Distribution functions for a two-dimensional non-interacting quantum electron gas in an external magnetic field

Distribution functions for a two-dimensional non-interacting

quantum electron gas in an external magnetic field

F. LADO*

Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202, USA

(Received 3 October 2002; revised version accepted 28 November 2002)

The exactn-body distribution functions are calculated for a two-dimensional, non-interacting quantum electron gas in an external magnetic field for any temperature and density. At low tempertures and filled lowest Landau level (LLL), these functions are identical to the exact distribution functions obtained by Jancovici [1981,Phys. Rev. Lett.,46,386] for the classical two-dimensional one-component plasma (2DOCP) at the special plasma parameter G¼2, thus establishing that the 2DOCP provides an exact classical Boltzmann factor which describes the ideal LLL quantum state associated with the integral quantum Hall effect.

1. Introduction

It was not so long ago that physical models in fewer than three dimensions were primarily of interest to the-orists seeking greater tractability for the mathematics used to describe the systems in three dimensions. Experi-ments carried out in recent decades, however, have demonstrated that a two-dimensional world is not an arcane corner of theoretical physics but a real presence with measurable and often fascinating properties. In the process, many once esoteric models have gained fresh immediacy. In particular, a system of non-interacting electrons in a plane subject to an external magnetic field, first studied by Landau [1] in 1930 to account for diamagnetism, became the basis for understanding the integral quantum Hall effect [2], discovered by von Klitzing et al. [3] 50 years later. The discovery of the fractional quantum Hall effect [4] not long after called for a different explanation and an important contribu-tion was made by Laughlin [5], who invoked the clas-sical two-dimensional one-component plasma [6, 7] in constructing his solution.

In this work, on the occasion of Dominique Levesque’s 65th birthday, we report a surprising con-nection between the same two-dimensional one-component plasma (2DOCP) and theintegralquantum Hall effect, as modelled by the simplest system, Landau’s non-interacting electrons in a plane, subject to an external magnetic field perpendicular to the plane. The connection is made after calculating the exact n-body distribution functions of this system and realizing that, in a certain limit, they are identical to the exact

distribution functions of the 2DOCP found by Jancovici [8] for plasma parameterG¼2.

2. General n-particle distribution functions 2.1. Preliminaries

We considerNelectrons at temperatureTin areaA. A uniform magnetic fieldB₀is applied perpendicular to the plane ofA. The canonical partition functionQNfor this system can be written

QN¼ 1 N!L2N

ð

WNðrNÞdr1 drN; ð1Þ

whereWNðrNÞis the (antisymmetrized) Slater sum,

WNðrNÞ ¼L2N X

P

ð1ÞjPjX

k

CkðPrNÞ

expðHÞCkðrNÞ; ð2Þ

which plays the role of Boltzmann factor. The conven-tional introduction of the thermal de Broglie wavelength

L¼ ðh2=2pmekBTÞ1=2in equation (1) serves to make the Slater sum dimensionless but has no effect on the results reported below. Here the Ck are a complete set of N-particle states distinguished by indexk representing 2N quantum numbers; the sum over permutations P, with even or odd parity jPj and antisymmetric weighting, accounts for the Fermi–Dirac statistics of the indistin-guishable electrons. The Hamiltonian H for non-interacting electrons in an external magnetic field is

HN¼ 1 2me

XN

j¼1

½ihrjþeAðrjÞ2; ð3Þ

Molecular PhysicsISSN 0026–8976 print/ISSN 1362–3028 online#2003 Taylor & Francis Ltd

http://www.tandf.co.uk/journals DOI: 10.1080/00268970310000755642

wheremeis the electron mass,eis its charge andAðrÞis avector potential which yields auniform magnetic field B0 (which we shall take to define thezdirection),

B₀¼B0^kk¼ r AðrÞ: ð4Þ As usual,¼1=kBTis the inverse temperature, withkB being Boltzmann’s constant. The system is taken to be fully spin-polarized, so that spin degrees of freedom may be ignored.

2.2. The Slater sum

Since interactions between electrons are being omitted, the Schro¨dinger equation for the energy eigen-states is readily solved and the sum over eigen-states in equation (2) can be carried out. The vector potential in the Landau gauge isA¼ B0y^{{, so the one-electron Hamiltonian becomes

H1¼ 1 2me

ih @

@xþeB0y

2

þ ih @ @y

2

" #

: ð5Þ

The Schro¨dinger equationH1 ¼ is then solved by [2]

nðx;yÞ ¼ A_n

L1x=2

exp ixðyl

2 0Þ

2 2l2

0 " #

Hn

yl2₀ l0

! ;

ð6Þ

with ¼2pj=Lx for j¼0;1;2;. . .;l0¼ ðh=eB0Þ1

=2 , andAn¼ ðpl20Þ1

=4

ð2nn!Þ1=2. Further,HnðxÞis the Her-mite polynomial of ordern andLxis the edge length of the rectangular area A in the x direction. The corre-sponding energy eigenvalues are _n¼ ð2nþ1Þ BB0, n¼0;1;2;. . .; where B¼eh=2me is the Bohr mag-neton.

We seek to determine the one-body Slater sum W₁ðx;yÞ ¼L2X

;n

expð_nÞ nðPx;PyÞ nðx;yÞ

¼ L

2

ðpl2 0Þ

1=2 L_x

X

exp½iðxPxÞ

exp½ðPyl20Þ2=2l20

exp½ðyl20Þ2=2l20

X

n

exp½ð2nþ1Þ BB0 2n_n!

Hn

Pyl2₀ l0

! Hn

yl2₀ l0

!

: ð7Þ

We note first that [9]

X1

n¼0

expðnÞ

2n_n! HnðxÞHnðyÞ

¼ expðÞ

2 sinh 1=2

exp ðx

2_þy2_{ÞexpðÞ þ2xy} 2 sinh

" #

:

ð8Þ

Using this in equation (7) with¼2 _BB0 and simpli-fying, we arrive at

W1ðx;yÞ ¼ ½L2=ð2pl20 sinhÞ1

=2 Lx

X

exp½iðxPxÞ

ðyPyÞ2=2l20 tanh

tanhð=2Þðyl20ÞðPyl20Þ=l20: ð9Þ

The second sum, over , is effected as an integral. For large L_x,L_x1P! ð2pÞ1Ðdand we get

1 2p

ð1

1

dexp½iðxPxÞ

tanhð=2Þðyl20ÞðPyl20Þ=l20

¼ ½4pl20tanhð=2Þ

1=2

exp½ðxPxÞ2=4l20 tanhð=2Þ

þtanhð=2ÞðyPyÞ2=4l20

þiðxPxÞðyþPyÞ=2l2₀; ð10Þ

and so finally

W1ðx;yÞ ¼ ½L2=4pl20 sinhð=2Þ

exp½ðrPrÞ2=4l2₀ tanhð=2Þ

þiðxPxÞðyþPyÞ=2l20; ð11Þ

after some further simplification.

The final result for theN-body Slater sum is

WNðrNÞ ¼ ðL2=4pl20 sinh BB0ÞN X

P

ð1ÞjPj

exp (

XN

j¼1

½ðr_jPrjÞ2=4l20 tanh BB0

þiðxjPyjyjPxjÞ=2l20 )

; ð12Þ

wherer_jlocates electronjin thexyplane, Bis the Bohr magneton andl0 ¼ ðh=eB0Þ1

=2

WNðrNÞcB0¼0¼ X

P

ð1ÞjPjexp p

L2 XN

j¼1

ðr_jPrjÞ2

" #

:

ð13Þ

2.3. The distribution functions

With the Slater sum in hand, we now seek to deter-mine then-body distribution function

ð_NnÞðr₁;. . .;r_nÞ ¼ 1 ðNnÞ!L2NQN

ð

WNðrNÞdrnþ1. . .drN

ð14Þ

for any n. One notes that each permutation P in equation (12) produces a disjoint set of ring-type inte-grands, with each particle appearing in one and only one ring. Particle positions 1;2;. . .;n are not integrated out and will be called root points. The remaining Nn particle positions are variables of integration, to be called field points. The bond linking adjacent particles iandjin aring is

fðr_i;r_jÞ ¼ ð4pl20 sinh BB0Þ

1

exp½ðr_ir_jÞ2=4l20 tanh BB0

þiðxiyjyixjÞ=2l20: ð15Þ LetIkðri;rjÞbe a chain of links beginning at root pointri and ending at root point r_j with k field points in between; i.e.

I_kðr_i;r_jÞ ¼

ð

fðr_i;r_n1Þfðr_n1;r_n2Þ fðr_nk;r_jÞdr_n1 dr_nk:

ð16Þ

The canonical distribution functionð_NnÞðr1;. . .;rnÞthen becomes [10]

ð_NnÞðr₁;. . .;r_nÞ ¼ 1 ðNnÞ!QN

X

P

ð1ÞjPj

X

Nn

m¼0

ð1Þm ðNnÞ! ðNnmÞ!

X

k1;k2;...;kn

m;X j

kj !

Ik1ðr1;Pr1Þ

I_k2ðr₂;Pr₂Þ I_knðr_n;Pr_nÞ

ðNnmÞ!Q_Nnm: ð17Þ Now the permutations P are among the n root points only, while mis the total number of field points in the rooted rings. The sums over number of field pointskjin individual chains are collectively constrained by the requirement Pn_j¼1kj¼m, expressed by the Kronecker delta ði;jÞ in the summand. This constraint can be

removed, and the individual sums over thekj factored, by transforming these expressions from particle number N to chemical potential as an independent variable using the grand canonical formalism. We have then

X¼X 1

N¼0

expð NÞQN ð18Þ

for the partition function and

ðnÞðr₁;. . .;r_nÞ ¼ 1

X

X1

N¼n

expð NÞQN

ðnÞ

Nðr1;. . .;rnÞ

ð19Þ

for then-body distribution function. Using equation (17) in the latter we now get simply

ðnÞðr₁;. . .;r_nÞ ¼X

P

ð1ÞjPjY

n

j¼1

X

1

k¼0

ð1Þk exp½ðkþ1Þ Ikðrj;PrjÞ

" #

:

ð20Þ

Starting fromI0ðri;rjÞ ¼fðri;rjÞ, one finds by induction that

Ikðri;rjÞ ¼ ½4pl02 sinhððkþ1Þ BB0Þ1

exp½ðr_ir_jÞ2=4l2₀ tanhððkþ1Þ BB0Þ

þiðxiyjyixjÞ=2l20; ð21Þ so that finally

ðnÞðr₁;. . .;r_nÞ ¼X

P

ð1ÞjPjY

n

j¼1

fðjr_jPrjjÞ

exp½iðxjPyjyjPxjÞ=2l20g; ð22Þ with

ðrÞ ¼ 1

4pl2 0

X1

j¼1

ð1Þj1 expðj Þ

sinhðj BB0Þ

exp½r2=4l20 tanhðj BB0Þ: ð23Þ The first of these distribution functions,

ð1Þðr1Þ ¼ð0Þ; ð24Þ is just the uniform density¼NN=A expressed in terms of the chemical potential ,

¼ 1

4pl2 0

X1

j¼1

ð1Þj1 expðj Þ

sinhðj BB0Þ

: ð25Þ

lnX¼ A

4pl2 0

X1

j¼1

ð1Þj1 expðj Þ

jsinhðj _BB₀Þ ; ð26Þ

using the thermodynamic relationNN ¼kBT@lnX=@ . The dimensionless forms

gðnÞðr₁;. . .;r_nÞ ðnÞðr₁;. . .;r_nÞ=n ð27Þ

are preferable in writing out higher distribution func-tions. Explicitly, we have for the next three,

gð2Þðr₁;r₂Þ ¼1D2ðr12Þ; ð28Þ gð3Þðr₁;r₂;r₃Þ ¼1 ½D2ðr12Þ þD2ðr13Þ þD2ðr23Þ

þ2Dðr12ÞDðr23ÞDðr31Þ

cosðjr₁₂r₁₃j=2l2₀Þ; ð29Þ

gð4Þðr1;r2;r3;r4Þ

¼1 ½D2ðr12Þ þD2ðr13Þ þD2ðr14Þ

þD2ðr23Þ þD2ðr24Þ þD2ðr34Þ

þ ½D2ðr12ÞD2ðr34Þ þD2ðr13ÞD2ðr24Þ þD2ðr14ÞD2ðr23Þ

þ2½Dðr12ÞDðr23ÞDðr31Þcosðjr12r13j=2l20Þ

þDðr13ÞDðr34ÞDðr41Þcosðjr13r14j=2l20Þ

þDðr₁₂ÞDðr₂₄ÞDðr₄₁Þcosðjr₁₂r₁₄j=2l2₀Þ þDðr₂₃ÞDðr₃₄ÞDðr₄₂Þcosðjr₂₃r₂₄j=2l2₀Þ 2½Dðr12ÞDðr23ÞDðr34ÞDðr41Þcosðjr13r24j=2l20Þ

þDðr12ÞDðr24ÞDðr43ÞDðr31Þcosðjr14r23j=2l20Þ

þDðr13ÞDðr32ÞDðr24ÞDðr41Þcosðjr12r34j=2l20Þ;

ð30Þ

where DðrÞ ¼ðrÞ=ð0Þ. The three-body distribution function is the first to show correlations due to phase.

A key result here, ðrÞ in equation (23), can be rewritten in a form that more directly displays the familiar Fermi degeneracy at low temperatures. We use the generating function of Laguerre polynomials,

exp½xz=ð1zÞ

1z ¼

X1

k¼0

L_kðxÞzk; ð31Þ

withx¼r2=2l20andz¼expð2j BB0Þin equation (23) and then sum overjto get

ðrÞ ¼expðr

2_=4l2 0Þ 2pl2

0

X

1

k¼0

Lkðr2=2l20Þ

1þexpf½ ð2kþ1Þ BB0g

; ð32Þ

where now the summation is over Landau levels, of energy _k ¼ ð2kþ1Þ BB0. For completeness, we note

that the expansion 1=sinhx¼2P1_k¼0exp½ð2kþ1Þx

used similarly in equations (25) and (26) gives

¼ 1

2pl2 0

X1

k¼0

1

1þexpð½ ð2kþ1Þ BB0Þ

; ð33Þ

lnX¼ A

2pl2 0

X1

k¼0

lnf1þexpð½ ð2kþ1Þ BB0Þg: ð34Þ

For low temperatures and large magnetic fields, i.e. for BB01, the summand in equation (33) acts like a discrete unit step function, so that for (integer) n filled Landau levels one has 2pl20 ¼n; similarly, using equation (32), the pair distribution function gð2Þðr12Þ ¼ gð2Þðr₁;r₂Þsimplifies in this regime to

gð2ÞðrÞc_T¼0¼1expðr2=2l20Þ 1 n

Xn1 k¼0

Lkðr2=2l20Þ " #2

;

ð35Þ

a result that has been obtained by Ciftja and Fantoni [11] and by Kamilla et al. [12].

2.4. A remarkable connection

We close this work by noting a remarkable fact. For physical conditions such that this non-interacting quantum electron system just fills the lowest Landau level, i.e. for BB01 a nd 2pl20¼1, so that DðrÞ ¼expðpr2=2), the exact distribution functions gðnÞðr₁;. . .;r_nÞ displayed above are identical to the exact gðnÞðr₁;. . .;r_nÞ of a classical two-dimensional one-component plasma (2DOCP) obtained by Jancovici [8] for the special plasma parameter value G¼2. Working backwards, we may then conclude that the quantum state for the filled lowest Landau level is pre-cisely described by an effective classical pair potential ðrÞ ¼ 2 lnðr=l0Þ. Adapting to the present notation the total potential V quoted by Jancovici [8] for 2DOCP particles confined to a disc of radius R, we get asystem potential

Fð_N1ÞðrNÞ ¼ 2X i<j

ln rij l0

þ1 2

X

i r_i l0

2

þN2 lnR l0

3

4

;

ð36Þ

or, equivalently, a Boltzmann factor W¼expðFÞ

given by

Wð_N1ÞðrNÞ ¼Y

i<j r_ij l0

2

exp 1 2

X

i ri l0

2 " #

; ð37Þ

understanding the integral quantum Hall effect [2]; we find that for the incompressible state at low tempera-tures with filling factor ¼1, this system is exactly described by the effective Boltzmann factor written above. Further, the wave function constructed by Laughlin [5] to explain the fractional quantum Hall effect for filling factor¼1=m, with man odd integer, produces just this same Boltzmann factor,

Wð_N1=mÞðrNÞ ¼Y

i<j rij l₀

2m

exp 1 2

X

i ri l₀

2 " #

; ð38Þ

for m¼1. One concludes that, at least for the filled lowest Landau level, ¼1=m¼1, Laughlin’s varia-tional wave function, obtained in the context of an electron gas in two dimensions interacting through the Coulomb e2=r potential, actually describes the non-interacting limit. This is consistent with the fact that the electronic charge e in these solutions appears only in a product with the magnetic field, aseB0, reflecting just the magnetic force on the electron.

References

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[2] Chakraborty, T., _and Pietila« inen, P., _{1995, The} Quantum Hall Effects(Berlin: Springer).

[3] Von Klitzing, K., Dorda, G.,andPepper, M.,1980, Phys. Rev. Lett.,45,494.

[4] Tsui, D. C., Sto« rm er, H. L.,_andGossard, A. C.,1982, Phys. Rev. Lett.,48,1559.

[5] Laughlin, R. B.,1983,_{Phys. Rev. Lett.},_50,1395.

[6] Caillol, J. M., Levesque, D., Weis, J. J.,andHansen, J. P.,1982,J. stat. Phys.,_28,325.

[7] Levesque, D., Weis, J. J., and MacDonald, A. H.,

1984,Phys. Rev.B,30,1056.

[8] Jancovici, B.,1981,Phys. Rev. Lett.,_46,386.

[9] Gradshteyn, I. S.,andRyzhik, I. M., 1980,Table of Integrals, Series, and Products (New York: Academic Press). Integral 7.374.10 and the orthogonality of the Hermite polynomials yield this result.

[10] Lado, F.,1967,J. chem. Phys.,47,5369.

[11] Ciftja, O., and Fantoni, S., 1997, Phys. Rev. B, 56,

13290.