Solid State Communications, Vol. 52, No. 12, pp. 985-988, 1984. Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

COMMENTS ON THE EQUILIBRIUM THEORY OF THE QUANTUM HALL EFFECT

J. Hajdu and U. Gummich

Institut fiir Theoretische Physik, Universitiit zu Kbin, D-5000 K6in 41, West Germany

(Received 2 August 1984 by B. Miihlschlegel)

The relation between the thermal equilibrium Hall conductivity generated by minimal gauge transformation and the isolated Hall conductivity given by the Kubo formula is investigated. The contribution of the edge states and some general questions concerning the definition of the equilibrium Hall conductivity are discussed. It is shown that, in the case of an additive electron-impurity system, the two Hall conductivities coincide as long as the Fermi energy is situated in an energy gap.

1. INTRODUCTION

IN THE FORMULATION of his gauge theory of the quantum Hall effect Laughlin [1 ] used the (zero tem- perature limit of the) current formula

el(° t where -- e (e > 0) is the charge of the electron, Ar = LxL~ is the area of the 2d system considered, B is the magnitude of the applied homogeneous magnetic field perpendicular to the system, ~2(T, ~, B; a) is the thermo- dynamic potential and a is a gauge parameter. Equation (1) gives the current density as generated by the minimal gauge transformation 6A = (0, Ba, 0). In a multi- connected system (1) is equivalent to the Byers-Yang current [2]. The expression (1) seems to be advantag- eous for numerical investigations [3].

The current density defined by equation (1) is equal to the thermal equilibrium average of the global current density operator,

- - e J~ = Ar Tr[p(/-/)vy], (2)

where v~ = i[H, y] is the (many electron) velocity operator (y-direction) and o(H) = exp ~(12 - -H + ~N) is the grand canonical density operator. H is the total Hamiltonian of the system including the potential energy ~ brought about by the exterial electric field E = (E, 0, 0), H = He + ¢(x). In the case of a finite single connected system j~ vanishes - in accordance with the requirement of gauge invariance.

In this paper we will be concerned with the following questions:

(1) How is the isothermal Hall conductivity oJ~x associated with equation (1) related to the isolated linear response Hall conductivity o~t~x (given by the Kubo

formula) and what are the general conditions for these two quantities to be equal? (Section 2)

(2) What is the answer to the last question in the special cases that, at zero temperature, the Fermi energy lies in an energy gap or moves in a range of bound (localized) states? (Section 3)

(3) How should one treat, in a finite or semi-f'mite system, the edge states and diamagnetic edge currents? (Section 4).

(4) What is the appropriate order of spatial average and thermal average in the definition of the macro- scopic Hall current density? (Appendix)

We hope that the answers to these questions (which are summarized in Section 5) will help to clarify the interrelations of some different theoretical approaches to the quantum Hall effect.

2. ISOTHERMAL VS ISOLATED HALL CONDUCTIVITY

Putting $ = eEx and expanding in equation (2) the density operator in powers of the electric field strength E we obtain in linear approximationj~ = oyxE with the isothermal Hall conductivity

e 2 _ith = ~ f Cx(--iJk)v~)dX, (3) Oyx 0

for a~x. Here ( . . . ) denotes equilibrium average at zero electric field and x(t) = exp (/H0 t)x exp (-- the t).

The thermal equilibrium conductivity (3) has to be compared with isolated Hall conductivity _iso given by I/~, x

- the Kubo formula otits~ -n + o

e2 ~ O o~ivis° (n) = -~r J ff nt ~ (vv(-- iX)vu(t)) d)k dt. (4)

0 0

985

986 EQUILIBRIUM THEORY OF THE QUANTUM HALL EFFECT Vol. 52, No. 12

Integrating by parts we get

_ i t h i so A = O y x - - O y x

e2 ~ e_nt r # = - - lim. r7 j j (x ( - - iX)v~( t ) )dX dt Ar n~o

0 0

e 2

= (5)

where the superscript d denotes the diagonal part with respect to H0. Introducing the variables X and ~ by

eB x = X + g , vy = ~--~ (6)

we conclude that the conductivities o ~ and O~Sx ° coincide if ~ is an ergodic variable in the sense that

lira (Xl~(t)) = 0, lira (~[~(t)) = 0, (7) t - - * ~ t "-+oo

e 2

a = y (ee - ( lO) ot

Thus, A vanishes if the Fermi energy lies in an energy gap and consequently the isothermal and isolated conductivities coincide. Furthermore, since bound states do not contribute to A ((~]vy [a) being zero for such states), the difference of these conductivities remains constant as long as the Fermi energy varies in a range of bound states.

4. CONTRIBUTION OF EDGE CURRENTS

In order to be more explicit we consider now free electrons with periodic boundary condition in the y- direction,

~k~(x,y + L y ) = ~k(y), (11)

and in the x-direction either the normalization condition

where the notation

IB> : f (8) 0

is used. This is the case if ~ does not contain a zero frequency Fourier component. I f open orbits in momen- tum space are accessible, conditions (7) may be violated. I f lim (gl~Q)) exists and the system is macroscopically

~;---). o o

isotropic (Oxx = oyy) then (eB/m) 2 times this limes is equal to lira ~Ox~07). On the other hand Oxx = 0 is not

~-~0 +

sufficient for the conditions (7) to be satisfied, i.e. to guarantee the equality of the isothermal and isolated Hall conductivities.

In the case of an infinite free electron system the ly ergodicity conditions (7) are satisfied and the isothermal and isolated Hall conductivities coincide. In the case of a semi-f'mite free electron system (with edges at x = -+ Lx/2), however, ~ is not ergodic and the two Hall con- ductivities are different. In fact _ith = 0 (see Section 4). o y x

3. ADDITIVE SYSTEM

In the following we consider an additive system (i.e. we neglect the Coulomb interaction between the elec- trom). For such a system

e 2 ~)

A = A---; 3-'--~ ~ f ~ x ~ v y ~ , (9) Ot

where fa = f (ea) is the Fermi distribution function and x and v~ are now single electron quantities. The matrix elements are taken in the energy representation, Ho J a) = ca I a). At zero temperature

f [~ba(x, 0)12 dx = 1, (12)

corresponding to an infinite system, or the vanishing boundary conditions

G - T , y = G ,y ---o, (13)

which characterize a semi-finite system. Using the single electron energy representation/-/ la) = e~la), e~ = e~ + eEX, ~ = (v. ky) , ky = -- X/ l 2 , l 2 = 1/eB, u = O, 1, 2 . . . . . ky = (2rr/Ly)#, ~u = 0, -+ 1 , . . . , equation (2) reads

- - e

LxLy ~ f~(otlv~lot), (14)

<~lvyla> : Des _ 12 Oe~ 3ky 3X

= - t 2 (15) ~X B

In the case of an infinite system [boundary con- ditions (11) and (12)] the energy spectrum{e°}is degenerate in X and discrete, e°a = ev = 6oc(v + ½), we = eB/rn and, from equations (14) and (I 5) with Z~f~ = N, aiy~ = en/B, n = N/Ar . Using v(t) = v(O) exp (io~et), v = vx + ivy, in the evaluation of equation (4) we obtain the same classical result O~x ° = en/B. Thus, for an infinite free electron system o ~ = oi~. A = 0 follows also immediately from equation (9) since, for an infinite system, (al v~ [ a) = 0.

In the case of a semi-finite system [boundary conditions (11) and (13) the spectrum {e ° } is non-

Vol. 52, No. 12 EQUILIBRIUM THEORY OF THE QUANTUM HALL EFFECT 987

degenerate, continuous and gapless, ea = ev(X). The first term on the r.h.s, of equation (15) - which is only significant near the edges at x = +- Lx/2 - contri- butes to the current. Since, in linear order in the electric field

~f[eX--\-~--ff]e=o] (16) f(ea) = .f(e °) + Oe o

we obtain from equation (14) (with reference to e ° (--X) = ¢ ( X ) a n d lim f[e°(X)] = 0 , ] y = 0 , i.e. oiytx h = 0.

X ~ : t oo

The current in termal equilibrium vanishes because balk and edge contributions exactly compensate each other. On the other hand oi~ ° is the same as for the infinite system since the diagonal part of vy with respect to Ho does not contribute to the Kubo conductivity (X is conserved for E = 0). Thus, we conclude that A = - - en/B. Of course this result follows also immediately from equation (9) using (c~lxl a) = X and (ot lvwlot) = -- t ~ ae o / ~x

e 1 ~ Of ~ e a x = en (17)

It has often been argued that, in the presence of a homogeneous electric field, the grand canonical density operator p(H) occurring in equation (2) is an ill-defined concept if the system under consideration is of infinite extension in the direction of the electric field. In the case of crossed homogeneous electric and magnetic fields, however, the reduced density operator

exists. Indeed, the integrals -Lxl2

im 1 dX e 1 e E X ,

L a:---~ o o d Lxl2

occur also in the partition function Z as factors and therefore the electric field drops out. Pr is just the grand canonical density operator for zero electric field. The corresponding distribution function is all we need to calculate the equilibrium average of the global current density. The diagonal matrix elements of this quantity are in the energy representation just (e/Ar)E/B (cf. equation (15) for an infinite system). Since these are constant, the problem actually reduces to the require- ment of normalizability which is obviously fulfilled - as anticipated in the discussion following equation (15).

5. CONCLUSIONS

The thermal equilibrium Hall conductivity and the isolated (Kubo) Hall conductivity coincide if the velocity in the direction of the Hall current (v~) is an ergodic variable [in the sense of equation (7)]. This condition is satisfied for an infinite free electron system - not, however, for a semi-finite system with edges (at x = +- Lx/2 ). In the first case both conductivities are equal to the classical value en/B. In the second the isolated Hall conductivity has still this value whereas the iso- thermal Hall conductivity vanishes (since the edge and the bulk contributions compensate each other exactly). Thus, the equilibrium Hall current definition (1) is not suitable for a semi-finite system. It is, therefore, confusing to use equation (1) and to argue with edge states at the same time [1,3] . If one wishes to apply equation (1) to a semi-finite system the contribution of the edge states has to be excluded. An alternative manipulation is to put E = 0 in the distribution function f(e~). By doing this, however, the equivalence of equations (1) and (2) is violated.

If, in an additive (e.g. electron-impurity) system, gaps in the energy spectrum exist (this is not the case for a semi-finite free electron system) and, at zero tem- perature, the Fermi energy lies in one of these gaps, then vr behaves ergodically and the two Hall conductivities coincide. The difference between these quantities remains constant as long as the Fermi energy moves within a range of bound states. The vanishing of the conductivity in the direction of the applied electric field is, in general, not a sufficient condition to ensure the equality of the isothermal and isolated Hall con- ductivities.

The non-existence of the canonical distribution in the case of an infinite homogeneous electric field is not a valid objection against the application of the equilibrium current definition (1) or (2) under such circumstances. In fact, a reduced canonical distribution exists which is sufficient to calculate the thermal average of the global current density. This is due to the fact that in the case of crossed infinite homogeneous electric and magnetic fields, a stationary (single electron) state exists.

If, however, the thermal equilibrium average and the spatial average of the microscopic current density are performed in the "physical" order, i.e. first the thermal average and then the spatial (which is opposite to the order taken in the current formulae (1) and (2)) then the average current density turns out to be just the diamag- netic current density (the curl of the Landau magneti- zation). Although proposed in the literature [4] this current cannot be associated with the Hall effect [5-7] .

We conclude that the thermal equilibrium current

988 EQUILIBRIUM THEORY OF THE QUANTUM HALL EFFECT Vol. 52, No. 12

equations (1) or (2) is not an unambiguous concept; it needs rather subsidiary interpretations and special care in application.

It should be pointed out that Halperin's [8] calcu- lation of the Hall conductivity of a f'mite double con- nected system (which is topologically equivalent to the semi-i~mite system considered in this paper) and his interpretation of Laughlin's gauge approach is essentially based on the assumption of an adiabatically disturbed local equilibrium distribution [9]. This stationary distribution function is different from both the equili- brium distribution [underlying equation (1)] and the adiabatically disturbed absolute equilibrium distribution which the Kubo formula is derived from, and plays presumably an essential role in the discussion about the validity (or completeness) of the Kubo formulae in the case of non-mechanical disturbances.

effect [1, 10]. Usually the opposite order of averaging, yielding

1 iv = f dx E :o:w(x), (i3)

is considered to be "physical". Since the order of integrations f dx f dX can not be interchanged,/v and iy are different. It is not difficult to show (cf. [11, 12]) that

i v = curl M, (A4)

where M = M [~'(X)] is the Landau magnetization which vanishes in the classical limit. Thus, in thermal equilibrium, the physical order of spatial and thermal averaging of the microscopic current density yields not the usual Hall current but the average diamagnetic current density.

APPENDIX: THE ORDER OF SPATIAL AND THERMAL AVERAGING OF THE CURRENT

DENSITY

In the case of free electrons the current density in the state la) is

/ya(x, y) = -- e~%(x -- X)l~(x)12/Lv = / w ( x ) . (A1)

The spatial average of this quantity Lxl2

_ 1 f e / w Lx ~ /,~(x) dx - A r (elvvl~), (h2)

-l.,x12

is the global current density, the thermal average of which, equation (14) [or, equivalently, equation (1)], has been used in some theories of the quantum Hall

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1. R.B. Laughlin, Phys. Rev. B23, 5632 (1981). 2. N. Byers & C.N. Yang, Phys. Rev. Lett. 7, 46

(1961). 3. H. Aoki, J. Phys. C15, L1227 (1982). 4. A. Widom, Phys. Lett. 90A, 474 (1982). 5. H. Oji, Phys. Lett. 98A, 127 (1983). 6. J. Hajdu & Ute Gummich, Phys. Lett. 99A,

396 (1983). 7. A. Widom, Phys. Rev. B28, 4858 (1983). 8. B.I. Halperin, Phys. Rev. B25, 2185 (1982). 9. J. Hajdu, Lecture Notes in Physics (Springer-

Verlag) 177, 23 (1983). 10. W. Brenig, Z. Physik B50, 305 (1983). 11. E. Teller, Z. Physik67,311(1931). 12. S.V. Peletminskii & V.G. Bar'yakhtar, Soy. Phys.

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