Desbois.Les.Houches. Impurities and quantum Hall effect

SEMINAR 5

RANDOM MAGNETIC IMPURITIES AND QUANTUM HALL EFFECT

J. DESBOIS Laboratoire de Physique Th´eorique et Mod`eles Statistiques, 91406 Orsay Cedex, France

Contents 1 Average density of states (D.O.S.)

897

2 Hall conductivity

901

3 Magnetization and persistent currents

904

RANDOM MAGNETIC IMPURITIES AND QUANTUM HALL EFFECT

J. Desbois

Abstract In this talk, we present works done in collaboration with Ouvry et al. [1–3]. Random magnetic fields have already been studied in several papers [4]. Here, we will consider a model where the disorder is contained in the definition of the magnetic field itself. By magnetic impurities, we will mean infinitely thin vortices carrying a flux φ (α ≡ eφ/2π), perpendicular to a plane. Those vortices are randomly dropped according to a Poisson’s law with average density ρ. In a first part, we will consider the average density of states (D.O.S.) of a charged particle coupled to the impurities. In particular, we will show that this system exhibits broadened Landau Levels for small α values. This fact has motivated us to study the Hall Conductivity (part II) and, finally, persistent currents and magnetization.

1

Average density of states (D.O.S.) [1]

We consider Hamiltonians for an electron minimally coupled to a vector ~ r ) with the additional coupling of the electron spin up or down potential A(~ σz = ±1 to the local magnetic field B(~r) (we set the electron mass me = ¯h = 1) 2 1 ~ r ) − eB(~r) σz . p~ − eA(~ (1) H= 2 2 It rewrites 1 (2) σz = +1 Hu = Π− Π+ 2 1 (3) σz = −1 Hd = Π+ Π− 2 where Π± = (px − eAx ) ± i(py − eAy ) ≡ vx ± ivy are the covariant momentum operators. In the homogeneous field case, the spin coupling is a trivial constant shift, but, in general, it has important effects. In the one vortex or magnetic impurity cases, it is a sum of δ(~r −~ri ) functions, which is c EDP Sciences, Springer-Verlag 1999

898

Topological Aspects of Low Dimensional Systems

needed to define in a non ambiguous way [5, 6] the short distance behavior of the wavefunctions at the location of the impurities ~ri . It can be attractive or repulsive and in the sequel we will only be concerned with the repulsive case (σz = −1, H = Hd ). ~ r ) = 2παδ(~r ), ~k ~ r ) = α~k × ~r/r2 , eB(~ For one vortex located in O (eA(~ is the unit vector perpendicular to the plane), it is easy to realize that the partition function reads: Zα (t) = Z0 (t)heiα2πn i{C}

(4)

where 2πn is the angle wound around O by the closed brownian curve C of length t. h· · ·i{C} stands for averaging over the set of all such curves and Z0 (t) is the free partition function. Zα (t) is unchanged when α → α + 1 and α → −α; so, we can restrict α to the interval [0, 1/2] when there is no additionnal magnetic field. The D.O.S. exhibits a depletion at the bottom of the spectrum: ρα (E) − ρ0 (E) =

α(α − 1) δ(E), 2

ρ0 (E) =

V 2π

(5)

(V is the (infinite) area of the system). ~ r) = Turning now to magnetic impurities located in ~ri , i = 1, 2, ..., N , (eA(~ PN PN ~ 2 ~ α i=1 k × (~r −~ri )/|~r −~ri | ), eB(~r) = 2πα i=1 δ(~r −~ri ), we get for a given configuration of the N vortices:  PN
 i 2πnj α j=1 (6) Zα (t) = Z0 (t) e {C}

2πnj is the angle wound around vortex j by C. Averaging over disorder, we are left with: P 2iπαn −1) i{C} Zα (t) = Z0 (t)heρ n Sn (e

(7)

Sn is the arithmetic area of the n-winding sector (Sn ≥ 0; −∞ < n < +∞). Remarking that the random variables Sn scale like t, we rewrite Zα (t) as: Zα (t) = Z0 (t)he−ρt(S−iA) i{C} S=

2X Sn sin2 (παn), t n

A=

hSi = πα(1 − α)

1X Sn sin(2παn), t n

hAi = 0.

(8) (9) (10)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

899

In this formalism, the rescaled algebraic area enclosed by C should be written: 1X nSn . (11) A= t n From (8), it is easy to deduce that the average D.O.S. ρ(E) is a function of E/ρ and α.

i) When α → 0, ρ → ∞ with 2πρα (≡ ehBi) fixed, we get after a careful analysis [1]: (12) Zα (t) →α→0 e−tehBi/2 ZhBi (t) ZhBi (t) is the Landau partition function forPthe average magnetic field hBi ∞ (hωc i ≡ ehBi/2). (12) shows that ρ(E) = n=1 δ(E − 2nhωc i) i.e. we get the Landau spectrum shifted by hωc i.

ii) α = 1/2. (10) shows that A ≡ 0 and (8) leads to: Z

E/ρ

ρ(E) = ρ0 (E)

P (S)dS

(13)

0

where P (S) is the probability distribution of S. ρ(E) is a monotonically growing function of E with a depletion of states at the bottom of the spectrum.

iii) 0 < α < 1/2. Using an argument based on the specific heat c (≡ kt2 d2 ln Z/dt2 , k is the Boltzmann constant), we can show that the D.O.S. surely oscillates when α is smaller than some value α0 . The argument runs as follows. With the expression (8), we get: c − c0 ∼t→0 kt2 (hS 2 i{C} − hSi2{C} − hA2 i{C} )

(14)

(c0 is the free specific heat). Numerically, one obtains that (c − c0 ) < 0 for ≈ .28. On another hand, Zα (t) is the Laplace Transform of 0 < α < αnum 0 ρ(E). Integrating by parts, we show that: Z c − c0 ∼t→0 kt 2π 2

2 0

∞

Z 0

∞

dEdE 0

dρ(E)/V dρ(E 0 )/V (E − E 0 )2 . dE dE 0

(15)

Thus, we deduce that ρ(E)/V is a non-monotonic function of E for 0 < α < . αnum 0

900

Topological Aspects of Low Dimensional Systems

Fig. 1. Average density of states of the random magnetic impurity model.

To precise the shape of ρ(E), we remark that variables |A| and |A| are strongly correlated, especially for small α values. Assuming the linear relation: p p |A| = (eBeff /ρ)|A| with eBeff /ρ = hA2 i/hA2 i = 12hA2 i, (Beff →α→0 hBi; Beff = 0 when α = .5), and introducing the new variable S 0 = S − µ|A| such that S 0 and |A| are uncorrelated, we can write: 0

Zα (t) ≈ he−ρtS i{C} he−ρtµ|A| cos(eBeff tA)i{C} .Z0 (t).

(16)

Performing the inverse Laplace Transform, we get ρ(E) as shown in Figure 1. Now, let us show briefly how the critical α0 value can be recovered analytically. With the concentration expansion: Z=

∞ X e−ρV (ρV )N ZN N!

(17)

N =0

ZN = hT r e−tHN i/V

(18)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

901

(ZN is the average N impurities partition function per unit volume), we see that α0 is given by the equation:  2 Z1 Z1 Z2 −2 +1= −1 . (19) Z1 Z0 Z0 Computing Z2 diagrammatically to fourth order in α, we finally get that α0 is solution of:  2   1 1 7 1 α(α − 1) 4 = (α(α − 1))2 + (α(1 − α))3 + 1 − ζ(3) (α(α − 1)) . 2 6 3 8 2 (20) In the interval ]0, 1/2], one obtains: α0 ≈ .29. To end up with this part, let us mention what happens when we consider correlated impurities that are spatially distributed like fermions at T = 0 [1]. Diagrammatic computations lead to: Zα (t) =

1 (1 + πα(α − 1)ρt + 0(ρt)2 + ...(ρt)3 + . . .) 2πt

(21)

and for the specific heat: c − c0 ∼t→0 −k(tρπα(1 − α))2 < 0

(22)

when 0 < α ≤ 1/2. From this, we conclude that the D.O.S. has always oscillations. 2

Hall conductivity [2]

For a review on the Integer Quantum Hall Effect, see, for instance, reference [7]. In this part, we develop a Kubo inspired formalism and compute the linear response of the system to a small uniform electric field applied in the ~ = δ(t)E ~ o . The local current ~x direction, E ~j(~r) = e {~v |~r ih~r | + |~r ih~r |~v } 2

(23)

~ is proportional to the conductivity (~v is the velocity operator ~v = p~ − eA) ρ[ji (~r, t), rj ]} σij (~r, t) = iθ(t)e T r {ˆ

(24)

where θ(t) is the Heaviside function. T rρˆ · · · is the thermal Boltzmann or Fermi-Dirac average. ji (~r, t) is the current density operator in the Heisenberg representation ~j(~r, t) = eiHt~j(~r)e−iHt .

(25)

902

Topological Aspects of Low Dimensional Systems

Considering the combination σ − (~r, t) = σxx (~r, t) − iσyx (~r, t)

(26)

and ρˆ = e−βH /Zβ (Boltzmann statistics), the global conductivity averaged over volume reads: Z 1 (27) d~r σβ− (~r, t) σβ− (t) ≡ V Z   e2 1 d~rd~r 0 Π− Git (~r, ~r 0 )x0 Gβ−it (~r 0 , ~r) − (it → it + β) = iθ(t) V Zβ (G is the thermal propagator). To deduce from (27) the conductivity of a gas of electrons at zero temperature and Fermi energy EF , one uses the integral representation of the step function θ(EF − H) Z ∞ 0 dt0 eiEF t Zβ→it0 +0 σβ→it0 +0 (t) (28) σEF (t) = lim η 0 ,0 →0+ −∞ 2iπ t0 − iη 0 where 0 and η 0 are regulators which have to be set to zero at the end. In general, it will be more convenient to calculate the derivative σ(t) ˙ of σ(t) with respect to time, rather than σ(t) itself. In the case of the thermal Boltzmann conductivity, one gets Z  e2 1 e2 − δ(t) − θ(t) d~rd~r 0 e B(~r) Π− Git (~r, ~r 0 )x0 Gβ−it (~r 0 , ~r) σ˙ β (t) = V V Zβ  −(it → it + β) . (29) To derive (29), the identity     e B(~r) ± V (~r) [H, Π± ] = ∓ eB(~r)Π± + Π± , 2

(30)

has been used, which is valid in general for an Hamiltonian H = (~ p− 2 ~ eA) /2 + V (~r). The appearance of the local magnetic field B(~r) in (29) – in the magnetic impurity case, it is a sum of δ(~r −~ri ) functions – greatly simplifies the space integrals. We now discuss some examples:

i) homogeneous magnetic field (29) rewrites as σ˙ βL− (t) =

e2 δ(t) + 2iωcσβL− (t) V

(31)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

903

leading to: σβL− (ω) =

1 e2 · V  − i(ω + 2ωc )

(32)

For a gas of electrons at T = 0: L (ω)|yx = −N (EF ) Re σE F

e2 2ωc · V 4ωc2 − ω 2

(33)

In (33), the limit ωc → 0 is properly defined only if one keeps ω 6= 0, in which case it vanishes, as it should. The Hall conductivity finally reads L (ω = 0)|yx = −N (EF ) Re σE F

e 1 · V B

(34)

This is the classical straight line, showing no plateaus in the Hall conductivity as a function of the number of electrons N (EF ), or of the inverse magnetic field 1/B.

ii) Hall conductivity for one vortex With the standard Aharonov-Bohm propagator  0 +∞ X 1 0 rr 1 − 2β (r 2 +r 02 ) e I|m−α| eim(θ−θ ) Gβ (~r, ~r ) = 2πβ β m=−∞ 0

(35)

(Iν (z)’s are the modified Bessel functions), one gets: e2 1 e2 sin(πα) α δ(t) + θ(t) 2 eiπα (t (t + iβ)1−α − t1−α (t − iβ)α ). V V β Zβ π (36) Its Fourier transform reads  i e2 1 iπα sin(πα)  Γ(1 + α)Ψ(1 + α, 3; β(ω + i)) e 1− σβ− (ω) = V (ω + i) Zβ π  −Γ(2 − α)Ψ(2 − α, 3; −β(ω + i)) (37) σ˙ β− (t) =

where the Ψ(a, b, z)’s are the unregular confluent hypergeometric functions. In the ω → 0 limit, the Hall conductivity reads:

Re σβ (ω)|yx =

 ¯ e2 sin(2πα)  h α(1 − α) 2 (¯ h βω) ln(¯ h βω) + · · · . (38) 1 + m2e V 2 ω2 2

904

Topological Aspects of Low Dimensional Systems

For a gas of electrons coupled to the vortex at zero temperature, one gets, in the limit ω  EF , Re σEF (ω)|yx = N (EF )

e2 1 e2 2πα sin(2πα) 2 'α→0 N (EF ) 2 2 2 V ω V ω

(39)

consistent with the homogeneous magnetic field result (Eq. (33) with 2πα/V = eB = 2ωc ). It is possible to generalize those results when an external uniform B field is added to the vortex. When α is small:   ie πα2 d − (ω = 0) = ) + N (E ) + ··· (40) N (E σE F F F V hBtot i V dEF (ehBtot i = eB + 2πα/V ).

iii) Perturbative hall conductivity for magnetic impurities Considering the nonunitary wavefunction redefinitions 1

ψ(~r) = e− 2 hωc ir

2

N Y

|~r − ~ri |α ψ˜0 (~r) ≡ U 0 d (~r)ψ˜0 (~r)

(41)

i=1

one obtains the Hamiltonian acting on the new wavefunctions ψ˜0 1 hLi hLi hLi H˜ 0 d = Π+ Π− − iα(Ω − hΩi)Π− (42) 2 P hLi z /2), Ω = N where Π− = −2i(∂z − hωc i¯ i=1 1/(z − zi ) and hΩi = πρz. It allows for perturbative computations. Skipping details [2], we get the final simple result: Re σEF (ω = 0)|yx = −N (EF + αhωc i)

1 e · V hBi

(43)

In Figure 2, σxy exhibits small oscillations above the classical straight line. It is worthwhile to notice that Hall plateaus shifted above the classical straight line have already been observed experimentally when the Quantum Hall device contains repulsive impurities [8]. 3

Magnetization and persistent currents [3]

Since the pioneering work of Bloch [9], several questions concerning persistent currents have been answered. The conducting ring case has been

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

905

Fig. 2. Hall conductivity in unit of e2 /h of the random magnetic impurity model h F) at first order in α for α = 0.01 as a function of the filling factor ν = N(E : V ehBi straight line = classical result, full line = perturbative result.

largely discussed in the literature [10], as well as the persistent current due to a point-like vortex [11]. Let us start with the definition of the total magnetization Z e 1 d~r (~r × h~j(~r)i) · ~k + hσz i (44) M= 2 2 where h i means average over Boltzmann or Fermi-Dirac distributions. In the Boltzmann 
case, one obtains the thermal magnetization Zβ ≡ T r e−βH n  o e T r e−βH (~r × ~v ) · ~k + σz · (45) Mβ = 2Zβ It is easy to realize that Mβ =

1 ∂ ln Zβ (B 0 ) lim 0 β B →0 ∂B 0

(46)

where Zα (B 0 ) is the partition function when an uniform magnetic field, B 0 , perpendicular to the plane, is added to the system.

906

Topological Aspects of Low Dimensional Systems

Dropping the spin term, the orbital part of the magnetization reads e Mβorb = Mβ − σz . 2

(47)

Let us now turn to the persistent current and consider in the plane a semi infinite line D starting at ~r0 and making an angle θo with the horizontal x-axis. The orbital persistent current I orb (~r0 , θ0 ) through the line is Z I orb (~r0 , θ0 ) ≡

D

d|~r − ~r0 |

(~r − ~r0 ) × h~j(~r)i ~ · k. |~r − ~r0 |

(48)

Consider now systems rotationnally invariant around ~r0 . I orb (~r0 , θ0 ) no longer depends on θ0 and, without loss of generality, can be averaged over θ0 . So Z 1 (~r − ~r0 ) × h~j(~r)i ~ ·k (49) d~r I orb (~r0 ) = 2π |~r − ~r0 |2 and for the Boltzmann distribution   e (~r − ~r0 ) × ~v ~ T r e−βH · k · Iβorb (~r0 ) = 2πZβ |~r − ~r0 |2

(50)

It is possible to show that [3]: Iβorb (~r0 ) =

e ∂ ln Zβ (α0 ) e Gβ (~r0 , ~r0 ) lim − σz 2πβ α0 →0 ∂α0 2 Zβ

(51)

where Gβ is the thermal propagator and Zβ (α0 ) the partition function when a fictitious vortex of strength α0 is added in ~r0 . The last term in equation (51) emphasizes the importance of the spin coupling in the Hamiltonian for a correct definition of persistent currents. For systems that are both invariant by translation and rotation, we can write, using (45, 50), Z 1 1 orb = (52) d~r0 I(~r0 ) = M orb . I V V Let us now discuss some specific examples:

i) Point vortex in O + uniform magnetic field case Using the corresponding partition function (b ≡ βωc )   e−b V be−b −(α−1)b sinh αb + α−e Zβ (B, α) = 2πβ sinh b 2 sinh b sinh b

(53)

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

one obtains Mβorb

=

  1 eπβ 1 sinh αb − coth b − α − e−(α−1)b b 2V b2 sinh b  b sinh αb e−(2α−1)b (α − e(α−1)b ) + ··· + sinh b sinh b e 2

907



Iβorb (~0)

e = 2V



1 2e−2αb − b 1 − e−2b

(54)

 (55)

with, of course, Mβorb 6= Iβorb V .

ii) Magnetic impurities After averaging over disorder, the system is invariant by translation and rotation, so it is sufficient to compute M orb to get the orbital persistent current. In the Brownian motion approach, we get:

−βρS  e sin(βρA)A {C} orb Mβ = − −βρS he cos(βρA)i{C}

(56)

where A, S and A have been defined previously in equations (9-11). (56) allows for numerical computations. Mβorb is actually only a function of βρ and α, odd in α. Thus, for α ∈ [0, 1/2], necessarily Mβorb = (1 − 2α)F (βρ, α(1 − α)) = (1 − 2α)

∞ X

(βρ)n

n=1

X

amn (α(1 − α))m

m≥n

(57) which can in principle be obtained in perturbation theory. Setting   1 e 0 orb − coth hbi Mβ |mean = (1 − 2α) 2 hbi0

(58)

0

with hbi = βπρα(1 − α) and using the previous example i), one obtains the result (mean field + one vortex corrections) [3]: Mβorb

=

Mβorb |mean + eα(1 − α)(1 − 2α)

×

hbi − 1 − e−2hbi 2hbi (1 − hbi )e−2hbi 1 + + 0 0 0 1 − e−2hbi (1 − e−2hbi )2 2hbi

0

0

0

0

0

! + · · ·(59)

Topological Aspects of Low Dimensional Systems

Mβorb

908

0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05

βρ =1

0

0.2

0.4

α

0.6

0.8

1

Fig. 3. The orbital magnetization in unit e = 1 in the magnetic impurity problem for βρ = 1. Comparison between the analytical computation, solid curve, and numerical simulations.

Figure 3 (βρ = 1), shows a rather good agreement between (59) – full curve – and the numerical simulations – points – based on (56). However, the situation becomes less transparent for higher βρ values. Clearly, the perturbative analytical approach needs more and more corrections coming from hBi + two vortices, . . . References [1] Desbois J., Furtlehner C. and Ouvry S., Nucl. Phys. B[FS] 453 (1995) 759; J. Phys. I France 6 (1996) 641; J. Phys. A: Math. Gen. 30 (1997) 7291. [2] Desbois J., Ouvry S. and Texier C., Nucl. Phys. B[FS] 500 (1997) 486. [3] Desbois J., Ouvry S. and Texier C., Nucl. Phys. B[FS] 528 (1998) 727. [4] Pryor C. and Zee A., Phys. Rev. B 46 (1992) 3116; Lusakopwski A. and Turski A., Phys. Rev. B 48 (1993) 3835; G. Gavazzi, J. M. Wheatley and A. J. Schonfield, Phys. Rev. B 47 (1993) 15170; Kiers K. and Weiss J., Phys. Rev. D 49 (1994) 2081; Emparan R. and Valle Basagoiti M.A., Phys. Rev. B 49 (1994) 14460; Geim A.K., Bending S.J. and Grigorieva I.V., Phys. Rev. Lett. 69 (1992) 2252; Geim A.K., Bending S.J., Grigorieva I.V. and Balmire M.G., Phys. Rev. B 49 (1994) 5749; Brey L. and Fertig H.A., Phys. Rev. B 47 (1993) 15961; Khaetskii A.V., J. Phys. C 3 (1991) 5115. [5] Bergman O. and Lozano G., Ann. Phys. 229 (1994) 416; Emparan R. and Valle Basagoiti M.A., Mod. Phys. Lett. A 8 (1993) 3291; Valle Basagoiti M.A., Phys. Lett. B 306 (1993) 307. [6] Comtet A., Mashkevich S. and Ouvry S., Phys. Rev. D 52 (1995) 2594; Ouvry S., Phys. Rev. D 50 (1994) 5296. [7] Janssen M., et al., “Introduction to the theory of the Integer Quantum Hall Effect”, edited by J. Hadju (VCH, Weinheim, 1994) and references therein. [8] Haug R.J., Gerhardts R.R., Klitzing K.V. and Ploog K., Phys. Rev. Lett. 59 (1987) 1349.

J. Desbois: Random Magnetic Impurities and Quantum Hall Effect

909

[9] Bloch F., Phys. Rev. 137 (1965) A787. [10] L´ evy L.P., Dolan G., Dunsmuir J. and Bouchiat H., Phys. Rev. Lett. 64 (1990) 2074; Cheung H.F., Gefen Y., Riedel E.K. and Shih W.H., Phys. Rev. B 37 (1988) 6050; Cheung H.F., Riedel E.K. and Gefen Y., Phys. Rev. Lett. 62 (1989) 587; Buttiker M., Phys. Scr. T54 (1994) 104; Buttiker M., Imry Y. and Landauer R., Phys. Lett. 96A (1983) 365; Avishai Y. and Kohmoto M., Ben Gurion University Report for a review and references see Narevich R., Technion Haifa Thesis 5757 (July 1997). [11] Akkermans E., Auerbach A., Avron J.E. and Shapiro B., Phys. Rev. Lett. 66 (1991) 76; Comtet A., Moroz A. and Ouvry S., Phys. Rev. Lett. 74 (1995) 828; Moroz A., Phys. Rev. A 53 (1996) 669; for a review and references see R. Narevich, Technion Haifa Thesis 5757 (july 1997); see also Sitenko Yu.A. and Babansky A.Yu., Bogolyubov Institute Report (1997).