n + a), e e J =

?^(a2-n2),

E = irC2(n + a).

(7.63)

Note that unlike the topological case, the angular momentum is no more equal to —Qcp/Air. Here a is a positive number but how much is it? The finiteness of energy requires that a > 1 but otherwise a seems to be completely arbitrary. However, it is not so and we now show [40] that a satisfies a rigorous lower bound of a > n + 2. To this end, consider the self-dual Eq. (7.51). On integrating both sides of this equation and using boundary conditions (7.60) to (7.62), one obtains (for n > 0) f°° da

-

1 f°°

-fdr = n + a=—

Jo

dr

2d

rdrf2(l - f) > 0.

(7.64)

Jo

Similarly, on using Eqs. (7.50) and (7.51) we have on integration

f 4* - V"2 -n2)-hC

r l{1 f) ir

' -l •

<765>

On integrating by parts and using the fact that r 2 / 2 and r 2 / 4 vanish as r —> oo (note / ( r ) ~ r~a with a > 1 as r —> oo), we then have

(a2-n^ = hj™rdr{f-\fA)-

(7 66)

-

On combining the two sum rules, we then have 1 f°° (a + n ) ( a - n - 2 ) = —j / rdrf4 > 0 ^o Jo which gives us a rigorous lower bound on a i.e. a > n + 2.

(7.67)

(7.68)

It turns out that this bound is never saturated in the relativistic case. However, as we shall see in the next section, it is indeed saturated in the case of the non-relativistic self-dual non-topological charged vortices. It may be noted here that there is however no upper bound on a. We thus conclude that the flux of the relativistic non-topological vortices must necessarily be greater than 4?r(n + l)/e. More remarkable is the fact that whereas

Ch 7.

Soliton as Anyon in Field Theories

203

the angular momentum of the topological vortices is always negative and proportional to n 2 , the angular momentum of the non-topological vortices, on the other hand, is necessarily positive and in general is not proportional to n2. Further, the magnetic moment of the non-topological vortices has also been computed analytically by using the sum rules and shown to be negative [40] Kz = - ^ - ( a + n)(a-n-l)<0. (7.69) e Note that the magnetic moment of the topological vortices is on the other hand always positive. Are these non-topological vortices stable or do they decay to the charged scalar meson? This question has been discussed [41,42] and it has been shown that as far as the decay to the scalar meson is concerned, these nontopological solitons are at the edge of their stability. In particular, using E and Q as given by Eq. (7.63) and noting that the mass m of the scalar particle in the symmetric vacuum is e2c2)/2/j1, it follows that E = mQ/e. Thus the stability does not impose any upper bound on the charge of the non-topological soliton. No analytic solutions of Eqs. (7.50) and (7.51) have been obtained as yet in the non-topological self-dual case. However, • 0 and for large r is easily obtained. In the behavior of the fields near r —> particular, using the boundary conditions (7.60) to (7.62), it is not difficult to show that for r —> oo, the n = 0 vortex solution has the behavior 9(r) = " « + C f(r) =

4 ( a

_ 1 ) < ^ ) a a - a + O((r/S)-^) C3 —5

L O((r/5)~5a+4)

(7.70)

("7 71)

On the other hand, as r —> 0, while g(0) must vanish so as to have a nonsingular solution, /(0) = bo is not so constrained. We find that as r goes to zero, 9(r) = -\bl(l

fir) =bo-

- b20){r/5)2 + - ^ ( 1 - b2)(l - 2b2)(r/5)* + ...

(7.72)

|&g(l - b2)(r/S)2 + ^ ( 1 - 6g)(2 - 3bg)(r/«5)4 + ... (7.73)

Thus for the n = 0 non-topological vortex, the magnetic field (—g'(r)/r) is maximum at the core of the vortex (r = 0) and falls off with a power

204

Fractional Statistics and Quantum Theory

law fall off as r —> oo. Note, however, that the magnetic field for the topological Chern-Simons vortices is zero at the core of the vortex (r —> 0), and is concentrated in a ring surrounding the vortex core. Finally, let us consider the behavior of the n ^ 0 (we as usual consider n > 0) non-topological self-dual charged vortex solutions. It is easily shown that these solutions are hybrids of the two previous cases i.e. their large distance behavior is the same as those of the n = 0 non-topological charged vortex solutions as given by Eqs. (7.70) and (7.71). On the other hand their short distance behavior is the same as those of the self-dual topological charged vortex solutions as given by Eqs. (7.58) and (7.59). Thus for n / 0 non-topological vortices, the magnetic field vanishes at the core of the vortex and falls off with a power law fall off as r —> oo. It is worth pointing out that since the (/>6-potential as given by Eq. (7.45) has disconnected but degenerate vacua at \\ = Co, hence, apart from the charged vortex solutions, they also possess one dimensional domain wall solutions [39,35]. So far, we have only discussed the self-dual rotationally symmetric Chern-Simons vortices. However, the self-dual solutions can in fact be obtained even without choosing the rotationally symmetric n-vortex ansatz (7.4). To this end, notice that the field energy corresponding to the pure Chern-Simons Lagrangian (no Maxwell term) and with the Higgs potential of the form (7.45), is given by (see Eq. (7.46))

E=\Jd^Dct>\2

+ \D04>\2 + ^\4>\2(\\2-C20)2y

(7.74)

Further, the Gauss law constraint equation which one obtains by varying the Lagrangian (7.1) (without the Maxwell term) with respect to Ao is liB = e2A0\\2.

(7.75)

As a result, for the time independent vortex solutions, the field energy can be rewritten, after an integration by part, as

E ^i/ d v(i(D 1±i D 2 )rf + [f^ T ||*|( C |-WI ±jC02».

2

)] 2 ) (7.76)

Now, in order to have finite energy, D must vanish at spatial infinity. If \\ is non-vanishing at spatial infinity, then it follows that asymptotically

Ch 7.

205

Soliton as Anyon in Field Theories

A = —-Vln^ and hence the flux $ is quantized i.e.

^=fBd2r=f

A-dl = -- [

dl-Vln= —

(7.77)

where n — 0, ± 1 , . . . . As a result, there is a lower bound on the energy as given by Eq. (7.49) which is saturated by the fields obeying the self-dual equations D i ^ = ^iD2

B B=±

yaw(

(7.78)

w\

-2^cs\}-cs)

(

}

where the upper (lower) sign corresponds to positive (negative) values of the flux $ and n. To investigate the properties of these solutions, we decompose the complex scalar field as 4> = e-^\<(>\

(7.80)

in which case the duality Eq. (7.78) implies that the corresponding vector potential is Ai = duo + Sijdj In \4>\

(7.81)

which when substituted in Eq. (7.79) yields the un-coupled second order equation

^ = -£f(.-!f). In the special case of the rotationally symmetric solutions, this un-coupled equation reduces to Eq. (7.52). It must be noted that this equation only holds away from the zeros of (f>; at these zeros there is an additional deltafunction contribution which results when one takes the curl of the gradient of the multiple valued function Arg(4>). The angular momentum of these self-dual vortices can be computed by using Eqs. (7.17) and (7.18). Using them, we have for the static configura-

206

Fractional Statistics and Quantum Theory

tions

jd2reijXiToj

J=

= - jd2r\D0*{r r

/

x D) + D04>{r x (Dtf>)*)l -i

d 2 r£ L r x A - r x V(.4r
L

(7.83)

J

(7.84)

On excluding from the integration surface the points where <>/ = 0 (since the integrand is non-singular on this set of measure zero, the result for J is unaffected), one can write using Eq. (7.81) (7.85) J = fj, I d2r(V x A ) . ( r x A ) where Ai = A* — diU>. On using Eqs. (7.79) and (7.81) and using the boundary conditions (oo) = 0 and (oo) = Co for the non-topological and topological vortices respectively we obtain J = i . [ d2r\cf>\2(2C2 - \ct>\2) for 0(oo) = 0

= - -

/ d 2 r ( C 2 - |0| 2 ) 2 for
(7.86)

Rigorous arguments have subsequently been given for the existence of the self-dual topological [43] and non-topological [44] charged vortex solutions even when the vortices are not superimposed on each other but lie at arbitrary positions in the plane. Let us note an interesting fact about the angular momentum of these charged vortices. For example, whereas the angular momentum of the n superimposed topological vortices is n2 times that of a single vortex, the angular momentum of the n topological vortices (each of which has unit vorticity) which are well separated from each other, is only n times the angular momentum of the single vortex. However, the energy, flux and the charge of the n vortices in both the cases is the same. Thus we see that whereas the energy, flux and charge, are the global quantities, the angular momentum of a configuration depends on the local behavior. A zero-mode analysis of the spectrum of small fluctuations [39] around the self-dual vortices indicates that whereas the number of zero modes in the case of the topological self-dual vortices is 2n, in the non-topological case, the same number is 2n + 2 [a] where [a] denotes the integer part of

Ch 7.

Soliton as Anyon in Field Theories

207

a. In the topological case, this number is identified with the number of parameters required to describe the location of the n vortices while the counting is less clear in the non-topological case. Interaction Between Self-Dual Chern-Simons Vortices The slow motion of the abelian self-dual Chern-Simons vortices has been analyzed [45] using Manton's technique [46]. In this approach, one constructs an effective quantum mechanical Lagrangian (not Lagrangian density) which describes the fluctuations about the static self-dual classical configurations. The underlying assumption here is that the slow motion of the self-dual vortices (which saturate the Bogomolnyi lower bound on energy) can only lead to small deformations of the classical fields with respect to the self-dual configuration. One considers the situation when these vortices are widely separated and allows the location of the vortices to have time dependence. On integrating over the spatial-coordinates, one obtains an effective Lagrangian for the fluctuation of these vortices about their self-dual configuration. For the slow motion of the vortices, it is possible to identify both the kinetic and the potential energy pieces of the Lagrangian. In particular, the kinetic energy term is for TV point particle vortices while the potential energy has an anyonic statistical interaction term arising due to the fact that the Chern-Simons vortices are in fact charged anyons. The interesting point is that in the slow motion limit, one has reduced the field theory problem with infinite degrees of freedom to a system with only finite degrees of freedom. We must warn here that, in this case, the analysis is somewhat involved due to the appearance of the terms which are first order in time derivative. This in turn indicates the presence of the velocity-dependent Magnus force [36]. One can show that this Magnus force is in fact necessary in order to have correct spin-statistics relation. This has some interesting implications for the scattering of Chern-Simons vortices [47]. Semi-Local Self-Dual Chern-Simons Vortices Semi-local self-dual Chern-Simons vortices have been obtained in an abelian Higgs model with pure Chern-Simons term [48] and with SU(N)giobai U(l)iocai symmetry. This was inspired by a similar result in the case of an abelian Higgs model with the Maxwell term [49]. The interesting point is that the semi-local vortices, even though topologically trivial, are stable under small perturbations due to the gradient energy term. Finally, it is worth enquiring whether the self-dual charged vortices can

208

Fractional Statistics and Quantum Theory

also be obtained in the original 4-type model itself which is of relevance in the context of the second order phase transition. The answer to the question is yes and the self-dual Maxwell Chern-Simons vortices have been constructed by adding a neutral scalar field to Eq. (7.1) and changing the ^-potential suitably [50]. Further, the self-dual solutions have also been obtained in the Chern-Simons models with additional magnetic moment interaction [51]. Besides, the self-dual non-abelian Chern-Simons vortex solutions have also been constructed [52]. 7.4

Non-relativistic Chern-Simons Vortices

Let us now discuss the non-relativistic limit of the abelian Higgs model with the pure Chern-Simons term. In the symmetric phase, we will see that this model provides us with a second quantized description of point particles moving non-relativistically in a delta function potential and interacting via a Chern-Simons term. We present classical, static, self-dual solutions of this model. It will turn out that the self-dual equation is equivalent to the Liouville equation, and hence is completely solvable [53]. The Lagrangian density for the abelian Higgs model with pure ChernSimons term is given by

£ = i(Z^)*CD'V) + fa^F^A* - J ^ p d ^ _ C2)2 f

(7 . 87)

where the Higgs potential is as given by Eq. (7.45). In this section we write all the factors of the velocity of light c explicitly since we are considering the non-relativistic limit of a relativistic theory. Let us first note that the quadratic term in the Higgs potential defines the mass through its coefficient m 2 c 2 /2. Comparison with Eq. (7.87) shows that CQ must have the value

Cl = » ^ so that the Lagrangian density (7.87) can be rewritten as

(7.88)

Ch 7.

209

Soliton as Anyon in Field Theories

The non-relativistic limit (c —> oo) now proceeds in the standard manner. On writing the mode expansion of the scalar field <j> as $=-$=

\e-imcH

ip + eimcHi>*\

0"L

(7.90)

J

and substituting it in Eq. (7.89), dropping all terms that either oscillate as c —> oo or are sub-leading in powers of c, the matter part of the Lagrangian density can be shown to be

C = it-Do* - ^ | D ^

+

^ f

+ Is^F^

.

(7.91)

Here p = ip*t/j is the matter density of particles and we have dropped the anti-particle part from the Lagrangian density (i.e. we are working in the zero anti-particle sector) by setting ip = 0 since the particle and the antiparticle parts are separately conserved. The remarkable fact is that one now has an attractive quartic (p2) self-interaction. This non-relativistic model can be looked upon either as a non-relativistic classical field theory or as a second quantized iV-body problem with two-body attractive delta-function interaction. The Euler-Lagrangian equations of motion which follow from the Lagrangian density (7.91) are

(7.93)

F?v = --EpupJ"

where J^ = (p, J) is a Lorentz covariant notation for the conserved nonrelativistic charge and current densities i.e. p = | V 2 | , Jk = -^bP*Dk1>

- {Dk1>)*1>] •

(7.94)

The field Eqs. (7.92) and (7.93) are together termed as the planar gauged nonlinear Schrodinger equations. The gauge field Eq. (7.93) can also be re-expressed as B = Fl2 = -p fj,

Ei = Fi0 = -—eikJk.

(7.95) (7.96)

210

Fractional Statistics and Quantum Theory

From here, we immediately obtain the fundamental relation between the Noether charge Q and the magnetic flux <3> i.e. Q — //$. As in the relativistic case, it is easily checked that, in the case of the static solutions, the second order field Eqs. (7.92) and (7.93) are solved by Eq. (7.95) and the self-dual ansatz Djtl) = ±iEjkDki>

(7.97)

A) = T^IVf-

(7.98)

with Ao chosen as 2mp,c Here one has to choose the + (-) sign depending on if \i < (>) 0. Note that with this constraint on Ao, the field Eq. (7.96) is automatically satisfied. Here we have made use of the following factorization identity D V = D±DTip =F -F12^.

(7.99)

We now show that the self-dual Eqs. (7.95) and (7.97) can be solved completely and explicitly. On writing the complex field ip as if, = e - i a y / 2 ,

(7.100)

the self-dual ansatz (7.97) yields the vector potential Ai = diUJ ± ^-eijdj

hip

(7.101)

which is valid away from the zeros of p. On inserting this form of A into the other self-dual Eq. (7.95) yields the famous Liouville equation V2ln,9=

2e 2 —-.p

c\n\

(7.102)

which is known to be integrable and completely solvable and which must be solved away from the zeros of p. It is worth noting that with our sign conventions, we have the Liouville equation with the correct sign in that only such an equation has real, positive, regular solutions. The most general such solution is known to be given by

" = e 2 [ l + |/(,)H

(7 103)

'

Ch 7.

211

Soliton as Anyon in Field Theories

where f(z) is any holomorphic function and z — re%e. Explicit, radially symmetric, solutions may be obtained by taking f(z) = (z/zo)±n

.

(7.104)

The corresponding self-dual charge density is P

~

% | n 2 c (r/rp) 2 ^- 1 )

eVg [l + (r/r o ) 2 «] 2

(7 105)

-

which behaves like r2^™"1) as r —> 0 while as r —> oo, it behaves like r~2~2". Thus p is regular at the origin if n > 1. From Eq. (7.101) it then follows that as r —> 0, the vector potential behaves as

Mr) ~ c ^ ± c ( n ~ 1 ) £ i j . ^

(7.106)

i.e. it is singular at r = 0. This singularity is removed if we choose u> = ±c(n — 1)0/e. Thus the profile of the self-dual ip field is given by er0

[1 + (r/ro)2n\

On requiring that tp be single valued, we then find that n must be an integer, and for p to have decaying behavior as r —> oo, we require that n must be positive. Several comments are in order at this stage. (1) Integrating p as given in (7.105) over all space yields n (the total number of particles) and hence the flux (in view of Eq. (7.95)). We obtain $ =

4TTC

e

n, n = l,2,...

(7.108)

which means that this configuration carries an even number of flux units. This is in contrast to the relativistic case where the flux unit need not necessarily be even. Further, note that unlike the relativistic non-topological case, here the lower bound on a as given by Eq. (7.68) is saturated. As has been shown [54], this is because of the special inversion symmetry of the Liouville equation. In particular, notice that the Liouville equation is invariant under the transformations r -+ 1/r, 9-*0, p(r) -» p(l/r) = rAp{r).

(7.109)

212

Fractional Statistics and Quantum Theory

As a result, the behavior of p at infinity is uniquely determined by its behavior at the origin thereby fixing a = n + 2. The corresponding Noether charge Q = A-nncn/e. (2) The angular momentum J is easily calculated and one finds that

J=^fn.

(7.110)

Note that unlike the relativistic superimposed topological vortices, here J is proportional to n (and not to n 2 ). (3) It is worth pointing out the Q, $ and J for the non-relativistic charged vortices are the same as those for the relativistic non-topological charged vortices as given by Eq. (7.63) provided one chooses a = n + 2 (note that in the non-relativistic case, n = 1, 2,... while n = 0,1,2,... in the relativistic case). (4) The radially symmetric solution (7.107) was obtained by choosing the holomorphic function f(z) oc (z)~n and corresponds to n solitons superimposed at the origin with common scale factor TQ. The most general solution corresponding to n separated solitons may be obtained by taking

/W = E ( ^ y

("")

where 2n real parameters Zi describe the location of the solitons and 2n real parameters a.i correspond to the scales and the phases of the solitons. Thus the solution depends on 4n parameters. Using an index theory calculation [55] it has been shown that this is the most general solution. (5) Solutions with a periodic matter density p have also been written down by choosing the function / to be a doubly periodic function [56]. (6) Time dependent solutions have also been obtained by applying a conformal transformation (which leaves the action invariant) to the static self-dual solutions [53]. Further, solutions in an external uniform magnetic field have also been obtained [57] by transforming the static solutions according to a certain symmetry operation of the system which leaves the action invariant. As a further generalization, the boosted time-dependent solutions have also been transformed and the resultant time-dependent solutions have been used in a semi-classical quantization of these self-dual Chern-Simons systems.

Ch 7.

Soliton as Anyon in Field Theories

213

(7) In the case of the static solutions, the left hand side of Eq. (7.92) with A0 evaluated as in Eq. (7.98), coincides with the Pauli equation with the gyromagnetic ratio equal to two which is known to be a system with hidden supersymmetry [58,59]. (8) By now, the non-relativistic self-dual vortices have been obtained in several other cases. For example, the non-relativistic Maxwell-ChernSimons vortices have been obtained [60]. Further, non-abelian, nonrelativistic, self-dual vortices have also been constructed and interesting connections have been obtained with the two dimensional integrable equations like the Toda equations [61,62,2]. Dynamical Symmetries of the Non-relativistic system The non-relativistic field theory with the Lagrangian density as given by Eq. (7.91) possesses several kinematical and dynamical symmetries at the classical level and these symmetries are valid for any value of the coefficient of the p2 term ( and not just the specific value as chosen in Eq. (7.91)). For example, apart from the usual Galilean symmetries (i.e. time translation, space translation, space rotation and Galilean boost), the Lagrangian density (7.91) is also invariant under conformal reparametrization of the time coordinate. As a result, the constants of motion corresponding to the Galilean symmetries are energy, momentum, angular momentum and Galilean boost. On the other hand, the conserved quantities corresponding to the additional conformal transformations are dilation and special conformal charge. In the quantum theory, one finds that there is a scale anomaly and as a result the Lagrangian density (7.91) is, in general, not scale invariant. It is only when the coefficient of p2 is as given in Eq. (7.91) that the quantum theory is scale invariant with no scale anomaly. 7.5

CP1 Solitons with Hopf Term

Finally, we discuss the 0(3) a-model [64,65,66] with Hopf term in 2 + 1 dimensions in which the first extended (neutral) anyon solutions in relativistic field theories were discovered [63]. The solitons of the 0(3) cr-model are bosons which become anyons in the presence of the Hopf term, which is an incarnation of the Chern-Simons term. Unfortunately, in this case, the Hopf term cannot be written down as a local function of the basic fields of the theory. Therefore, we shall discuss the essentially equivalent example

214

Fractional Statistics and Quantum Theory

of the CP1 model with the Hopf term since in this case the Hopf term can be written down as a local function of the basic fields of the theory [67]. The action for the CP1 model in 2+1 dimensions is given by /=

Id3x

(D^,z)*(D^z)

(7.112)

where D^z = (<9M —iA^z with z = (z\, z-i) being a complex vector fulfilling z\2 = 1. Note that A^ here does not represent independent degrees of freedom, but is entirely determined in terms of z(x) through the constraint equation (7.113)

Ali = -iz* d^z.

The action (7.112) is invariant under the local U(l) transformations za{x) -> za{x)eiA^\

A^x) -» An(x) + 0MA(a:).

(7.114)

As is well known, the CP1 model admits self-dual soliton solutions. To obtain them, let us first note that the field equation is obtained by extremising the action (7.112) with respect to z(x) subject to the constraint z\2 = 1. This constraint is best introduced in the variational formalism by using a Lagrangian multiplier i.e. one extremizes / + J d3x\(x)(z*z - 1). The resulting field equation is (D^

+ X)z = 0.

(7.115)

The Lagrange multiplier X(x) is eliminated by using A = \z*z = -z*DllDliz.

(7.116)

Let us now consider the static solutions. In this case, the field equation (7.115) reduces to V2z-(z* -V2z)z = 0.

(7.117)

The energy of a static solution as obtained from the action (7.112) is clearly E=

f(Dizy(DiZ)d2x,

i = 1,2.

(7.118)

Finiteness of energy requires that as r = |x| —> oo Diz = diz - iAiZ = 0.

(7.119)

Ch 7.

215

Soliton as Anyon in Field Theories

Let us start from the topological inequality which follows from

[(Diz)* ± ieyiDjzy]

• [DiZ =F ieikDkz] > 0.

(7.120)

Because of the constraint \z\2 = 1, this inequality can be re-expressed in the form (Piz)* • (DiZ) > SijiDiz)* • (DJZ)

(7.121)

so that the energy is bounded from below by the topological charge Q E > 2TT|Q|

(7.122)

where

In any Q-sector, the energy reaches its minimum when the fields minimize the energy in that sector and satisfy the first order self dual field equation Diz = ±ieijDjz.

(7.124)

Note that the solutions of Eq. (7.124) automatically solve the second order field Eq. (7.117) while the converse need not be true. The most general solution for z can be written down in terms of (anti) holomorphic function w

* = ^rmi(i)2

1/1 + lwl W

(7 125)

-

These solutions are characterized by the energy E = 2ir\Q\ where Q is as given by Eq. (7.123). One can in fact define a topological current J^ J» = -^e^x(Dvz)*(Dxz)

(7.126)

which is conserved by construction, and the topological charge Q as given above, is related to it by Q = J J°d2x. One can easily show that for the soliton solutions, Q is just the winding number i.e. Q clearly describes the homotopy of the mapping S2 —> S2. Since JM is the topological conserved current, hence one can clearly add the following gauge invariant action

IH=fd3x^A^

(7.127)

216

Fractional Statistics and Quantum Theory

to the original action (7.112) . This action is nothing but the Hopf term which is related formally to the Chern-Simons term since from Eqs. (7.113) and (7.126) it follows that

A>iJfl = £>LVX A FvX

h

»

•

(7 128)

-

Note however that here A^ is not an independent gauge field but is entirely determined in terms of z(x) through the constraint Eq. (7.113). As a result, unlike the Chern-Simons term, the Hopf term is locally a total divergence and hence does not contribute to the equations of motion. To see the total divergence, let us put zi=Vi+ so that X)t=i Vi written as

=

-*-• * n

e^\z*d^z)(d,z*)(dxz)

iy-2,

terms

-22 = 2/3 + iyA

(7.129)

°f 2/is> the Hopf density (7.128) can be

= e^x [z1(d^z1)(duz*2)(dxz2) + (1 <-> 2)] -e^x(d^yi)(d,y2)(dxy3)

= 2/4

= 2e^xdx[f(y)dfiy1duy2]

(7.130)

where

/(,) = ±arcsin[ ( i _^^ 2 2 ) 1 / 2 ].

(7.131)

Note that unlike the Chern-Simons term, the Hopf term has no dynamics. Besides, for the CPl soliton solutions (which are time independent solutions of the equations of motion), the Hopf term is identically zero because of the time derivative and the relationship (7.113). Thus the way the Hopf term imparts fractional spin and statistics to the soliton is similar to that in quantum mechanics. However it is very different than the way the Chern-Simons term imparts fractional spin and statistics. In particular, since the Hopf density is a total divergence, hence the Hopf action can be expressed n terms of the surface terms, namely two integrals at the initial and final times so that in the path integral formalism, the contribution of this action is essentially in terms of the phases of the initial and the final wave functions. Since the configuration space in question is multi-connected, the Hopf action depends on the homotopy classes of the path and, therefore, the converted phases are multi-valued which in turn

Ch 7. Soliton as Anyon in Field Theories

217

gives rise to the fractional spin (= 0/2ir) and the solitons obey fractional statistics characterized by 6 [63,67]. References [1] For a detailed review of the various properties of the Chern-Simons term, see for example, A. Khare, Fort. der. Physik 38 (1990) 507 ; Proc. Indian Natn. Sc. Acad. A61 (1995) 161. [2] For self-dual charged vortices see, G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Phys. m 36 (Springer-Verlag, Berlin, 1995). [3] A.A. Abrikosov, Sovt. Phys. JETP 5 (1957) 1174. [4] H.B. Nielsen and P. Olesen, Nucl. Phys. B61 (1973) 45. [5] B. Julia and A. Zee, Phys. Rev. D l l (1975) 2227. [6] S.K. Paul and A. Khare, Phys. Lett. B174 (1986) 420; B177 (1986) E453. [7] J. Frohlich and P.A. Marchetti, Comm. Math. Phys. 121 (1989) 177. [8] E. Witten, Phys. Lett. B86 (1979) 283. [9] D. Buchholz and K. Fredenhagen, Comm. Math. Phys. 84 (1982) 1. [10] D. Boyanovsky and R. Blankenbeckler, Phys. Rev. D34 (1986) 612. [11] M.B. Paranjape, Phys. Rev. Lett. 55 (1985) 2390 ; 57 (1986) 612. [12] D.P. Jatkar and S. Rao, Phys. Lett. B229 (1989) 67. [13] A.N. Redlich, Phys. Rev. Lett. 51 2007 ; Phys. Rev. D29 (1984) 2366. [14] See for example A. Niemi and G.W. Semenoff, Phys. Rep. 135 (1985) 99. [15] V.I. Inozemstsev, Euro. Phys. Lett. 5 (1988) 113. [16] G. Lozano, M.V. Manias and F.A. Schaposnik, Phys. Rev. D38 (1988) 601. [17] L. Jacobs, A. Khare, C.N. Kumar and S.K. Paul, Int. J. Mod. Phys. A6 (1991) 3441. [18] S.K. Paul and A. Khare, Phys. Lett. B171 (1986) 244. [19] S. Deser and Z. Yang, Mod. Phys. Lett. A4 (1989) 2123. [20] G. Dunne, R. Jackiw and C. Trugenberger, Phys. Rev. D41 (1990) 661. [21] L. Jacobs and C. Rebbi, Phys. Rev. B19 (1979) 4486. [22] R. Jackiw and P. Rossi, Nucl. Phys. B190 (1981) 681. [23] E.J. Weinberg, Phys. Rev. D19 (1979) 3008 ; D24 (1981) 2669. [24] N. Ganoulis and M. Hatzis, Phys. Lett. B198 (1987) 503. [25] R.B. MacKenzie and A.M. Matheson, Phys. Lett. B259 (1991) 63. [26] H.J. de Vega and F.A. Schaposnik, Phys. Rev. Lett. 56 (1986) 2564 ; Phys. Rev. D34 (1986) 3206. [27] C.N. Kumar and A. Khare, Phys. Lett. B178 (1986) 395, B182 (1986) E415; Phys. Rev. Lett. 59 (1987) 377 ; Phys. Rev. D36 (1987) 3253. [28] D.P. Jatkar and A. Khare, Phys. Lett. B236 (1990) 283. [29] J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230. [30] R. Jackiw and E.J. Weinberg, Phys. Rev. Lett. 64 (1990) 2234. [31] E. Bogomolriyi, Sov. J. Nucl. Phys. 24 (1976) 449. [32] H. J. de Vega and F. Schaposnik, Phys. Rev. D14 (1976) 1100.

218

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65]

Fractional Statistics and Quantum Theory

P. di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93. C. Lee, K. Lee and E.J. Weinberg, Phys. Lett. B243 (1990) 105. S.N. Behera and A. Khare, Pramana (J. Phys., India) 15 (1980) 245. Y. Kim and K. Lee, Phys. Rev. D 49 (1994) 2041. A. Khare, Phys. Lett. B277 (1992) 123. A. Khare, Phys. Lett. B255 (1991) 393. R. Jackiw, K. Lee and E.J. Weinberg, Phys. Rev. D42 (1990) 3488. A. Khare, Phys. Lett. B263 (1991) 227. D.P. Jatkar and A. Khare, J. Phys. A24 (1991) L1201. D. Bazeia, Phys. Rev. D43 (1991) 4074. R. Wang, Comm. Math. Phys. 137 (1991) 587. J. Spruck and Y. Yang, Comm. Math. Phys. 149 (1992) 361. S.K. Kim and H. Min, Phys. Lett. B281 (1992) 81. N. Manton, Phys. Lett. B110 (1982) 54 ; Phys. Lett. B154 (1985) 397 ; P.I. Ruback, Nucl. Phys. B296 (1988) 669 ; T.M. Samols, Phys. Lett. B244 (1990) 285. J. Dziarmaga, Phys. Rev. D49 (1994) 5469 ; D51 (1995) 7052. A. Khare, Phys. Rev. D46 (1992) R 2287. T. Vachaspati and A. Achucarro, Phys. Rev. D44 (1991) 3067 ; M. Hindmarsh, Phys. Rev. Lett. 68 (1991) 1263 ; G.W. Gibbons, M.E. Ortiz, F. Ruiz Ruiz and T.M. Samols, Nucl. Phys. B385 (1992) 127. C. Lee, K. Lee and H. Min, Phys. Lett. B252 (1990) 79. P.K. Ghosh, Phys. Lett. B326 (1993) 264; Phys. Rev. D49 (1994) 5458 ; J. Lee and S. Nam, Phys. Lett. B261 (1991) 437; M. Torres, Phys. Rev. D46 (1992) R2295. K. Lee, Phys. Rev. Lett. 66 (1991) 553 ; L.F. Cugliandolo et al., Mod. Phys. Lett. A6 (1991) 479. R. Jackiw and S-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969 ; Phys. Rev. D42 (1990) 3500. S.K. Kim, W. Namgung, K.S. Soh and J.H. Yee, Phys. Rev. D46 (1992) 1882. S.K. Kim, K.S. Soh and J.H. Yee, Phys. Rev. D42 (1990) 4139. P. Olesen, Phys. Lett. B265 (1991) 361 ; B267 (1991) E541. Z. Ezawa, N. Hotta and A. Iwazaki, Phys. Rev. D44 (1991) 452. A. Khare and J. Maharana, Nucl. Phys. B244 (1984) 409. F. Cooper, A. Khare, R. Musto and A. Wipf, Ann. Phys. 187 (1988) 1. G. Dunne and C. Trugenberger, Phys. Rev. D43 (1991) 1323. B. Grossman, Phys. Rev. Lett. 65 (1990) 3230. G. Dunne, R. Jackiw, S-Y. Pi and C. Trugenberger, Phys. Rev. D43 (1991) 1332. F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. A.A. Belavin and A.M. Polyakov, JETP Lett. 22 (1975) 245. For detailed exposition of solitons see, R. Rajaraman, Solitons and Instantons, (North-Holland, Amsterdam, 1982) and references therein.

Ch 7.

Soliton as Anyon in Field Theories

219

[66] For detailed discussion of sigma models see, W. Zakrzewski, Low Dimensional Sigma Models, (Adam Hilger, Bristol, 1989) and references therein. [67] Y.-S. Wu and A. Zee, Phys. Lett. B147 (1984) 325 ; A.M. Din and W. J. Zakrzewski, B146 (1984) 341.

Chapter 8

Anyons as Elementary Field Quanta

Nothing was visible, nor could be visible, to us, except straight lines — E.A. Abbott in Flatland 8.1

Introduction

In the last chapter we have discussed the soliton solutions in several relativistic and non-relativistic models with the Chern-Simons term (or its incarnation, the Hopf term) and shown that these solitons are in fact anyons. Being solitons, these are extended (and not point) objects and hence are not very useful in case one wants to construct local field theory of anyons. It is then worthwhile to enquire whether one can construct local quantum field theories where the fundamental fields represent the creation and annihilation of anyons. The purpose of this chapter is to discuss such models in 2+1 dimensions where this is possible provided one also adds the ChernSimons term to the theory. In particular, in Sec. 8.2, we discuss the case of the non-relativistic matter field which, for definiteness, we take to be bosonic and couple it minimally to an abelian gauge field. The key point is that the kinetic energy of this gauge field is given purely by the ChernSimons term (instead of the usual Maxwell term). We show that because of the Gauss law constraint, the gauge field is not an independent degree of freedom but is entirely determined in terms of the matter field. On eliminating the gauge degrees of freedom, one is left with new matter fields whose action is that of a free (complex) scalar field. However, such a field does not obey ordinary commutation relations but rather obeys anyonic commutation relations. The relativistic case is discussed in Sec. 8.3, where 221

222

Fractional Statistics and Quantum Theory

we briefly mention what is known so far and the difficulties involved in this case.

8.2

Non-relativistic Field Theories

Let us consider a complex bosonic non-relativistic matter field ip(x,t) of mass m (of course a similar discussion can also be done for the fermionic matter field). Let us minimally couple it to an abelian gauge field A^ with a Chern-Simons kinetic term [1,2,3,4]

S = J dzx L+^oV- + ^+(Di

+ DD^ + f e""AAAA J

(8.1)

where D^ — d^ — ieA^ is the covariant derivative. For simplicity, in this chapter we shall set H = c = 1. It is worth pointing out that the same action was used by us in the last chapter to obtain the Chern-Simons soliton solutions. On varying the action with respect to A^, we obtain e»vXFuX = —J» M where the current J^ is explicitly given by

(8.2)

p = J° = xp+ip

(8.3)

Jk = ^~. W+Dki> - (D V)+V>].

(8.4)

Arm Here p and J are the number density and the current density operators respectively which satisfy the continuity equation dtp + V.J = 0.

(8.5)

As also seen in the last chapter, Eq. (8.2) is a remarkable relation indicating that the Chern-Simons field strength is completely determined by the particle current. Even more remarkable is the fact that the gauge potential Ap itself is not an independent degree of freedom. Let us consider the JJ, = 0 component of Eq. (8.2) (8.6) M where B = V x A is the Chern-Simons magnetic field. This equation is clearly the second quantized version of the Gauss law constraint obtained in B=-p

Ch 8.

Anyons as Elementary Field Quanta

223

the last two chapters (except that whereas in those cases p was the charge density, here p is the matter density, hence the extra factor of e in Eq. (8.6) compared to those cases). Now, in the Weyl gauge diA1 = 0, hence, one can invert Eq. (8.6) without any ambiguity and solve for the vector potential A. We obtain

A\x) = £v^-(^-Jd2yG(x

- y)p(y))

(8.7)

where G is the two-dimensional Green function V2G(x - y) = <5(x - y)

(8.8)

whose solution is well known to be G(x-y) = i-ln(M|x-y|)

(8.9)

where M is an arbitrary scale. Thus A1 can be written as

--i/^s?* 1 -^

(8 10)

-

where is the winding (polar) angle i.e. (x - y) = arctan (*x~ V ^) • y J v^

(8.11)

Note that while writing the second line of Eq. (8.10), we have used the Cauchy-Riemann equations

in|x y| =

y)

^ - -L^~ -

(8 i2)

-

It is worth pointing out that ^^jG(x-y)

(8.13)

is ill-defined at x = y. Thus one has to supplement Eqs. (8.8) and (8.9) with a regularization prescription. One such prescription is

224

Fractional Statistics and Quantum Theory

where the regulated Green function G^ (x) is

G(a)(x) = ~ J d2y ( ^ l n | x - y | ) e ^ 2 / a .

(8.15)

This has the desired property that lim G (o) (x) = G(x) = — In |x| 2?T

a—>0

(8.16)

while for any nonzero a lim sij -^-G(a\x)

=0

(8.17)

so that once Eq. (8.15) is systematically used, all ambiguities are eliminated. If one is now allowed to move the derivative operator outside the integral (8.10), then one could express A as a gradient. However, (f>(x — y) is a multi valued function. Hence one must first fix a branch-cut in the y-plane starting at a; so as to make it single-valued. No matter what choice is made for this cut, the resulting range of integration of y will depend on x and hence extra contributions are produced in moving d/dxl outside the y integral. Thus, in general one cannot write

A(x) = -^-V s [|d 2 2 / 0(x-y) / 9( 2 /)] .

(8.18)

As a result, in general A is not a pure gauge and hence it cannot be removed by a gauge transformation. However, in the special case when p(y) is a sum of J-functions, A(x) can indeed be written as (8.18) and hence is a pure gauge. Such a situation arises in the case of non-relativistic localized point particles [2]. Let us assume that in the context of our non-relativistic model (8.1) too, p(y) is a sum of <5-functions and hence A is a pure gauge. Let us now show that AQ is also a pure gauge i.e. A{x) = VA(x,t), AQ{X) = -dtA(x,t)

(8.19)

where

A(x) = -^Jd2y(x-y)p(y).

(8.20)

Ch 8.

Anyons as Elementary Field Quanta

225

To this end, notice that from Eq. (8.2) we have Fio = diA0 - doAi = ~eikJk . (8.21) M On using this relation and the continuity Eq. (8.5) one can solve for Ao and obtain

Ao(x) = -- [d2yG(x-y)Vx3(y).

(8.22)

We thus have explicitly shown that the Chern-Simons gauge field A^ is entirely determined by the matter configuration i.e. p and J. In order to show that the above Ao can be written as a pure gauge, let us perform an integration by parts and rewrite it as

A0(x) = J Jd2ye^-£jG(x - y)J\y).

(8.23)

On integrating by parts, and using Eqs. (8.5), (8.9) and (8.10) we then obtain the desired result Ao{x) = ^ ^ r f 2 ^ ( x - y ) p ( y , t ) = - ^ A ( x , t ) .

(8.24)

While deriving the above relation, we have dropped the surface terms in the integration by parts. This is clearly justified in the case of localized densities. Thus, in the case of localized densities, A^x) = — <9MA(:r) i.e. the Chern-Simons field is a pure gauge. Hence it can be removed by the gauge transformation AM —-> A'p = A^ + dMA = 0.

(8.25)

Thus, under such a singular transformation, covariant derivatives turn into ordinary derivatives, and the action (8.1) becomes

S' = f dsx liTp+dotP + - ^ + { d l + #f )]

(8-26)

where the new matter field i/> is defined as j>(x) = e- i e A ^V(z), i>+{x) = ^ + (z)e i e A ( x ) .

(8.27)

The above action (8.26) is that of a free, complex, non-relativistic, scalar field i/>. However, we now show that such a field does not obey the conventional commutation relations as satisfied by ip.

226

Fractional Statistics and Quantum Theory

Consider the non-relativistic action (8.1). We can quantize the action by imposing the equal-time commutation relations for the bosonic field ip [V>(x,t), V + (y,i)]=<*(x-y) foKx, t), V(y, t)]=0=

(8.28)

[V + (x, t), V+(y, t)].

(8.29)

Since the gauge field A is a function of the number density operator p (=tp+tp), hence the commutator of A and V is n o t trivial. In fact using Eqs. (8.10) and (8.28) we obtain

On using the regularized Green function as given by Eq. (8.15), it then follows by using Eq. (8.17) that [4'(x,i),V(x,*)] = 0.

(8.31)

This is interesting because it means that there are no ordering ambiguities in the quantum theory as given by Eq. (8.1). We are now ready to show that when ip obeys ordinary commutation relations, ip obeys anyonic commutation relations. To this end, let us first notice that foKx,i),A(y,t)] = - ^ - 0 ( y - x ) ^ ( x , t )

(8.32)

which follows from Eqs. (8.28), (8.29), (8.20) and (8.3). Hence for x ^ y we have ^(x,t)^(y,*) = e- ieA ( x -*)v(x,t)e- ieA ( y ' t )v(y,i) =

e-«e[A(x,t)+A(y,t)]ei^(y-x)^(X)

^ ( ^ ^

=

e -ie[A(x,t)+A(y,t)] e i^(y-x)^( y)t ^( X) ^

=

e-ie[A(x,t)+A(y,t)]ei^(y-x)eieA(y,t)^^y)

=

e *«[*(y-x)-0(x-y)]^( y ) t ^( X j t )

^eieA(x,t)^^X) ^ (833)

where a = e2/2nfi. Thus, the whole discussion depends crucially on the winding angle . As stated above, it is a multi-valued function a priori, but it can be made single-valued by introducing cuts across which jumps by 2TT. Taking the cut in the direction of the positive z'-axis so that (ei +

Ch 8.

227

Anyons as Elementary Field Quanta

E&2) —> 0, (ei — E&2) —* 2TT as e —> 0, we have

0(y - x) - (x - y) = -K sign{x2 - y2), x2 ^ y2 = ir signix1 - y1), x2 = y2 .

(8.34)

Thus we have shown that ^(x, t)j>(y, t) = e^(y,

t)^(x, t)

(8.35)

i.e. the matter field ip obeys anyonic commutation relations of statistics a (= e2/2n/j,). If instead, we make a cut along the negative x'-axis, then we would obtain an opposite value for the difference of the argument in Eq. (8.34) and hence a phase factor (e~lna), opposite to that in Eq. (8.35). Proceeding in the same way, it is easily shown that
(y,t)i>(x,t)+5(x-y)

(8.36)

so that if x ^ y then we have ^(x,t)^+(y,t) = e- i i r Q ^+(y,*)^(x,t).

(8.37)

It must however be noted that for x = y, the phase proportional to a vanishes and hence the canonical commutation relations remain unchanged. We can in fact also show that so far as the problem of N anyons is concerned, the non-relativistic field theory as given above is equivalent to the first quantized approach as discussed in Chapter 2. To this end, note that the Hamiltonian corresponding to the Lagrangian (8.1) is H= [d2xH{x)J where

II(:E)

(8.38)

n(x) = ^-U+(x)U(x) 2m

is the momentum operator associated with ip, i.e. n(x) = [V - ieA(x)]ip(x) = DV>(a;)

(8.39)

and II + (x) = (Dtl>(x))+ is its hermitian conjugate. One can then show that H=—

[d2x\-tp+y2ip+ieA-(t/j+\7ip-(Vtp+)ij)

2m J

I

+ e2A2iP+ip\ . (8.40)

J

We now want to obtain the first quantized Schrodinger equation from our second quantized theory. To that end, let us first note that this H commutes

228

Fractional Statistics and Quantum Theory

with the number operator i.e. [H,N} = 0,

N=fd2xp(x)

(8.41)

and hence H and TV can be simultaneously diagonalized i.e. there will exist states \E, TV) which will simultaneously satisfy H\E, TV) = E\E, TV), N\E, TV) = N\E, TV).

(8.42)

Further, let us assume the existence of a vacuum state |0) annihilated by ip{x),H and TV •0(a;)|O> = 0 = (0|V+(a;);

H|0) = 0 = JV|O>.

(8.43)

The states for a fixed number (TV) of particles in the configuration space are then given by In,..., r j v ) = V + (ri)..-V + (riv)|0)

(8.44)

and hence the TV-body wave function with the energy eigenvalue E is given by * ( n , . . . , vN) = (n,..., rN\E, TV) = <0|V(ri)...V<(rjv)|£, TV).

(8.45)

Thus (r 1 ,...,r JV |tf|£;,TV) =

(0\^(ri)..^(rN)H\E,N)

= {0\Mr1)...(r1)...il>(TN)\E,N) = £?*(r 1 ) ...,rj V ).

(8.46)

Notice that in the second line we have a commutator which we now have to compute. To motivate this computation, first consider the case of N = 1 in which case we need to compute the commutator [ip(ri), H]. On using

Ch 8.

229

Anyons as Elementary Field Quanta

Eqs. (8.40), (8.28), (8.29) and (8.31) we find

mri),H\ = -l-Vfyin)

+ ^A 2 ( r/ )V(r/)

+ i l [A(r7) • VMn) + Vj • (A(r/)V(r/))j +—

/d2^'^-G(x-rj)V(r/)

x ^ + (x)^V(x) - (^V+(x)M(x)]

+ ^ ( x ) ^ - G ( x - r/)V(r7)p(x)] .

(8.47)

In view of Eqs. (8.7) and (8.23), this commutator can be simplified yielding

Mr,), H] = - A-VfV(rz) +

eA^n)^)

+ 2^-2 /" d2WG(x - rj) • VG(x - rj)p(x)^(rj).

(8.48)

(0|^( r / ),/f] = -^-V 2 (0|V(r/)

(8.49)

Hence

since both A0 and A (see Eq. (8.7)) involve expressions that contain operator ip+ standing on the left and hence annihilating the vacuum (0|. Proceeding in the same way and using Eqs. (8.48) and (8.49) for 1 = 2 and 1 respectively, we obtain (0\[IP(V1)^(T2),H}\E,N I

= 2)

2

= ~^n Z)(V/ - »eA(n, r 2 )) 2 (0|V(riMr 2 )|£, N = 2) where

(8.50)

230

Fractional Statistics and Quantum Theory

Here Eq. (8.9) has been used in obtaining the second line from the first. In view of Eq. (8.50), Eq.(8.46) for JV = 2 takes the form !

2

Z m

i=i

-7T- J2lVi

- *eA 7 (n, r 2 )] 2 *(r 1 , r 2 ) = E*(Tl, r 2 ).

(8.52)

Recall that this is precisely the first quantized Schrodinger equation for two particles in the non-local potential (8.51). The generalization to arbitrary TV is straightforward and we obtain exactly the same equation as obtained in the first quantized version in Chapter 2 thereby showing the equivalence of the field theory formulation and the first quantized formulation for the problem of iV-anyons. In fact, the equivalence goes even further. For example, it is easily shown that the action S as given by Eq. (8.21) is invariant under space-time translations as well as rotations [2]. In particular, the rotational invariance gives rise to the conserved angular momentum J. One can show that J is given by j

=

°N(N

- 1)

(8.53)

which is in general fractional and is identical to that derived in the first quantized theory in Chapter 2. It is worth noting that the angular momentum J obtained here is only the orbital one; for example J = 0 for N = 1. Thus what we have really shown is that in the non-relativistic field theory with Chern-Simons term, there is a fractional orbital angular momentum and particles carry no spin. Some clarification is called for at this stage. What we have shown is that the fields ip(x.,t),tp(y,t) satisfy anyonic commutation relations with phase factor e +r/ra or e~l7ra depending on how we choose the cut to make the winding angle <j> single valued. However, this is not enough. What is really required is that the phase of the wave function changes both by +ira and — wa in response to which way we braid in interchanging x and y. No one has been able to show this so far. In fact, what we have shown above is the best that one can achieve with the local operators rp, tp+. Local information, like the initial and the final positions of particles, is simply not sufficient to code the braiding, where we also have to specify which way the particles have passed around each other in interchanging their positions. As I see it, the only way to take care of this problem in this formalism is to choose such a definition of the multi-valued function (f> which will make •0 a non-local operator [5].

Ch 8.

Anyons as Elementary Field Quanta

231

Summarizing, it appears that within the non-relativistic field theory formalism, anyons can only be described by non-local operators, which are hard to deal with. If one insists on a local formulation, then one has to hide the statistics in an interaction with a Chern-Simons field.

8.3

Relativistic Field Theories

There is no doubt that ideally the various effects of fractional spin, such as the spin-statistics theorem should be understood only in a full fledged relativistic quantum field theory. However, relatively little is known in this respect. In this section we shall briefly discuss few attempts to address some of the issues involved [6]. The simplest approach to a field theory of excitations with fractional spin and statistics consists of looking at field theories with Chern-Simons term (or its incarnation, the Hopf term) that support soliton solutions. This has already been done in the last chapter. The other possibility is to construct fundamental field theories with fractional spin and statistics. In this context, it might be worthwhile to first discuss the general features of relativistic particle mechanics from a general, group theoretic point of view. Group Theory Aspects As we have seen in Chapter 2, fractional spin in two spatial dimensions is possible because the rotation group, SO{2) is infinitely connected; its group manifold is the circle S1 and the first homotopy group of S1 is non-trivial i.e. 7Ti (51) = Z. Now in general, a wave function may provide a multi-valued representation of rotations, provided the multivaluedness is contained in a phase. A wave function that carries fractional spin provides a multi-valued representation of the group, in that a rotation of 2TT does not leave the wave function invariant, but rather, it multiplies it by the phase e2**'-7. Clearly, this is only possible if the group is multiply connected. Then the wave function provides a multi-valued representation of the group or equivalently, a true representation of its universal cover [7,8]. In a relativistic theory, the rotation group is a subgroup of the Lorentz group, which in 2+1 dimensions is 50(2,1). Thus the wave function must either provide a multi-valued representation of S0(2,1) with the multi-valuedness contained in a phase, or a true representation of its universal cover SO(2,1). Thus, a necessary condition for fractional spin is that the manifold of the Lorentz group be

232

Fractional Statistics and Quantum Theory

infinitely connected. This is enough to ensure that the group admits multivalued representation. Further, it is also necessary that the restriction to the rotation subgroups of a multi-valued representation be multi-valued too, i.e., that the corresponding sub-manifold also be infinitely connected. Let us consider the group 50(2,1) [9]. The generators in the fundamental representation are the 3 x 3 matrices L(/»0" p = -i(g^g"

p

- g™g» p) .

(8.54)

Here the operator j(L^12^ — l/ 2 1 )) = R generates the compact rotation subgroup, while the operators ^(Z,(oa) — Z/ao)) = Ba generate the noncompact boosts and they satisfy the Lie algebra [Ba, R] = -ieabBb

, [Ba, Bb] = ieabR.

(8.55)

In covariant notation, this can be written as [!>"), £>")] = i (g^L{up)

+ gupL^a)

- g^L^

-g"aLM)

(8.56)

which is the same as the SL(2, R) Lie algebra ; hence the two groups admit the same universal cover. Now an SL(2, R) matrix is

where a, b, c, d are real numbers that satisfy ad — be = 1 which can also be written as

This is precisely the equation of a three-dimensional one-sheeted hyperboloid. Its section by the plane a — d = b + c = Oisa, real circle, which in the SO(2,1) language corresponds to the SO(2) subgroup generated by R. Thus it follows that there indeed exist noncontractible paths on the group manifold, namely, those that wind around this circle 7ri(5ri(2,JR)) = 7r 1 (6'O(2))=Z.

(8.59)

We thus conclude that the group 50(2,1) is infinitely connected, and its multi-valued representations correspond to multi-valued representations of rotations. It may be noted here that if, instead, we consider theories on the Euclidean space-time, then the Lorentz group would be 5O(3) whose

Ch 8.

Anyons as Elementary Field Quanta

233

group manifold is only doubly connected (the universal cover is SU(2)) i.e. TTI(5O(3)) = Z2. This implies that the representations of 5O(3) can only be single or double valued, i.e. the spin may be either integer or half-integer. In conclusion, the Minkowski nature of the matrix is essential in order to consider fractional spin. Coming back to the group 50(2,1), it is well known that 50(2,1) has no finite-dimensional unitary representations and more important, its finite-dimensional representations are at most double valued. Thus in the case of fractional spin, one necessarily has to work with an infinite dimensional representation of the Lorentz group in general, or the rotation group, in particular. What it means is that in order to write the wave function explicitly, we have two alternatives-either we define an infinite component wave function or we stick to a wave function that gives the trivial representation of 50(2,1) (a scalar) and supplement it with a phase that has multi-valued transformation properties upon the action of SO(2,1). If we are interested in constructing relativistic one particle states, then the relevant group is Poincare (rather than Lorentz). In particular, a physical on-shell state must provide a unitary irreducible representation of the universal covering of the Poincare group. In the 2+1 case, the relevant Poincare group is R3 (g> 50(2,1) which is the semi-direct product of the Lorentz group and the translation group. Its universal cover is /5O(2,1) = R3 ® 5O(2,1) whose unitary irreducible representations have been constructed [10]. The simplest approach to fractional spin in a relativistic treatment is to add the Chern-Simons term to a system of ./V-particles. For example, the propagator for a relativistic spinning particle has been obtained in terms of a bosonic path integral [11]. The quantization of relativistic particles with arbitrary fractional statistics has also been discussed [12]. Wave Equation For Anyons An entirely different approach to the relativistic theory is to construct a wave equation for anyons in the same spirit as the Dirac equation for spin\ particles [13]. One straightforward way of writing this equation is to postulate, a la Wigner, that the one particle states u(p) with mass m and spin s provide irreducible representation of the Poincare group. In that case, the equation of motion reduces to PpP^uip) = m2u(p)

(8.60)

234

Fractional Statistics and Quantum Theory

which is the mass shell condition and ^s^xP»M»xu(p) = msu(p)

(8.61)

is the spin condition i.e. the Pauli-Lubanski equation. The infinitesimal transformation law for the states u(p) under translation, rotation and boost can now be immediately written down. A dynamical equation that satisfies these requirements has been proposed in [13]. The wave functions u(p) are chosen to be of the spinor-vector form: F£(x), where /J, is the vector index and 0 < n < oo runs over an infinite dimensional irreducible representation bounded either below or above of SO(2,1). On imposing the Pauli-Lubanski Eq.(8.66) on F£, we obtain

s^xP» [ M ^ W + M^6a0j F£(x) = msFZ (x)

(8.62)

where we have set P^ = —id^, the generator of space translations, and JI/(M") the Lorentz generators in the tensor product of the spaces spanned by the indices /i and n in Eq. (8.61). Note also that the value of the spin s here is fixed in terms of the value of d that characterizes the infinitedimensional irreducible representation ; in particular s = 1 — d. It turns out that in order to project out the physical degrees of freedom, two additional subsidiary conditions i.e. PMM("v)FW(x)=0

(8.63)

u) vXPaM^

(8.64)

and e"

F^{x) = 0.

have to be imposed. By expanding the solutions to the spin equation in plane waves, it is easily shown that the solutions that satisfy the above subsidiary conditions also satisfy the transversality condition PflF^(x) = 0.

(8.65)

The general solution to these equations can be explicitly constructed. Only one component of Fn is non-vanishing, the highest-weight one (n = 0) and this satisfies the mass-shell conditions automatically by iteration. The explicit solution in the rest frame of the particle is /0"(P0) = JV" A

0

(8.66)

Ch 8.

Anyons as Elementary Field Quanta

N*

v

=

~V2 0 0" O i l _ 0 -ii_

235

(8.67)

where PQ = (m,o,o) and ip(po) is a function that provides a one dimensional positive energy Poincare irreducible representation with zero spin and given momentum. This solution is valid for representations bounded below. The solution for arbitrary p is easily constructed from here by Lorentz boosting. Similarly, the negative energy solution can be constructed from representations bounded above. Fundamental Fields as Anyons Finally let us discuss the question of construction of quantum field theories whose fundamental fields carry multi-valued representations of 50(2,1). Clearly, such fields must provide a linear representation of the Lorentz group with a generally reducible Poincare representation content. Now if the fundamental fields are to carry fractional spin, they must carry a multivalued irreducible representation of SO(2,1). We then have the following two options and we discuss them one by one. The first option is that we define infinite component fields and from them construct the one particle dynamics by imposing equations of motion that satisfy the requirement that one-particle states provide multi-valued Poincare Eqs. (8.60) and (8.61) and have appropriate infinitesimal transformation laws under translation, rotation and boost. The most difficult part is the derivation of an action that reproduces these equations of motion. This is because, the only available set of equations of motion with fractional spin is given by Eqs. (8.62) to (8.64). The finite component wave function Fj£(x) as introduced in these equations may now be promoted to a relativistic field whose classical solutions coincide with those given by Eq. (8.66). However, these equations, as they stand, do not satisfy the integrability condition that would follow if they were the variation of something. However, it is possible to introduce a nonlocal set of equations containing a further auxiliary, infinite component field whose classical solutions coincide with Eq. (8.66). Further, it is possible to obtain these non-local equations by varying a non-local action [13]. But then no one really knows how to quantize such a non-local theory. Besides, its physical interpretation is also not clear. Let us now discuss the second option. We work with multi-valued fields by adding the Chern-Simons term to the action and essentially repeat what

236

Fractional Statistics and Quantum Theory

we have done in the previous section for the non-relativistic case. Thus, instead of the non-relativistic model (8.1), one could consider a relativistic field theory, say a complex scalar field theory, coupled to an abelian gauge field with a Chern-Simons kinetic energy term (and no Maxwell term). As a side remark, we mention that the fields must be complex if we want to allow for fractional statistics. This is because, anti-particles which are generated by complex conjugate operator have spin and statistics, equal in magnitude but opposite in sign to that of particles. In other words, a real field is necessarily bosonic. Coming back to complex fields, one again wants to know if one can construct local quantum field theory where the fundamental fields represent the creation and annihilation of anyons. On proceeding exactly as in the last section, one again obtains Eq. (8.6). However, unlike the last section, it is not possible now to write A as a pure gauge and hence it cannot be removed by a gauge transformation. This is because, unlike the non-relativistic case, here p(y) cannot be a sum of delta functions, since, now the particles are not point particles but are extended objects [14]. Thus, it is not at all clear whether in the relativistic case the only effect of the gauge field is to endow the particle with arbitrary spin or if residual interactions are present. A similar problem also arises in models which emerge from the relativistic theory in the non-relativistic limit. In particular, one obtains different results depending on which limit is taken first i.e. the size of the extended object going to zero vis-a-vis the regulator parameter going to zero [15]. Attempts have been made to tackle these problems by quantizing the theory with Chern-Simons term on a lattice [16,17,18] with or without the Maxwell term. So far, these attempts have met with only a limited success. Thus it is fair to say that, so far, we do not have a model in relativistic local quantum field theory where the fundamental (non-interacting) field quanta are themselves anyons. In fact it appears unlikely that one can obtain a simple, local (relativistic) Lagrangian for anyons. This is because, even in 2 + 1 dimensions, spin has to be integral or half-integral for local fields. On the other hand, fractional spin is admissible for fields which carry charges associated with gauge symmetries (with accompanying flux integrals at infinity) which are typically localizable only in space-like cones [19,20]. This is what happens for example, when one generates fractional spin by coupling point particles to a Chern-Simons gauge field [4] (which has non-trivial long-ranged properties). A similar thing also happens in the case of the Jackiw-Nair wave equation in that when one tries to eliminate

Ch 8.

Anyons as Elementary Field Quanta

237

the auxiliary fields or tries to enforce the corresponding constraints, the non-localities arise in the theory. References [1] R. Jackiw, Ann. Phys. 201 (1990) 83. [2] R. Jackiw and S-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969 ; Phys. Rev. D 4 2 (1990) 3500. [3] Z. Ezawa and A. Iwazaki, Phys. Rev. B 4 3 (1991) 2637 ; Z. Ezawa, N. Hotta and A. Iwazaki, Phys. Rev. D 4 4 (1991) 452. [4] For a detailed exposition see A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics, Lecture Notes in Phys. m 14 (SpringerVerlag, Berlin 1992) and references therein. [5] G.W. Semenoff, Phys. Rev. Lett. 61 (1988) 517 ; G.W. Semenoff and P. Sodano, Nucl. Phys. B328 (1989) 752. [6] For a detailed exposition see S. Forte, Rev. Mod. Phys. 64 (1992) 193 and references therein. [7] A.P. Balchandran, G. Marmo, B.-S. Skagerstam and A. Stern, Gauge Theories and Fiber Bundles (Springer-Verlag, Berlin, 1983). [8] A.O. Barut and R. Raczka, Theory of Group Representations and Applications (World Scientific, Singapore, 1986). [9] B.G. Wybourne, Classical Group for Physicists (Wiley, New York, 1974). [10] B. Binegar, J. Math. Phys. 23 (1982) 1511. [11] A.M. Polyakov, Mod. Phys. Lett. A 3 (1988) 325. [12] P de Sousa Gerbert, Nucl. Phys. B346 (1990) 440 ; Int. J. Mod. Phys. A 6 (1991) 173. [13] R. Jackiw and V.P. Nair, Phys. Rev. D 4 3 (1991) 1933. [14] S. Forte and T. Jolicoeur, Nucl. Phys. B350 (1991) 589 ; S. Forte, Int. J. Mod. Phys. A 7 (1992) 1025. [15] G.V. Dunne, A. Lerda, S. Sciuto and C.A. Trugenberger, Phys. Lett. B 2 7 7 (1992) 474. [16] M. Luscher, Nucl. Phys. B336 (1989) 557. [17] J. Ambjorn and G.W. Semenoff, Phys. Lett. B226 (1989) 107. [18] V. Miiller, Z. Phys. C47 (1990) 301. [19] D. Buchholz and K. Fredenhagen, Comm. Math. Phys. 84 (1982) 1. [20] J. Frohlich and P.A. Marchetti, Comm. Math. Phys. 121 (1989) 177.

Chapter 9

Anyon Superconductivity

Why should the thirst for knowledge be aroused, only to be disappointed and punished? — E.A. Abbott in Flatland 9.1

Introduction

The concept of (anyonic) fractional statistics received a lot of attention when it was realized that the anyons could provide a mechanism for superconductivity. It is worth noting that till then (and even now) no mechanism was known to explain the high-temperature superconductivity as observed in the cuprate and other materials. Subsequently, however, the experiments have indicated that the anyons do not provide the mechanism for high-Tc superconductivity. However, the fact remains that anyons do provide a novel mechanism for superfluidity (and superconductivity after coupling to the electro-magnetism) and the current picture is that a gas of semions (also called half fermions i.e. 9 = TT/2) shows superfluidity and becomes superconducting if the anyons are electrically charged [1]. It is of course an open question whether nature has made use of this mechanism in some materials. In this chapter, we shall discuss some elementary ideas about the anyon superconductivity. In particular, we shall show that a gas of anyons exhibits superfluidity (and superconductivity after coupling to the electromagnetism). It is also worth pointing out here that the mechanism of anyon superconductivity seems very different from the spontaneous symmetry breaking and specifically from the BCS pairing theory. Two basic issues are involved while discussing the anyon superconduc239

240

Fractional Statistics and Quantum Theory

tivity. (i) One must really start from the microscopic condensed matter physics and get some kind of effective field theory in which the excitations turn out to be semions. (ii) One then has to show that a gas of semions exhibits superfluidity and also superconductivity in case the semions are charged. I may add here right away that the first issue is very hard and much less established than the second one. We shall therefore not discuss the first issue here. As far as the second issue is concerned, it is indeed a very attractive and challenging problem to figure out the behavior of an ideal anyon gas. As far as the high temperature, low density behavior is concerned, we have already addressed it in Chapter 4, when we discussed the virial coefficients and hence the equation of state of such a gas. However, it did not really address the central issue regarding these ideal gases. This is because, as is well known, the most important effects of the quantum statistics occur at low temperature and high density. Let us therefore discuss the behavior of a non-interacting anyon gas at T = 0 K. Before we discuss more formal arguments, it may be worthwhile to give a rather simple minded and naive argument as to why a gas of anyons can show superfluidity. The point is, it is well known that fermions with arbitrarily weak attractive interaction form superfluids at T = OK. Now, as we have discussed in the previous chapters, there is a real sense in which anyons can be considered as fermions with an additional attractive interaction, and hence the expectation that an anyon gas will become superfmid at low temperatures. While certainly naive, this argument clearly points to the fact that the anyon superconductivity is highly plausible. Put differently, one knows that fermions like to stay apart while the bosons like to stick together. Semions being half-way between the two, are clearly more likely to form a pair than two fermions. When two semions do form a pair, they form bosons whose condensation can then lead to superfluidity (and superconductivity in case anyons are charged). One might wonder as to why two semions make a boson and not a fermion. This is easily seen as follows. If we move a bound state of two semions half-way around the other bound state (i.e. interchange the two bound states), then the phase we get is i4 = 1. This is because, each semion which goes round the 2-semion bound state gets a factor of i2 and hence the net phase factor is i2 x i2 = 1. It is worth adding here that, whereas in the usual superconductors, it is the pairing interaction which holds the key, in anyon superconductors it is

Ch 9.

Anyon Superconductivity

241

the gauge interaction which holds the key. Unlike the pairing forces, which are weak, the gauge force originating from fractional statistics is large and the creation energy of a particle-hole excitation is, in general, larger than the typical Coulomb energy. Hence, one has superconductivity inspite of the repulsive Coulomb interaction between the anyons. One of the main difficulty in discussing the behaviour of an ideal anyon gas is that the wave function for N non-interacting anyons cannot be written in terms of the single particle wave functions. Thus we have here a fascinating example of a strongly correlated quantum many body system in two dimensions. The most popular approach is to work near the Fermi statistics

so that n —> oo corresponds to fermions, n = 1 corresponds to bosons while n = 2 corresponds to semions. In order to establish that the statistical attraction (relative to fermions) gives rise to the superfluidity, it is crucial to show that even a weak statistical attraction among a system of (otherwise) free fermions leads to superfluidity. Once that is established, it is reasonable to expect the same for the strong statistical attraction that arises for semions (6 = TT/2) which are believed to be of interest in the context of the anyon superconductivity. The underlying assumption here is that the effect of the weak and strong statistical attraction are, at least qualitatively similar. The advantage of this approach is that since near 8 = IT, the statistical attraction is weak, hence the onset of superfluidity can be studied in a controlled manner. The other advantage of perturbing around the fermions is that the crucial features of the ground state for low energy properties like incompressibility and absence of low lying excitations (equivalently absence of gap) are easy to see in a fermion picture, where they occur when the bands are exactly filled. On the other hand, it is not easy to see these features in a boson picture. To begin with, we shall discuss the mean field theory in Sec. 9.2, and show how the Meissner effect can be explained rather easily in such a theory. Mean field theory, however, is inadequate in explaining the central feature of superfluidity i.e. the existence of a sharp Nambu-Goldstone mode. In Sec. 9.3, we briefly discuss the random phase approximation (RPA) [2,3,4], where one calculates the effect of adding back the residual interaction and show that these interactions produce the necessary pole in the currentcurrent correlation function. In Sec. 9.4, we discuss some salient features of

242

Fractional Statistics and Quantum Theory

an effective Lagrangian containing a massless scalarfieldinteracting with an electro-magnetic field. The interesting point about this effective Lagrangian is that it reproduces many of the results of the RPA calculations [5]. In Sec. 9.5, we discuss the obvious and interesting question whether the highTc superconductors are indeed anyon superconductors. We argue that the answer to this question seems to be no since these superconductors do not seem to violate the discrete symmetries of parity (P) and time reversal invariance (T) which is one of the hallmark of anyon superconductivity. Apart from superconductivity, anyons also seem to predict the existence of novel metals which is briefly discussed in Sec. 9.6. 9.2

Mean Field Theory

The basic idea of mean field approach is to replace the effect of many particles by an average or mean field and to accommodate deviations from the mean field as residual interaction. Let us start with the Hamiltonian for the non-interacting anyon gas as derived in Chapter 2 but in the fermion representation i.e. H

= Y,^\Pi-M*i)\2

(9-2)

where the gauge potential is given by

A,(r,) = ( l - a ) £ ^ . k*j

'

(9.3)

jkl

Here Vjk = r^ — r^ and a = | so that a = 1(0) corresponds to fermions (bosons). The central assumption of the mean field theory is that for 6/TT = (1 — 1/JI), n being an integer, one can replace the anyon gas, to a first approximation, by a gas of fermions moving in the presence of a fictitious uniform external magnetic field b whose magnitude is such that n Landau levels are exactly filled. When is the mean field approximation likely to be correct? Obviously, configuration by configuration, the fictitious magnetic field looks extremely inhomogeneous. Recall that the anyons are point charged particles with a thin flux tube attached at the site of each anyon so that at each anyon site, the (fictitious) magnetic field is concentrated in delta functions. As we have seen in Chapters 3 and 4, it is quite awkward to deal directly with the

Ch 9.

Anyon Superconductivity

243

resulting long range interaction. However, it is sometimes valid to replace the effect of many distant particles by a mean field, with the deviation from the mean represented by a residual weak interaction. In our case, we are thus replacing the many singular flux tubes by a smooth magnetic field with the same flux density. In particular, for 6 as given by (9.1), the fictitious magnetic field is given by 8 = 2Tr~p/n, when ~p is the average particle density. In such a magnetic field, the particles will move along in the cyclotron orbits. Clearly, if the number of particles inside a typical orbit is greater than one, then it is indeed a valid first approximation to replace the field generated by the particles by its average value. One can show that in the limit of large n, this approximation is at least self consistent. It is also clear that this approximation is valid when the density of flux tubes is high and fluctuations are small, i.e. at high density and low temperatures. Coming back to the mean field approximation, it is reasonable to expect that the exact filling of the Landau levels is especially favorable energetically and produces a particularly stable state like noble gas atoms, magic nuclei or insulating solids. In other words, one expects that the states formed at these special fractions are superconducting. This hypothesis can be tested by finding out as to what it costs to have configurations for which n levels are not exactly filled. Now, since in the Chern-Simons theories, the magnitude of the fictitious magnetic field is directly related to the local density, hence in order to have a situation when n Landau levels are not exactly filled, one has to introduce an extra real magnetic field B and check if a Meissner effect exists. Two cases are possible in that case. (1) The real field B is in the same direction as the fictitious field b so that the net magnetic field is more and hence the density of states per Landau level is somewhat greater. As a result, we will not quite completely fill n levels any more. Let the fractional filling of the highest level be (1 — x). Then the particle number conservation gives (9.4)

(b + B)(n-x)=bn.

Now, it is well known that the energy levels of a charged particle moving in a uniform magnetic field b are grouped into Landau levels and the corresponding energy eigenvalues are (see Chapter 10.2 for the details)

£ = Z+

' ( 0S'

Z = 012

9 5

' ' '- ( - )

244

Fractional Statistics and Quantum Theory

while the degeneracy in each Landau band is

Thus the total energy is

.^dt + ^-fx-iyi.

(9.7)

4-7rm [ n \ n) J It is worth noting that the coefficient of the linear term in B is positive so that for small B, the energy relative to the ground state is positive and grows linearly with B. In other words, the anyon gas is a perfect diamagnet and tends to expel any external flux. (2) If, on the other hand, the real field is in the opposite direction to the fictitious one, then the density of states per Landau level will be somewhat smaller and some particles will be pushed to the (n + l)'th level. If x denotes the fractional filling of this level, then from particle conservation we have (9.8)

(b-B)(n + x) = bn. Hence, the total energy in this case is

Anm [

n

\

n)

J

Rather remarkably, even in this case, the energy relative to the ground state is positive, and for small B, grows linearly with B. We thus find that irrespective of whether the real field is along or in the opposite direction to the fictitious one, the coefficient of the linear term in B is always positive and the same ; however, the quadratic terms differ. This result is rather remarkable and it suggests that the anyon gas, at 6 values close to the Fermi case, will strive to exclude the external magnetic field. This, in essence, is the germ of the Meissner effect which is one of the key characteristics of superconductivity. These arguments also suggest the existence of energy gap in the charged particle spectrum. Besides, it is clear that the production of the quasi-particles and quasi-holes is closely

Ch 9.

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245

related to the presence of a real magnetic field. This is because if the quasiparticles and quasi-holes are spatially separated, then the quasi-particle (or the quasi-hole) excitation is analogous to that of applying an external magnetic field B which is aligned parallel (or anti-parallel) to the fictitious magnetic field B. Further, these arguments also suggest a close connection between the charged vortices and the fermion excitations in the case of anyon superconductivity. This can be understood as follows. Since the fictitious field is uniquely tied to the particle density and is appropriate to n Landau levels being exactly filled, to accommodate any additional real magnetic field, we will necessarily have to excite particles across the gap. Similarly, if the particles do not fill the Landau levels exactly, there must be a real magnetic field present to account for the mismatch. In short, in anyon superconductivity, unlike the conventional case, the charged quasiparticles and the charged vortices do not constitute two separate elementary excitations but rather they are one and the same. This calculation raises several questions. For examples, what happens to the anyon gas at the other value of the statistical parameter i.e. when 6 =fi TT(1 — 1/ra)? Is the ground state always homogeneous? These questions have more or less remained unanswered. Further, one of the major criticism against the mean field theory (or, for that matter, many other approximations) is that they do not take into account one of the important property of the multi-anyon system. In particular, we saw in Chapter 3 that due to the three-body forces, the effective N-anyon interaction near the fermionic end (in an oscillator potential) is repulsive (attractive) depending on whether the fermionic shells are closed (not closed). Unless this fact is taken into account, the reliability of most of these approximate calculations will always remain suspect. 9.3

The Random Phase Approximation

There is one feature of superfluidity which is not at all obvious within the mean field approximation. This feature is the existence of a sharp NambuGoldstone mode, or more concretely, an excitation with the dispersion relation us2 oc k2 at low frequency and small wave vector. It was Fetter, Hanna and Laughlin [4] who showed that this feature comes out nicely within the RPA approximation provided one calculates the effect of adding back the residual interactions in this approximation. Let us start from the many anyon Hamiltonian as given by Eqs. (9.2) and (9.3). In the mean field approximation we have assumed that the

246

Fractional Statistics and Quantum Theory

density of anyons is almost uniform so that we can separate the Hamiltonian H into an average (mean field) part Ho and a fluctuating part Hj. On denoting the spinless field by ip, in second quantized notation we have H = H0 + Hi

(9.10)

jyo = y ^ r ^ + ( r ) ^ | p - a ( r ) | 2 ^ ( r )

(9.11)

while Hj is the remaining part. Note that here a(r) is the mean field i.e. a(r) = -bzxr, Zi

b= ^

.

(9.12)

til

Here, while defining the mean field, we have replaced the sum (see Eq. (9.3)) by an integral for some almost constant p of compact support, and then taken the limit as the support of p extends over all space. This defines a(r) uniquely modulo an integration constant. With this definition of Ho, the analysis becomes an expansion in the perturbation Hamiltonian H[. On carefully analyzing H/, one can show that for large n, when the fluctuation in density is relatively small, one of the important terms of Hj is the effective Coulomb interaction in two dimensions i.e. ln|r' — r"|, which is repulsive between the like charged particles and attractive between the oppositely charged particles. This has two important consequences. Firstly, it generates an effective long range repulsion between two particles or two holes which is responsible for the anyon superconductivity being of type / / . This is because of the identification of the charge excitations with the vortices. Secondly, this interaction also generates an effective long range attraction between the particles and the holes which is responsible for the formation of zero-mass bound state. The key observation of Fetter et al. [4] was that Hi contains potentials with long range, similar to those that occur in the problem of the electron gas. Therefore taking the cue from the electron gas problem, these authors summed the repeated bubble diagrams (the RPA) in which the same momentum transfer q appears on each interaction line. Using these results, they calculated the current-current correlation function and hence also the response function. The Meissner effect followed immediately from the static limit of the response function. In particular, one finds that in this limit, the response function has contribution from both the diamagnetic and the paramagnetic parts and that the paramagnetic piece vanishes

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247

for long wave lengths, leaving a full Meissner effect. We might add here that the relativistic models have also been used to obtain the massless mode [6]. Subsequently, people have also considered the effect of the Coulomb repulsion between the anyons by including the term

ly—A—

(9.i3)

where eo is the charge of the anyon and £o is the dielectric constant. Treating this Coulomb interaction as a perturbation, it has been shown within the RPA approach that the Meissner effect is stable against the repulsive Coulomb interaction. Further, the collective mode of the system becomes the plasma mode in the presence of the Coulomb interaction. Of course, the fractional statistics gas will cease to be a superfluid when the inter-particle repulsion becomes too strong and then the system will form a Wigner crystal. Thus, the RPA clearly bring out the existence of a sharp NambuGoldstone mode. However, it is not clear from these calculations why the anyon gas is compressible and hence supports sound waves with w2 oc k2 for small w and k? However, conceptually the reason is the following. The worst case is that of the fermions since they have the largest repulsion and hence exert the greatest pressure. But we know that an ideal Fermi gas is compressible and hence the ideal anyon gas must certainly be compressible. What is the origin of these massless modes? Is it the same as in the usual superconductivity i.e. due to the Nambu-Goldstone mechanism? Recall that in the usual case, some symmetry of the microscopic theory is spontaneously broken in the effective theory. Put differently, in this case, the associated charge operator does not commute with the Hamiltonian, even though the local conservation law is guaranteed on microscopic grounds. As is well known, this mismatch is repaired by the addition of the massless modes to the theory. It is very important to realize that in anyon superconductivity, a similar thing does not seem to happen. This is easily seen as follows. Normally when a symmetry is spontaneously broken, one has to make some arbitrary choice among a set of energetically degenerate possibilities. However, in the RPA calculation discussed above, one had simply started from the mean field Hamiltonian as Ho and treated the remaining part Hj as perturbation and had not made any such arbitrary choice. What then is the origin of the massless modes in the anyon superconductivity? While the answer to this question is not very clear, one possibility

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Fractional Statistics and Quantum Theory

[5,1] is that the translation symmetry which is an exact symmetry at the microscopic level, although not broken, is realized in an unusual way in the mean field theory. In particular, the translation generators do not commute in the mean field theory any more even though they do so at the microscopic level. The point is, even though the mean magnetic field b, and all gauge invariant quantities are invariant under translations, yet the gauge covariant derivatives which generate translations in the two directions do not commute with one another. For example, one can choose a gauge where ax = —by,ay = 0 so that the translation symmetry in the x direction is manifest or one can choose ax = 0, ay = bx so that the translation symmetry in the y direction is manifest, but it is impossible to choose a gauge in which both are manifest at the same time. Put differently, one can choose single particle eigenfunctions, which are plane waves in the x direction but localized in the y direction, or vice versa. However, genuine plane waves in both the directions are definitely not the eigenfunctions in this case. The question then arises as to how symmetry commutators can be altered, without the symmetry being lost? This is possible due to a famous theorem of Wigner concerning the realization of a symmetry in quantum mechanics. The deep point of Wigner's theorem is that in the physical interpretation of quantum mechanics, what occur are not amplitudes themselves, but absolute squares of amplitudes. As a result, when symmetries are realized by unitary transformations in Hilbert space, these transformations are only determined up to a phase. For example, if S\ and S2 are symmetry operations which are realized by the unitary operators [/(Si), [/(S2) respectively, then the unitary operator for the combined operation Si • S2 will in general be U(S1 • S2) = r ? (S 1 ,5 2 )[/(5i)f/(5 2 )

(9.14)

where rj(Si,S2) is a phase factor of unit modulus. Now there are two possibilities. (i) One could redefine U's, by multiplying by phases, such that 77 reduces to the identity. In this case one has a linear representation of the symmetry algebra. (ii) It is possible that TJS cannot be denned away. In that case, one has a (properly) iprojective representation and the infinitesimal form of Eq. (9.14) will give a modification of the commutation relation between the symmetry generators. Thus, in the case (ii) one finds that the integrated charge operator does

Ch 9.

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249

not commute with some operator, even though the local conservation law is guaranteed on the microscopic grounds. This mismatch can be repaired, by the addition of the massless particles to the theory. Here it must be made clear that, we cannot really prove the existence of massless particles on such grounds since there could be other possibilities to eliminate this mismatch.

9.4

The Effective Lagrangian

We thus have seen that within the random phase approximation, the electro-magnetic response of the charged anyon gas (for 6 = TT(1 — 1/n)) at low frequency and wave vector is dominated by a single collective mode, essentially a sound wave and that there are no other low energy excitations. In this case, it is possible to write down an effective Lagrangian which reproduces the electro-magnetic response in full detail. Such a Lagrangian contains the electro-magnetic field and a neutral scalar field representing the sound wave. It turns out that such a Lagrangian (which is also consistent with the Galilean and the gauge invariance) is [5,1] C=\{j>-

CA0)2 - y(di - CAif + aeijdiAjM - CA0)

+ feijidoAi - diAo)(dj - CAj)

(9.15)

where we have restricted ourselves to terms of low order in space and time since one is interested in the behavior at low frequency and wave vector. Here <j> and A^ represent the sound wave and the electro-magnetic field (treated as an external source) respectively. Several comments are in order at this stage. (1) Notice that in the absence of electromagnetism, this Lagrangian would simply describe a neutral scalar field with velocity of propagation being v. (2) The first two terms in the above Lagrangian have minimal coupling. Further, the entire Lagrangian is invariant under the local gauge transformation cj> -> cf> + Cf, A» -> A , , + d^f.

(9.16)

Note that we have an additive transformation law for the neutral scalar field 4>.

250

Fractional Statistics and Quantum Theory

(3) The last two terms are not invariant under the discrete transformations of parity (P) and the time reversal invariance (T) separately even though they are invariant under the combined operation of PT. These two terms are however invariant under rotations and the gauge transformations. What is perhaps most remarkable is the fact that there are two possible Chern-Simons like terms and not one as in the relativistic theory. In other words, if one drops the demand of Lorentz invariance but demands only Galilean invariance, then there are two possible Chern-Simons like terms and both are separately gauge invariant. This may have some interesting consequences which are worth looking into. This model exhibits a variant of the Higgs mechanism due to Stuckelberg. The scalar field , which in the absence of electromagnetism represents a scalar degree of freedom (i.e. a sound wave with the speed of sound v), loses its independent significance when coupled to electromagnetism as above. In the Landau gauge (0 = 0), the first two terms in the Lagrangian reduce to the photon mass terms while the next two are the Chern-Simons like terms. Since these terms are of higher order in gradient, hence, formally, they are sub-dominant. On adding the interaction term J^A^ to the Lagrangian, one can calculate the charge density and electric current by varying the Lagrangian with respect to AQ and Ai. We get p = -C(4> - CAQ) - (a +

ftCeijdiAj

(9.17)

Jfc = Cv2(dk4> - CAk) - pCsjkidoAj - djA0) + aCejkdjA0 + pCsjkdoAj - (a + (3)ejkdjd0.

(9.18)

Using these relations, the last two terms in the Lagrangian (9.15), to the lowest order in a and f3 can be written as

-~cPB-Jc3xE

^

where B = e^diAj, Ei = d0Ai — diAo. Thus we see that the term proportional to a correlates the electric charge density with the magnetic field while the f3 term correlates the electric current with the perpendicular electric field in a manner reminiscent of the Hall effect. Thus the a-term represents an intrinsic magnetic moment for anyons in the superfluid state. Since one has consistently ignored the spin in the analysis, hence this magnetic moment must be generated entirely by the orbital motion. Since there

Ch 9.

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251

is a universal gyromagnetic ratio e/2m for the orbital motion, hence the magnitude of the a term is directly proportional to the angular momentum per particle. On the other hand, the (3 term (representing a coupling between the current and the transverse electric field) causes a currentflowin the perpendicular direction in case an electric field is applied in the plane. However, such a flow should not occur for the anyon superfluid and one should expect (3 = 0 in this case. This is because, all the charge carriers in the anyon superfluid possess the same ratio of electric charge to mass. As a result, a uniform electric field in the plane (even if time dependent) couples directly to the center of mass coordinate of the underlying fluid. Hence, so long as the translation invariance is not broken, we expect that the response of the fluid to the uniform electric field will simply be to translate rigidly as a whole and hence there will be no current transverse to the applied field i.e. (3 = 0.

9.5

Anyons and High-Tc Superconductivity

Some time ago, there was a tremendous upsurge in interest in the fractional statistics in general and the anyonic theories of superconductivity. This was because of the speculation by Laughlin and others that anyons provide a mechanism for the high-Tc superconductors. In particular, the speculation was that the quasi-particles in the copper oxide (CuO) plane, which presumably are the key actors in the high temperature superconductivity, are in fact anyons. Is the speculation correct? How can one test it experimentally [7,8,9]? As we have seen so far, anyons inherently violate the discrete symmetries of parity (P) and time reversal invariance (T). Now it is well known that materials that violate parity (P) and time inversion (T) symmetries, cause optical effects. For example, one effect caused by the parity (P) violation alone, is the rotation of linearly polarized light as it passes through a solution of molecules with a definite handedness such as sugar water. In this case, though, the rotation unwinds, returning to its original polarization if the light reflects back on itself. Another example which breaks both the parity (P) and the time reversal (T) symmetries is the Faraday effect in which the linearly polarized light is rotated when it goes through the matter in the direction of an applied magnetic field. The field destroys the time reversal (T) invariance by establishing a definite direction within

252

Fractional Statistics and Quantum Theory

the sample. In this case, the reflection of light on itself does not cause unwinding. It was suggested [8] that the presence of anyons in the high-Tc materials might cause behavior analogous to the Faraday effect that would re-orient the direction of the linearly polarized light after it is reflected at right angles from the copper oxide planes. The basic reason for the parity (P) and the time reversal (T) symmetry violation, e.g. these optical effects, is easily understood in terms of the parity (P) and the time reversal (T) violating Chern-Simons term. As seen in Chapters 6 and 7, the effect of the Chern-Simons term when combined with the conventional Higgs mechanism, is essentially to give unequal masses to the two circularly polarized states of the photon. Below threshold, they will be exponentially attenuated with distance but at different rates. Great sensitivity and sophistication are required in measuring these magneto-optical effects to ensure that all extraneous factors that might rotate the polarization are eliminated. The non-reciprocal magneto-optical experiments might measure either dichroism or birefringence. One has circular dichroism when the material absorbs right handed circularly polarized light either more or less than the left handed circularly polarized light, i.e. the circular dichroism changes the ellipticity of light. Lyons et al. [10] used this technique to look for time reversal (T) violation in YBa^Cu^O-j-x as well as Bi2Sr2CaCu2Os crystal samples. They obtained a positive result. However, a subsequent careful experiment by Spielman et al. [11] has found no signal of time reversal (T) violation. Spielman et al. also worked on a thin film of YBa^Cu-iO-j-x but they compared the polarizations of two light beams of the same handedness transmitted through the sample in opposite directions. In this case circular birefringence occurs. The advantage is that this arrangement eliminates all effects except those that violate time reversal (T) invariance. The group modified a fiber-optic gyroscope and created an instrument that could measure optical rotations with exceedingly high sensitivity. It is worth noting that whereas this experiment was highly sensitive to the Faraday rotation, it completely rejects any contribution due to optical activity, linear birefringence, linear dichroism or magneto-electric effect. Thus, it eliminates all reciprocal effects. The fiber-optic gyroscopic is delicately sensitive to the rate of rotation. The earth's rotation is in fact used for the calibration. They found no rotation within the two micro-radian sensitivity of their apparatus in the case of YBa2Cus0j-x as well as Bi-zSrzC aCuzOs,There is one problem with the analysis of all these experiments. This has do with the fact that the anyonic theories strictly apply in two space

Ch 9.

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253

dimensions while the high-Tc materials have several layers. So one has to worry about the nature of the coupling between the layers [12,13]. Unfortunately the anyon theories are not sufficiently developed to predict whether the sign of the symmetry breaking is the same in successive layers, in a kind of ferromagnetic ordering, or whether it is opposite, as in an antiferromagnetic ordering. Clearly, the detected signal would be much more in the former case while it could even be null in the latter case. How does one proceed if the sign of the parity (P) and the time reversal (T) symmetry breaking alternates from layer to layer so that there is a net cancellation of the anomalous contribution in the average bulk properties of the material? In such a case, one has to conduct surface sensitive tests. One such test is with muon spin relaxation. This is related with the fact that the anyons possess a magnetic moment which creates a small internal magnetic field. One can probe this field with a charged particle such as a muon. As the anyons rearrange to screen this charge probe, the disturbed density distribution should produce a finite internal magnetic moment irrespective of whether one has ferromagnetic or anti-ferromagnetic ordering of layers. The resulting local magnetic field at the muon site would cause the muon spin to precess and the precession would affect the distribution of its decay products. Such an experiment was indeed undertaken by Kieff et al. [14] where they undertook muon spin relaxation measurements to measure the internal magnetic field in both vitrium and bismuth superconductors. The group found a magnetic field of 2.5 Gauss of which they would attribute no more than 0.8 Gauss to any time reversal effect. On the other hand, crude theoretical estimates predict [7] an effect of about 10 Gauss if the anyons are present. Taking all the available experimental evidence into account, it appears that anyons do not provide the mechanism for superconductivity in the high—Tc compounds. However, this does not rule out anyon superconductivity per se. It is quite possible that there are other, yet to be discovered, compounds in nature which are examples of anyon superconductors. Only time will decide whether this hope is realized in nature or not. 9.6

Anyon Metal

Let us consider the case when there is no gap in the electron spectrum, either from the exact filling of the Landau levels or collectively. In such a case, we will have many low energy excitations. In that case one would

254

Fractional Statistics and Quantum Theory

obtain a sort of metal in case one is not in a superconducting state [1]. Put another way, loosely speaking, the speed of sound could either be infinite, finite or zero. When it is infinite, then we have an incompressible state such as the fractionally quantized Hall states (see the next chapter). On the other hand, when the velocity is finite and when the sound mode does not mix or decay into other low energy modes, we get an anyon superconductor. Finally, when there are an abundance of low energy modes, with a dispersion relation softer than linear, then the sound mode mixes with these and disappears into the continuum. We then have dissipation and something like an ordinary metal. One would expect that the anyon metal would be an anomalous metal in the sense that it would resemble the intermediate or dissipative states of the fractionally quantized Hall effect, i.e. the states lying in between the plateaus but occuring in the absence of the external magnetic field (see next chapter). Generalizing even further, one would expect that every phenomenon that occurs for fermions or bosons, say Kondo lattices, metal insulator transition etc. should have an anyon generalization. What is not clear, though, is how many of these generalizations involve qualitatively new features. Finally, anyon superconductors raise the following interesting question. Can anyon superconductivity and BCS pairing coexist? Or, are there two different kinds of superconductors in nature, one for weak attraction or repulsion, and quite a different kind for strong attraction, with nothing in between? Hopefully, answers to some of these questions will be found in the near future. References [1] For a detailed discussion see Fractional Statistics and Anyon Superconductivity, ed. F. Wilczek (World Scientific, Singapore, 1990). [2] R.B. Laughlin, Phys. Rev. Lett. 60 (1988) 2677 ; Science 242 (1988) 525. [3] V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59 (1987) 2095. [4] A.L. Fetter, C.B. Hanna and R.B. Laughlin, Phys. Rev. B39 (1989) 9679. [5] Y.H. Chen, F. Wilczek, E. Witten and B.I. Halperin, Int. J. Mod. Phys. B3 (1989) 1001. [6] T. Banks and J. Lykken, Nucl. Phys. B336 (1990) 500. [7] B.I. Halperin, J. March-Russell and F. Wilczek, Phys. Rev. B40 (1989) 8726. [8] X. G. Wen and A. Zee, Phys. Rev. Lett. 62 (1989) 2873 [9] D.M. Gaitonde and S. Rao, Mod. Phys. Lett. B4 (1990) 1143. [10] K.B. Lyons et al, Phys. Rev. Lett. 64 (1990) 2949.

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[11] S. Spielman et al., Phys. Rev. Lett. 65, (1990) 123. [12] A.G. Rojo and G.S. Canright, Phys. Rev. Lett. 66 (1991) 949 ; A.G. Rojo and A.J. Leggett, 67 (1991) 3614. [13] D.M. Gaitonde, D.P. Jatkar and S. Rao, Phys. Rev. B46 (1992) 12 026. [14] R.F. Kieff et al., Phys. Rev. Lett. 64 (1990) 2082.

Chapter 10

Quantum Hall Effect and Anyons

0, my Lord, my Lord, behold, I cast myself in faith upon conjecture, not knowing the facts — E.A. Abbott in Flatland 10.1

Introduction

In this chapter, we discuss the role played by anyons in the fractional quantum Hall effect (FQHE) [1]. As mentioned before, this is the only physical system where there exists incontrovertible evidence for the existence of anyons. In particular, it has been shown that the quasi-particle and the quasi-hole excitations over the fractional quantum Hall ground state are anyons. In fact, most of the interest in anyon theories in recent years is because of their relevance to a better understanding of the fractional quantum Hall effect. The integer quantum Hall effect (IQHE) was experimentally observed in 1980 [2] while the fractional quantum Hall effect was observed in 1982 [3]. While the integer quantum Hall effect was readily understood at the level of an independent particle model, it also became obvious that the fractional quantum Hall effect must essentially be a collective, many-body effect. Great insight into its nature followed due to a seminal paper by Laughlin [4], where he proposed a variational wave function to describe the fractional quantum Hall effect [5,6]. Around the same time, Haldane [7] and Laughlin [8] pointed out that the observed fractions could be understood as a hierarchy of quantum Hall effects. However, whereas Haldane conjectured that the quasi-particle excitations over the fractional quantum Hall ground 257

258

Fractional Statistics and Quantum Theory

state were bosons, Laughlin conjectured that they were fermions. It was however Halperin [9] who argued that the only statistics for the quasiparticles and the quasi-holes that makes sense is the anyonic statistics. The essential correctness of Halperin's idea was subsequently confirmed by the work of Arovas, Schrieffer and Wilczek [10]. They computed the Berry phase [11] associated with the adiabatic interchange of two quasi-holes and showed that provided the two quasi-hole state is well approximated by the Laughlin wave function, the Berry phase is exactly that of anyons i.e. of particles obeying fractional statistics. The wave function approach, based on the seminal paper of Laughlin, is a microscopic approach. Though fairly successful, it is not completely satisfactory since it fails to illuminate all the symmetries and does not provide a complete understanding of the problem. To give an analogy, it is as if, soon after the superconductivity was discovered, the ./V-body projected BCS wave function in the coordinate representation was directly written down. Although correct, in that case a complete understanding of the phenomenon of superconductivity would not have been possible without the discovery of the phenomenological Landau-Ginzburg theory and the phenomenon of Cooper pairing. Historically of course, the microscopic BCS theory came after the discovery of the phenomenological Landau-Ginzburg theory and the phenomenon of Cooper pairing. Hence, for the fractional quantum Hall effect too, several attempts have been made to find an analogue of the Cooper pair that condenses and also an analogue of the Landau-Ginzburg theory. This is the so called Landau-Ginzburg-Chern-Simons (LGCS) theory which has been initiated a few years back [12,13,14]. This approach also makes use of the anyon ideas in a fundamental way. It must, however, be made clear here that in the context of the fractional quantum Hall effect, one is not really talking of anyons in vacuum but the contention is that they arise as quasi-particles in a medium. The plan of the chapter is the following. In Sec. 10.2, we discuss the Landau level problem. In Sec. 10.3, we briefly discuss the basics of both the integer and the fractional quantum Hall effect. Sec. 10.4, is, in a sense, the heart of this chapter where we discuss the Laughlin wave function, its plasma analogy and show that the excitations over the fractional quantum Hall ground state are indeed anyons. Finally, in Sec. 10.5, we briefly discuss the Landau-Ginzburg-Chern-Simons theory of the fractional quantum Hall effect.

Ch 10.

10.2

259

Quantum Hall Effect and Anyons

The Landau Levels

The behavior of a charged particle in a plane in an external uniform magnetic field B (at right angles to the plane), is an interesting problem which reveals many important features which are of very general character. Even though we have briefly mentioned it in Chapter 3, we are discussing it in detail here, since we believe that a thorough understanding of this problem is a must for the proper appreciation of the fractional quantum Hall effect. It is well known that in this case, the spectrum consists of equally-spaced levels called Landau levels. The Hamiltonian in this case is given by (IO.I)

H = ^-(P--A) 2m \

c /

where the vector potential Ai is related to the magnetic field B by SijdiAj = B (10.2) whose general solution is

where the scalar function <j> determines the gauge in which we are working. Note however, that most of the results in this section are actually independent of the choice of (f>. The two most frequently used gauges for this problem are (i) Landau gauge in which <j> = X\X2 and (ii) symmetric gauge in which <j> — 0. We shall discuss the solution of this problem in the symmetric gauge. It is very illuminating to discuss the solution of this problem in terms of the complex coordinates z and ~z denned by z = x1 + ix2, z = x1 - ix2, d=—, d=~.

CJ Z

OZ

(10.4)

In terms of these coordinates, the Hamiltonian (10.1), in symmetric gauge, takes the form

H=-*#8d+?!£\z\>-*±Je 777.

O

Z

(10.5)

where UJC = e\B\/mc is the cyclotron frequency while J c is the total (canonical) angular momentum operator defined by Jc = zd-zd.

(10.6)

260

Fractional Statistics and Quantum Theory

It is worth noting that Jc and H commute with each other so that we can simultaneously diagonalize both of them. The Landau level problem can be solved in many different ways. One elegant way is to define the annihilation and creation operators a and a+, i.e. a = -i\

I ft / -=- mu)c \ , / ft / _ mu>c \ , . 2<9 H -z), a+ = -id 25 -z) K (10.7)

V 2mujc V

2ft ) '

V 2mw c V

2K )

'

in terms of which the Hamiltonian as given by (10.5) takes the form H = (a+a+^jhuc.

(10.8)

It is easily checked that [a, a+\ — 1 so that the spectrum of the Hamiltonian is

En=(n+^jhcoc.

(10.9)

The corresponding eigenfunctions are easily obtained, as in the oscillator case, from the ground state eigenfunction |n) = ^ H 0 >

(10.10)

where the ground state |0) obeys the equation a\0) = 0.

(10.11)

On using the operator form of a as given by Eq. (10.7), this reads

whose solution is Mz,z) = f(z)exp(-^]

(10.13)

where f(z) is any analytic function of z. The corresponding properly normalized eigenfunctions are

tiW - J^t + J^^) ^ (" ^)

(1 14)

°-

Ch 10.

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261

where the magnetic length I is given by

I = yg

(10.15)

while n and j are integers such that n > 0 and j > —n. Note that the above tp3n is also an eigenfunction of the angular momentum operator J c with eigenvalue j . Here LJn, are the generalized Laguerre polynomials. From Eq. (10.9), we find that the energy level spectrum is purely discrete, is independent of j , and is similar to that of the one dimensional harmonic oscillator. These discrete levels are known as Landau levels [15]. In free space, each of the Landau levels has an infinite-fold degeneracy given by all states with the same quantum number n and an arbitrary angular momentum j > —n. However, for samples of finite area, this degeneracy is finite, since in this case, the angular momentum is bounded from above. For example, if A is the area of the sample, then the degeneracy is given by

where <j> = AB is the magnetic flux through the area A and 4>0 = hc/e is the flux unit. The degeneracy of the Landau levels is also clear from the expression for the ground state wave function as given by Eq. (10.13) where f(z) is any analytic function of z. It is well known that the degeneracy of the Landau levels is because of the invariance of the Hamiltonian under a magnetic translation group. 10.3

Basics of Quantum Hall Effect

Both the integer and the fractional quantum Hall effect were discovered in the very special context of semiconductor hetro-structures subjected to very large magnetic fields at very low temperatures (T = 2 K or even less). Let us first briefly discuss just what the effect is. The electrons are trapped in a thin layer at the interface between two semiconductors or between a semiconductor and an insulator. Low temperature and a strong magnetic field perpendicular to the plane freezes the motion along the perpendicular to the plane so that the relevant dynamics takes place in the plane. Thus, electrons in this layer can, to a good approximation, be idealized as a twodimensional gas with three-dimensional Coulomb repulsion.

262

Fractional Statistics and Quantum Theory

The quantized Hall effect has to do with the behaviour of such a two dimensional electron gas at very low temperatures and in very strong magnetic fields. It is experimentally found that under these conditions, the transverse Hall conductance behaves in a most unexpected way. In particular, the transverse Hall conductance instead of varying inversely with the magnetic field as is expected classically, in fact stays constant over finite intervals i.e. the Hall coefficient shows plateaus. It is also found that these plateaus are separated by intervals of more normal continuous behaviour. Another remarkable fact is that the value of the Hall conductance an on the plateaus is expectedly constant and reproducible to a very high degree of accuracy. In particular, one finds i/e 2


( 10 -!7)

where v is a rational number with odd denominator. Notice that the Hall conductance is expressed solely in terms of fundamental physical quantities h and e. This is quite remarkable, considering the fact that the measurements are made directly on a macroscopic material with all the complexity and dirt that comes with it! The quantum number v is found either to be an integer or a simple fraction with odd denominator. The former is called the integer quantum Hall effect while the latter is known as the fractional quantum Hall effect (FQHE). One of the most astonishing facts is that the relation (10.17) is experimentally observed with extremely high accuracy (the precision is roughly 10~7 to 10~8 for the integer quantum Hall effect and 10~4 to 10~5 for the fractional quantum Hall effect). To appreciate how unusual the quantum Hall effect is, let us contrast it with the behavior expected from essentially classical considerations. Let us suppose that the electrons move in the (x,y)-plane with velocity vx. Since the magnetic field is directed at right angles to the plane, the electrons feel a Lorentz force in the y-direction Fy ~ -vxB

(10.18)

which is then compensated by an emerging electric field Ey such that eEy

~ -vxB.

(10.19)

Now, the transverse Hall conductance is defined as

OH = axy = §-

Ey

(10.20)

Ch 10.

263

Quantum Hall Effect and Anyons

where jx ~ epvx is the electric current in the x direction. Here, p = N/A is the electron density i.e. N is the total number of electrons and A is the area of the sample. Using Eqs. (10.19) and (10.20), we then have PC

°H = ^P

(10.21)

which is the well known classical result. Thus, classically, we find that at fixed electron density p, the Hall conductivity varies linearly with 1/B. On the other hand, as mentioned above, experimentally at low T and high B (with p held fixed), one finds that an is in fact unexpectedly constant on the plateaus. On comparing the
< 10 - 22)

% =v h

Thus, we can also say that the essence of the quantum Hall effect is that on the plateaus, p/B is frozen at the rational values. This relation can also be written as V=

A~B^Jhc = Wo

(1

°'23)

where = AB is the magnetic flux through the area A and o = hc/e is the flux unit. In fact, v can also be shown to be the same as the filling factor which plays a central role in the quantum Hall effect. A heuristic proof of this is as follows. The filling factor is defined as the number of electrons (N) per number of Landau levels available. As seen in the previous section, in free space, each Landau level is infinite-fold degenerate, but for samples of finite area A, this degeneracy is finite. In particular, the number of available Landau levels in area A are 2^

(10-24>

where I is the magnetic length as given by Eq. (10.15). Hence, the filling factor is given by

which is identical to v as given by Eq. (10.22). In this way we see that the v that appears in Eq. (10.17) is nothing but the filling fraction. Thus the experimental fact that the Hall conductance an is constant to a very

264

Fractional Statistics and Quantum Theory

degree of accuracy on the plateaus can be interpreted to mean that certain filling fractions are especially stable i.e. energetically favorable.

10.4

Trial Wave Function and Anyons

How does one theoretically understand the integer and the fractional quantum Hall effects? Consider first the case of the integer quantum Hall effect. To be specific, let us consider the case when v = 1 so that the first Landau level is completely filled. In this case, the phenomenon becomes essentially a direct manifestation of the Landau level problem for the non-interacting electrons in a constant magnetic field. As seen in the last section, when an integer number of Landau levels are completely filled, there is a gap to single particle excitations given byfou>c= he\B\/mc. Since in this case B is independent of the density p, there is no reason why one should have massless collective excitation. Further, the system is particularly stable when the magnetic field (or equivalently the density) is such that v is an integer and this naturally explains the integer quantum Hall effect at the midpoint of the plateaus. Why does the Hall conductance remain fixed even when B (or p) is changed from the midpoint value? This is readily understood at the level of an independent particle model. Due to impurities, the weak disorder in the system leads to the formation of some localized states while others are extended states. Now current can only be carried by the extended states. Hence, when the Fermi energy is in the region of the localized states, varying the number of electrons only adds or subtracts the localized states which carry no current. As a result, the magnitude of the current is stuck at the full Landau level value. Thus, as the impurity potentials are turned on, the current will remain at its value in the absence of the impurities (i.e. at the full Landau level value). The integer quantum Hall effect is thus an inevitable consequence of the independent particle behavior. What about the fractional quantum Hall effect? Clearly it is hard to understand it due to the independent particle behavior. In fact, the independent particle behaviour used above to understand the integer quantum Hall effect predicts that the system would be highly degenerate and not at all stable in case v is not an integer but is a fraction. Hence, the fractional quantum Hall effect must essentially be a collective, many-body effect. In particular, the electron-electron interaction which was neglected in the understanding of the integer quantum Hall effect, must be playing an

Ch 10.

265

Quantum Hall Effect and Anyons

important role here. Thus, the appropriate Hamiltonian for the fractional quantum Hall effect is expected to be H

=E

(Pi

}

~2l

+ E «('O + \ E

V

^ - r,-)

(10.26)

where V(rj — rj) denotes the inter-electron Coulomb repulsion while v(r^) is a neutralizing background potential. It may be noted here that the specific form of the electron-electron interaction is not important here. This Hamiltonian cannot be solved exactly, being a highly non-trivial many-body problem. However, in a seminal paper, Laughlin [4] proposed a variational wave function to describe the ground state of the fractional quantum Hall state. This trial function approach uses the anyon concept in a fundamental way. Laughlin's Trial Wave Function To motivate how Laughlin arrived at his celebrated trial wave function, consider the case when (a) only sufficiently many electrons are there to just partially occupy the lowest Landau level, (b) the splitting between the electron states having spin aligned or anti-aligned with the magnetic field is so great that only the former need to be considered in the first approximation; in this case the relevant electrons are all identical particles and (c) The cyclotron energy >> the characteristic Coulomb energy i.e. hujc » e2/I. With these assumptions, Laughlin arrived at his famous trial wave function by demanding that (i) the wave function should be antisymmetric under the exchange, (ii) it should comprise of single particle states in the lowest Landau level and (iii) it should be an eigenstate of the total angular momentum operator. Now, complete degeneracy of states in the Landau level means that there is effectively no kinetic energy associated with the motion in the band (i.e. particles have infinite mass). At the same time, they are subjected to Coulomb potential. Clearly, the most favourable many-body wave function is one that treats all the electrons symmetrically and which prevents the electrons from coming close together. Under these conditions, Laughlin proposed the following specific form for the trial wave function to describe the ground state of the fractional quantum Hall system when the filling factor v (= 1/m) is fractional and less than one

MM)

= Nm Y[(Zi - Zjr exp (--L £ \ZkA . i<j

^

k

'

(10.27)

266

Fractional Statistics and Quantum Theory

Here, Zi is the complex coordinate for the i'th electron, Nm is the normalization constant and m = 3,5,7,.... Note that for odd m, this wave function is totally antisymmetric and hence describes ordinary fermions (or better super-fermions if m = 3, 5,...). Some salient features of this wave function are worth pointing out. The prefactor Yl izt — zj)m is purely analytic i.e. all the particles are in the i<j

lowest Landau level. However, ipm is not simply the product of the single particle wave functions but is a complicated superposition of such products. The prefactor is of Jastrow type i.e. it has a zero of order m at coincident points (zi = Zj) showing that electrons tend to repel each other very strongly in a way that is appropriate to minimize the Coulomb interaction. In this sense, these electrons are called super-electrons. In particular, if the particle at Zi goes round the one at Zj by an angle A<j), then the wave function acquires a phase elmA^, it is as if each particle carried m units of flux. How good is this approximate variational trial function? There is good numerical evidence that it is an excellent approximation in the case of the Coulombic form of the electron-electron interaction. Further, the approximate wave functions works even better when the range of interaction is reduced. In fact, if the electron-electron Coulomb repulsion term is replaced by an infinitely short ranged potential then the Laughlin trial function is known to be the exact ground state. These arguments show that the Laughlin trivial wave function captures the essential and the universal features of the fractional quantum Hall effect since it is largely insensitive to the specific form of the repulsive interaction among the electrons. Analogy with classical plasma There is an interesting connection of the Laughlin trial wave function with the two dimensional gas of classical particles. Many of the phenomenological consequences of the trial function are in fact obtained from this plasma analogy. In particular, the quantum probability density IV'ml2 can be interpreted as the probability distribution function of a classical statistical mechanics problem. On writing \1>m(z1,...,zN)\2 = e~*

(10.28)

we obtain

$ = -2m £ In |* - zi\ + 2J2 E I2* I* • i<3

k

(10-29)

Ch 10.

267

Quantum Hall Effect and Anyons

This is the potential energy of a one component plasma. Now it is well known that a two-dimensional one-component plasma of particles with charge q moving in a neutralizing background of density p is characterized by a potential energy

$ = -g 2 £mk-^| + ^ £ k | 2 i<j

(10-30)

k

where the first term is due to the Coulomb repulsion among the particles while the second term is due to the interaction of particles with the neutralizing background. On comparing these two expressions, we can regard <E> as the potential of a plasma of particles with charge q — \/2m in a neutralizing background of density p where npq2 = i .

(10.31)

In this way, we see that the Laughlin trial wave function is a quantum liquid of density

>=db*-

(1032)

-

One knows that the state is a liquid because the temperature of the equivalent plasma is sufficient to melt it. The potential energy <3? is minimized (i.e. I^^J is maximized) by the uniform distribution of charged particles and is electrically neutral everywhere when the average density of electrons p as given by Eqs. (10.22) and (10.15) exactly equals p i.e, when the filling fraction v = 1/m. These arguments provide good justification for identifying the Laughlin wave function as the ground state of the fractional quantum Hall state at filling fraction v = 1/m. Further, these arguments suggest that the Laughlin wave function represents an incompressible quantum liquid and that incompressibility is, in a sense, the essence of the fractional quantum Hall effect. Hence, at these densities, the many electron wave function peaks when the coordinates Zi are uniformly distributed and is expected to be energetically favourable since in that case the Coulomb repulsion term is minimized. What happens when the electron density p deviates from p as given by Eq. (10.32)? Indeed, the remarkable plateaus seen experimentally mean that as p or B is varied (keeping the other one fixed), the filling fraction stays fixed. How can one understand this? What it means is that the effect of adding a particle is not to change the filling fraction by a small amount over a large volume, but rather to leave the filling fraction pinned at its

268

Fractional Statistics and Quantum Theory

favorable value macroscopically, with the deviation from this value carefully localized. Thus, any deviation from this p must lead to the creation of localized quasi-particle or quasi-hole excitations with a gap in the spectrum corresponding to the finite energy cost of these excitations. We shall now show that these excitations have fractional charge and they obey fractional statistics i.e. they are anyons. Before we rigorously prove that these excitations are anyons, let us give some qualitative arguments as to why these are anyons and why these excitations have a gap. To that end, let us first see how such excitations can be obtained. Consider the following idealized situation. In the Laughlin state (10.27), we introduce an innnitesimally thin flux-tube at the point za> and then turn on adiabatically the flux <j> from zero to the final value of one unit (i.e. ±0 = ±hc/e) in such a way that the system remains an (instantaneous) eigenstate of the changing Hamiltonian. At the end of the process, however, the Hamiltonian must return to the original Hamiltonian, since one unit offluxthrough an infinitely thin solenoid can be gauged away. However, the state has not returned to the original state. As the flux is adiabatically added, every single particle state acquires one unit of angular momentum i.e. zme-\zf/rf

_^ zm+le-\zf/4l* _

(1Q

33)

The state with the highest angular momentum moves over to the next Landau level and a new state appears at m = 0. Similarly, if a single unit of flux is removed adiabatically then each state loses one unit of angular momentum so that a state from the next Landau level moves down and the m — 0 state disappears. Thus, the effect of adding (or removing) one quantum of flux and then gauge transforming it is to increase (or decrease) the angular momentum of the single particle states by one unit. But since the original state, i.e. the Laughlin wave function is non-degenerate (at least to date no other states with the same energy have been found), hence the new state, after evolution of the flux, has to describe an excited state of the original Hamiltonian with a higher eigenvalue, thereby proving that the quasi-particle or quasi-hole excitations have a gap. The electric charge of the excitation can also be easily computed. Let us again consider the case when one unit flux has been adiabatically turned on through a very thin solenoid. Now, far away from the solenoid, this state will be indistinguishable from the ground state, except that every level of the single particle state will have moved over to the next level. Hence, if the flux point is surrounded by a large circle, then the charge that enters

Ch 10.

269

Quantum Hall Effect and Anyons

or leaves the circle is just the average charge per state, provided of course the total charge and the total number of states are uniformly distributed. Now the total charge is given by epA while the total number of states d is as given by Eq. (10.16). Hence, using Eqs. (10.16) and (10.24), we find that the charge per unit state is given by he

which is identified as the charge of the excitation. We can thus say that a quasi-hole (or quasi-particle) is formed in the incompressible fluid described by ipm as a bubble of size such that a fraction 1/m of an electron is removed (or added). Once the charge of the excitation is known, the statistics of these quasiparticles can be obtained simply by noticing that these excitations have a flux of 2n/e and a charge of e/m. Hence, they are anyons with fractional spin i

J

I

e_ 2TT_

j=W=

rn_!L_

4TT

=

4TT

— 2ro

y

(10.35) '

and hence the corresponding statistics is characterized by 6 = 2TTJ = TT/TO

(10.36)

which is fractional. The quasi-hole and the quasi-particle excitations at rilling v = 1/rn are approximately described by the wave functions

^+z" = N+ exp ( - - L J2 \zk\2) U(zi ~ za) Y[(Zj - zk)m ^

4t

k

J

(10-37)

j
i

and

w=N- -p ( - ^ E i^i2) n ( ^ - J ) n ^ - ^m (io-3g) ^

k

'

i

^

*

'

j
where za denotes the center of the excitation and N± is a normalization factor. This ansatz is expected to be good except when it is very near the solenoid. The quasi-hole anzatz corresponds closely to our heuristic description of a quasi-hole as the result of adiabatic insertion of unit flux which increases the orbital angular momentum of each electron around za by one which is essentially what the factor (zi — za) does. Note that this way, one is able to maintain the analyticity of the wave function and hence stay in the ground state. Thus these linear factors are the cheapest way to

270

Fractional Statistics and Quantum Theory

push up the angular momentum. The fact that the quasi-holes have charge e* = e/m can also be proved by applying the plasma analogy directly to Eq. (10.37). The wave function (Eq. (10.38)) describing a quasi-particle excitation above the ground state is more complicated and in some sense less obvious than that for the quasi-hole. However it can be motivated by noting that the creation operator for a particle (electron) is as given by Eq. (10.7). Note that here the derivative operators do not act on the exponential factor of tpm. As in the quasi-hole case, one can prove that these quasi-particles have charge —e/m. Computation of Charge We shall now give a more direct and rigorous way of calculating the charge of the excitations of the fractional quantized Hall effect [10]. The advantage of this method is that it is very elegant and it also enables us to compute the statistics that is obeyed by these excitations. Let us first briefly recall the concept of Berry phase and see how this phase is computed. This simple but profound phase is a geometric phase and arises due to a refinement of the adiabatic theorem in quantum mechanics [16]. In fact it is usually unobservable but in some cases, when the wave functions are not single valued, it has interesting and observable consequences. Anyons, whose wave functions are multi-valued, are therefore the ideal place to compute the Berry phase. The adiabatic theorem states that if we slowly vary the parameters in a Hamiltonian, then a state occupying an isolated energy level remains fully in that level and there are no quantum jumps. Berry made a profound contribution concerning the change in the phase of the wave function if the Hamiltonian, after a sequence of adiabatic changes, returns to its starting value [11]. He showed that in the adiabatic approximation, apart from the usual dynamic phase / E(t')dt', there is an additional phase which remains finite in the adiabatic limit. This phase is given by

i=-i J * dt(m\ft\m) •

(10.39)

It is worth noting that here the integral can be taken over time t, or alternatively over any coordinate used to parameterize the path of the function in the function space. Since dt appears both in the numerator and the denominator in Eq. (10.39), the integral depends only on the geometry of this path and not on the coordinate used to describe it or on the way in which

Ch 10.

Quantum Hall Effect and Anyons

271

the evolution actually occured in time. Let us now compute [17] the Berry phase 7 for the case of quasi-holes by using the wave function (10.37). We compute 7 under the assumption that the quasi-hole is slowly transported around a closed loop V so that za becomes a time-dependent parameter and the adiabatic approximation can be used. Since the entire time dependence of the problem comes solely from za, hence using Eq. (10.37) we have

^ t f ' W = E ^ ( l n ^ - z«m^Za(t) i

(10.40)

and hence 7

= -i T dt(^{t)\ Jto

£ -£(lnfc - za(t)])\^+"-(t)). i

al

(10.41)

Here the integral over t is from initial time to when the quasi-hole begins its tour, to the final time t\ when it returns to its original position after traversing a non-trivial loop F in the anti-clockwise direction. This expression for 7 can be written in a more transparent way by noting that the electron density p in the state V'm2'* *s

p(z) = (^(^^(z-^lVv^W)i

(10.42)

Using Eqs. (10.41) and (10.42) we have

7 = - i f dt [d2z^-(Hz - za(t)])p(z) dt Jto

J

tl

=

d

- [ >z[ dt( -?fl)-J--p(z) J l dJt0 V dt J za(t) - z ^ >

= -i J

d2z I dza Jr

p(z).

(10.43)

za — z

If we now assume that p(z) is a regular function (a reasonable assumption), then using the residue theorem, it follows that all points z outside T give a vanishing contribution to the contour integral. In this way, we find that the Berry phase for the quasi-hole wave function (10.37) is given by f

f

1

f

7 = -i I d2z (b dza p(z) = 2?r / d2zp(z) = 2TTNT . (10.44) J
272

Fractional Statistics and Quantum Theory

Now, we have seen before that if a particle of charge q is moved along a closed loop F encircling a flux (j>r, then its wave function acquires a phase

exp[-.*]

(10.45)

due to the Aharonov-Bohm effect. On identifying the Berry phase exp (—ry) with this Aharonov-Bohm phase, we have = ^ . (10.46) he Using Eq. (10.23) to express the flux r in terms of the filling factor v (= 1/m), we finally obtain 2-KNT

^

hcNr e ^ = m

<10-47>

thereby showing that the quasi-holes really have charge e/m. Similarly, one can prove that the quasi-particles have charge —e/m. We thus have shown using several different arguments that the charge of the excitations above the ground state in the fractional quantum Hall effect is ± e / m when the filling fraction v = 1/m. C o m p u t a t i o n of Statistics The issue of what statistics these excitations obey is quite subtle. Historically, all possible statistics (i.e. bosonic, fermionic and anyonic) were proposed. Even though all these descriptions are mathematically equivalent, it turns out that only the anyonic description is physically correct. Using Berry phase arguments one can show that both quasi-hole and quasiparticle excitations in the fractional quantum Hall state with filling fraction v = 1/m obey fractional statistics with anyon parameter a = 6/TT = 1/m. Again, for simplicity we only consider the quasi-hole case. Let us consider a state having two quasi-hole excitations, one located at za and the other at zp. In the special case, when the two excitations are located far apart so that one can neglect the interaction between them, the wave function of the system can be written as ^Za'+Zp

= Na0 Y[(zi - za)(Zi - z0)Tpm

(10.48)

i

where tpm is the Laughlin trial wave function as given by Eq. (10.27) and Nap the normalization constant. Now let us move the quasi-hole at za adiabatically on a full loop F, keeping the quasi-hole at zp fixed. During

Ch 10.

Quantum Hall Effect and Anyons

273

this process, the wave function i)^c"+z'3 acquires a Berry phase 7 which can be easily calculated. On exactly following the steps as given in Eqs. (10.39) to (10.44), we obtain

7 = -ip

dt<^-+z"(*)|||^z-+Zfl(t))

= 2TT f d2zp{z) (10.49) J
2-KNV

.

(10.50)

This is of course the trivial case when the two quasi-holes are not exchanged so that the loop T is homotopically trivial. On the other hand, if the loop F traversed by za contains zp and is homotopically non-trivial then we will get 7 = 2TT /

d2zp(z) = 2ir(Nr - — ) .

(10.51)

The point is, if a quasi-hole is present inside F, then the number of electrons have to be diminished by the fraction I/TO that is needed to build the quasihole. On comparing Eqs. (10.50) and (10.51), we then conclude that when a quasi-hole at za encircles another quasi-hole at zp, the wave function picks up an extra phase given by exp(-zA7) = exp(27ri/m)

(10.52)

which can be interpreted as a statistical effect. Now, as one quasi-hole encircles the other quasi-hole, the two quasi-holes are exchanged twice and hence we conclude that the statistics obeyed by the quasi-particles is a=e-

=^

= l/m.

(10.53)

In other words, whereas for m = 1 (i.e. v = 1 so that the lowest Landau level is completely filled) the quasi-holes are fermions, when m = 3,5,... then the quasi-holes are anyons satisfying fractional statistics with a = 1/m i.e. the statistics is numerically equal to the filling fraction of tpm. Thus we see that the statistics satisfied by the excitations is directly related to the fraction of electron which forms a quasi-hole or a quasi-particle. This

274

Fractional Statistics and Quantum Theory

fraction is a well-defined number as a consequence of the incompressibility of the ground state tjjm of the fractional quantum Hall system. Note that this derivation does not depend on the details of the loop F but only on its homotopy class so that the extra phase as given by Eq. (10.52) can be interpreted as a statistical effect.

Hierarchy Scheme We thus see that the Laughlin wave function ipm nicely explains the plateaus at the filling fraction v = 1/m where m is an odd integer. However, experimentally, plateaus are also seen at many other rational fractions of the form v — p/q with q being an odd integer. This was explained by Halperin [9] by deriving the so called hierarchy scheme. This scheme is easily understood in the anyon language. Halperin's point was that the quasi-particle or the quasi-hole excitations over the Laughlin state themselves behave like particles in a magnetic field and could form new correlated many body states which would represent the fractional quantum Hall state at other fractions. Let us recall that the wave function for the quasi-hole excitation remains analytic (see Eq. (10.37)), and further, the quasi-holes have charge e/m and obey anyonic statistics as given by Eq. (10.53). Hence the simplest possible wave function for a collection of many quasi-holes would be given by N

/

i<j

^

1

* + , (*oi, *02,.-, Z0N) = Y[(zOi -zOj)m> exp 1-—Y,

k

M

2

\

) • (10-54) '

The factor of ml2 instead of I2 in the exponential is easily understood since I2 oc 1/e (see Eq. (10.15)) and the quasi-holes have charge e/m. Here mi =

h 2pi (10.55) m where pi is a positive integer. This is easily motivated since, under the exchange of two quasi-hole coordinates zOi and zOj, the factor (zoi — zOj)mi produces the required anyonic phase factor i.e. exp(iTrmi) = exp ( i?r (

h 2pi I I = exp(i7r/m).

(10.56)

Let us now recall that in the Laughlin wave function (10.27), whereas m = 1 are fermions, m = 3,5,... are sometimes called super-fermions. Using the same analogy, in the quasi-hole wave function (10.54), we can call the p\ = 0

Ch 10.

Quantum Hall Effect and Anyons

275

quasi-holes of V>mi as anyons while pi ^ 0 quasi-holes can be called as super-anyons. Thus the wave function (10.54), for a collection of quasi-holes, is the straightforward generalization of the Laughlin wave function ipm. Now, recall that ipm describes an incompressible electron fluid of charge — e, fermionic statistics (i.e. a = 6/n = 1) and filling factor v = I/TO. It is then worth enquiring about the properties of tp^. By analogy, ^+ ± also describes an incompressible system of quasi-holes with charge +e/m and anyonic statistics (with a = O/TT = 1/m). Let us now compute the new electron filling factor of the system by again applying the plasma analogy to V'mi- I n particular, we interpret IV'mJ2 a s the probability distribution function of a classical statistical mechanics problem with the corresponding potential being (see Eqs. (10.28) and (10.29))

- zoj\ + ^—pYL\z°k\2 •

$ m i = -2mi^\n\zoi i<j

(10.57)

k

On exactly following the argument as given after Eq. (10.30), we find that the density of the neutralizing background is (10-58)

P=1TT2-^— •

As in the electron case, the variational arguments again suggest that we identify this p with the density of the quasi-holes. Hence the number of quasi-holes in area 2nl2 are Ni = 2nl2p = —!— . (10.59) mmx But since each quasi-hole has a charge e/m, hence the total charge carried by the quasi-holes is qi = — —!— -

(10.60)

TO TOTOi

Clearly this charge has to be removed from the underlying electron system; hence the electron filling factor changes from v = 1/m to V1 = I/ _«. = e

I__^_ TO

= TO^TOI

_J_. .

(10.61) J-

TO+-—

2Pl Thus we see that due to the formation of an incompressible system of quasihole excitations above the ground state, the filling fraction has changed

276

Fractional Statistics and Quantum Theory

from v = 1/m to V\ as given by Eq. (10.61). For example, when m = 3 and Pi = 1 we have the new filling fraction ui = 2/7 which is indeed one of the experimentally observed fractions in the fractional quantum Hall systems. It is now straightforward to generalize this construction and obtain the hierarchy scheme by considering excitations over excitations over excitations in a recursive way. For example, one can construct second generation quasi-hole excitations from the quasi-hole wave function ip^. These excitations would be characterized by a charge e** = — e*/mi = —e/mmi and by a statistics v\ — \jm\. Thus the wave function for such a system of particles would be

*£ = W** - **T> exp (-~-p V

i<j

£ Wa)

k

'

(10-62)

where m2 = — + 2p2

(10.63)

with p 2 being a positive integer. By following the same argument as given above, it is easy to see that the new filling factor is fqi \e


1 m

1 m%i

1 mzmfm2

L_

= m H

.

(10.64)

T-~

Note that there is a minus sign in front of qil& since we have added the "holes of holes". Clearly, this construction can be iterated further and in this way one generates the whole hierarchy whose filling fraction is represented by a continued fraction. Further, if the quasi-particle excitations are also taken into account, then the generic filling factor is given by the continued fraction v=

L_ 2P2 +

.

2^T7

(10.65)

Ch 10.

Quantum Hall Effect and Anyons

277

Here a* = +1(—1) if the i'th generation consists of quasi-holes (quasiparticles) with respect to their parent fluid. In this way, one is able to reproduce all the fractions which have been experimentally observed. Unfortunately, however, this hierarchy construction also predicts fractions which have not been experimentally observed. In fact, what is even more mysterious is the fact that whereas some of the sixth stage hierarchical states have been clearly seen (such as 6/13 for which m = 3, a.\ = 0:2 = 0:3 = CX4 — CK5 — —l,pi = pi = Pi — Pi — P5 — +!)> some of the second or third stage hierarchical states have either not been observed or have been barely observed. Thus it is fair to say that whereas the explanation of Laughlin about the fractional quantum Hall effect at the filling fraction v = 1/m (m odd) is quite sound, the same cannot be said of the hierarchical picture which tries to explain the fractional quantum Hall effect at other filling fractions. In recent years, Jain [18] has proposed an alternative explanation of the fractional quantum Hall effect atfillingfractions v other than 1/m (m odd). His key observation is that phenomenologically, the fractional quantum Hall effect is very similar to the integer quantum Hall effect, and hence theories of both phenomena should also be related. In particular, according to him, the fractional quantum Hall effect for ordinary electrons occurs because of the integer quantum Hall effect for composite electrons i.e. the same kind of correlations between the electrons is responsible for both the integer and the fractional quantum Hall effect. In particular, the central idea of Jain is that the composite fermions, each made of an electron and 2p flux quanta (where p is any integer), can also favour the statistical correlation of n filled Landau levels, just like ordinary electrons. Now, within the mean field approach, the flux of the flux tubes can be spread out and we have ordinary fermions moving in an effective magnetic field. Thus, following the discussion above, one has the fractional quantum Hall effect with filling factor i/"1 = 2pn ± 1. This picture of Jain depends only on two integer parameters (n and p) and Jain is able to account for almost all of the observed fractional quantum Hall states and also explain why some states are experimentally seen while some are not. It must however be admitted that Jain's approach also has a few unsatisfactory features. For example, it is quite puzzling why he needs to involve the basis states of the higher Landau levels in order to describe states with filling fraction v < 1 which is entirely in the subspace of the lowest Landau level.

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Fractional Statistics and Quantum Theory

10.5 Mean Field Theory In the last section we have discussed the Laughlin wave function approach, which is a microscopic approach, to explain the fractional Hall effect and have seen that it is fairly successful in explaining the phenomenon. However, this approach is not completely satisfactory, because it fails to illuminate all the symmetries and does not provide a complete understanding of the problem. To give an analogy, it is as if soon after the normal superconductivity was discovered, the iV-body projected BCS wave function in the coordinate representation was directly written down! Although correct, a complete understanding of the phenomenon of superconductivity would not have been possible without the discovery of the Landau-Ginzburg mean field theory and the phenomenon of Cooper pairing. Let us recall that due to the attractive interaction, electrons with opposite spin, pair together to form composite particles of charge 2e which are called Cooper pairs. These Cooper pairs subsequently acquire phase coherence and condense into a superconducting state. In the mean field theory, this superconducting fluid is described by a complex scalar order parameter (which describes the Cooper pair) and obeys Ginzburg-Landau equations. One of the central issues in the fractional quantum Hall effect is thus to enquire if there exists an order parameter associated with some type of symmetry breaking. It is to be hoped that identifying an order parameter for the fractional quantum Hall effect, would significantly deepen our understanding. Further, one can then construct a Landau-Ginzburg type field theoretic description which would hopefully, not only capture the basic essence of the phenomenon in all its simplicity and beauty, but would also lead to new experimental predictions which are inaccessible to the Laughlin wave function approach. To that end, let us start from the microscopic Laughlin wave function as given by Eq. (10.27) and see whether there is a long range order in the system. Let us first compute the one-body density matrix as given by p(z, z') = j [

d2z2...d2zN

il>*m{z, Z2,..., zN) il>m{z', z2,..., zN)

(10.66)

where Z is the norm of the Laughlin wave function ipm. Using ipm as given by Eq. (10.27), it can be shown that

p(z, z') = ± exp ( - \\z - z f ) exp[(Z*' - l'z)}

(10.67)

Ch 10.

279

Quantum Hall Effect and Anyons

where we have set the magnetic length to unity. We clearly see from here that the one-body density matrix is short ranged with a characteristic scale given by the magnetic length. In particular, notice the Gaussian decay of the density matrix which is faster than the usual exponential decay in a localized electron phase in the absence of the magnetic field. Thus, (as is also expected from Yang's theorem [19]), long range order is not present in the one-body correlation function. How about n-body correlation functions? Recall, for example, that in a BCS superconductor, while the one-body density matrix is short ranged, the pair correlation function exhibits long range order. Can the same thing happen in the Laughlin state? Unfortunately, the n-body correlation functions cannot in general be computed exactly. Nevertheless numerical computations as well physical arguments suggest that all n-body (n finite) correlation functions have a Gaussian decay and none of them show any long range order. For the special case of m = 1, which corresponds to the case of the integer quantum Hall effect, one can in fact compute all the (diagonal) correlation functions exactly for the Laughlin wave functions. In particular, for m = 1, \ip\2 has precisely the same form as the joint probability density function of the eigenvalues of matrices from an ensemble of complex matrices [20,21]. The normalization constant is easily determined and the expression for |-0|2 (normalized to unity) is given by

exp(-f>| 2 > )ni*-*;l 2

\H{zi})\2 = UN f[p\] \

P=l /

V i=l

(10.68)

/ i<3

where the magnetic length I has been set to unity. One can now compute the n-point correlation functions (for any finite n) which are defined as AT]

i^,...,^)^——T^

n

>- J

r

r

JL

... M{Zi})\2 I ] l^l2 J

(10-69)

j=n+l

One can perform these integrals and show that

*.(*,...,*.) = ^exp ( - £ N 2 W £ ^ j V *=1

/

Lp=0

P- J{»,j=l,2

As N —> oo, these correlation functions tend to well-defined limits

Rn(z1,...,zn) ~ ^ e x p f - ] T M 2 ) det[eZiZ5}{iJ=1,2_tn}.

n}

(10.70)

(10.71)

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Fractional Statistics and Quantum Theory

For example, the one-point correlation function, defined as R1(z)=N

[

N

Jc -^

\iP({zi})\2dz2dz3...dzN

(10.72)

and interpreted as the density of the TV-particle system, is given by i?1(C) = ^ e x p ( - C 2 ) ^ ^

(10.73)

where we have put \z\ = £. Notice that the density is isotropic and does not depend on the angular coordinate 6 = Arg(z).

Further, this density

has been normalized as /

R(()(d(d0 = N

(10.74)

Jo the total number of particles. On the other hand, the pair correlation function for the system is given by ^2(C) = - [ l - e x p ( - C 2 ) ]

(10.75)

where C = r i ~ r 2 • One can similarly write down explicit expressions for the higher correlation functions i? 3 ,i?4,.... One finds that all the correlation functions are isotropic i.e. they do not depend on the angular coordinate. Further, they do not show any long range order and in fact all of them have a Gaussian fall off rather than the exponential one. We thus see that corresponding to the Laughlin wave function, there is no diagonal long range order. However, following Girvin and MacDonald [12], we now show that the Laughlin wave function has a novel type of offdiagonal algebraic long-range order (ODLRO). To this end, we first apply a singular gauge transformation on the Laughlin wave function (10.27) which has the net effect of merely changing the phase of the wave function so as to obtain a bosonic wave function from the fermionic wave function. Consider the transformation t/jnew = exp[-im ^2 91n(zj - Zj)\ipold .

(10.76)

i<j

Applying this transformation to the Laughlin wave function (10.27) yields

ipnevttZi}) = exp C - \J2 N 2 ) I I I* - ^ ^

k

'

i<j

( 10 - 77 )

Ch 10.

Quantum Hall Effect and Anyons

281

which is purely real and symmetric under the particle exchange for both even and odd m. Thus, under this singular gauge transformation, both fermion and boson systems map into bosons. Let us now calculate the singular gauge density matrix pnew(z,z') as denned by Eq. (10.66) but with ipm being replaced by tpnew One finds that pnew decays algebraically with distance i.e. as \z — z'\ —> oo Pnew{z,z')

oc \z-z'\-m'2

(10.78)

thereby proving the existence of the off-diagonal long range order characterized by an exponent — y . Notice that pnew differs not just in the phase but also in the magnitude from p. What is basically happening is that the additional phases introduced by the singular gauge transformation cancel the phases in ipm nearly everywhere and produce the off-diagonal long range order in p if and only if the zeros of ip (which must necessarily be present because of the magnetic field) are bound to the particles. Thus the off-diagonal long range order in pnew always signals a condensation of the zeros onto the particles. This off-diagonal long range order properly captures the fundamental correlation of the Laughlin wave function and hence should be viewed as an order parameter of the fractional quantum Hall effect. As a side remark, we wish to point out that the ground state of the two-dimensional Calogero-Sutherland model exhibits similar features [22], i.e. it too does not show any long range order but exhibits an off-diagonal long range order. Now that we have obtained an order parameter for the fractional quantum Hall system, it is natural to try to construct the corresponding LandauGinzburg type mean-field theory . In particular, starting from the microscopic Hamiltonian of the interacting electrons in an external magnetic field, we shall construct an interacting boson problem with an additional ChernSimons term. The key point of this construction is that an electron can be viewed as a composite of a charged boson and a flux tube with an odd number of fundamental flux units <j>o(= hc/e) attached to it. As seen before, this can be accomplished by introducing a statistical gauge field a(x) which is determined by the particle density p(x). The dynamics of the statistical gauge field is given by the Chern-Simons rather than the Maxwell term and hence such a theory is known as the Chern-Simons-Landau-Ginzburg theory. We start from the microscopic Hamiltonian of the two dimensional gas of polarized electrons as given by Eq. (10.26). Here A is the vector potential

282

Fractional Statistics and Quantum Theory

of the external uniform magnetic field. In the symmetric gauge, Aa can be expressed by Aa = ^eapxp. On the other hand, Ao is the scalar potential of the external electric field i.e. Ea — —daAo. Now, since the electrons are assumed to be completely polarized, their space wave function ip(ri,..., rjv) must be totally antisymmetric under the exchange of coordinates. The Schrodinger equation (10.79)

HiP(r1,...,rN)=E*l>(r1,...,rN)

with H given by Eq. (10.26) along with the total antisymmetry requirement defines the quantum eigenvalue problem. Now we shall perform a singular gauge transformation to map this problem into a bosonic one. In particular, we shall show that an equivalent bosonic problem is characterized by the Hamiltonian

H

' = ^ E [p< - \A^ - -c
The Hamiltonian H' defines the Schrodinger equation H'4>(TU ..., vN) = E'cPin,..., rN)

(10.81)

where <> / is now totally symmetric under the exchange of coordinates. Every term in H' has the same meaning as in the H given by Eq. (10.26) except that the new vector potential a in H' is the statistical gauge field and is given by a r

( ') - 2 ^ - 1 .

|

(1

,2 - ^^^V6ij

°- 8 2 )

where 9ij is the angle made by (r» — Tj) with an arbitrary axis, 6 is a parameter unspecified at the moment and 0 = ^f is the unit of flux quantum. Let us now show that the two systems as denned by (if, \I>) and (H1, $) are really equivalent in case the parameter 6 — (2p + 1)TT with p being any integer. To this end, let us define a unitary transformation

^(ri,...,r Ar ) = i7^(ri,...,r J v); t/ = exp - i ^ J ] % )

.

(10.83)

Ch 10.

283

Quantum Hall Effect and Anyons

It is easily checked that U \Pi - ^A(rO - ^ a ( r i ) | C/"1 = p< - ^Afo)

(10.84)

and hence UH'U~l = H where H is as given by Eq. (10.26). Thus if (J)(TI,...,TN) obeys the Schrodinger Eq. (10.81), then (ri, ,r/v) will obey the Schrodinger equation H(f> = E'(f>. Besides, since (ri,..., TN) must be totally antisymmetric if 9 — (2p + 1)TT. This is easily seen from Eq. (10.82) and the fact that 6ij = 6ji + 7T (from the definition of the angle). Thus <j> obeys the same Schrodinger equation as given in Eq. (10.79) and is also a totally antisymmetric wave function. Hence its eigenspectrum E' must coincide with that of the original electron problem, i.e. E. In this way the equivalence of the two problems as given by Eqs. (10.79) and (10.81) is established in case 9 = (2p + 1)TT. Now that one has established the exact equivalence between these two problems, one can exclusively work with the boson representation. Using Eq. (10.82) we find that the magnetic field coming from the statistical gauge field is given by 9 N b(n) = V x a = 0 o - V IT

*—'

9 5(n - Tj) = 0o-p IT

(10.85)

where p is the density of particles. But as seen in the previous chapters, this is precisely the form of the equation of motion that one derives from a Chern-Simons Lagrangian density. We are now in a position to write down the Chern-Simons-Ginzburg-Landau effective scalar field theory of the fractional quantum Hall effect by introducing the second quantized bosonic field operators which satisfy the equal time commutation relation [0(r),^+(r/)]=5(r-r/).

(10.86)

In terms of cp, the effective action can be shown to be

S = Sa + S4>= I cPxCa + I (PxCtj,

(10.87)

where c

« = ^r£^Pa^ap

(10-88)

284

Fractional Statistics and Quantum Theory

C* = 4>+ ([ihdt - e(A0 + ao)})