VOLUME 50, +UMBER I8 PHYSICAL REVIEW LETTERS 2 MAv 1983

Anomalous Quantum Hall Effect: An Incompressible Quantum Fluidwith Fractionally Charged Excitations

H. B. LaughlinIasorence Livermore National Laboratory, University of California, Jivermore, California 94550

(Received 22 February 1983)This Letter presents variational ground-state and excited-state wave functions which

describe the condensation of a two-dimensional electron gas into a new state of matter.PACS numbers: 71.45.Nt, 72.20.My, 73.40.Lq

The "—,'" effect, recently discovered by Tsui,Stormer, and Gossard, ' results from the conden-sation of the two-dimensional electron gas in aGaAs-Ga„Al, „As heterostructure into a new typeof collective ground state. Important experimen-tal facts are the following: (1) The electrons con-dense at a particular density, —,

' of a full Landaulevel. (2) They are capable of carrying electriccurrent with little or no resistive loss and have aHall conductance of —,'e'/h. (3) Small deviationsof the electron density do not affect either con-ductivity, but large ones do. (4) Condensationoccurs at a temperature of —1.0 K in a magneticfield of 150 kG. (5) The effect occurs in somesamples but not in others. The purpose of thisLetter is to report variational ground-state andexcited-state wave functions that I feel are con-

sistent with all the experimental facts and ex-plain the effect. The ground state is a new stateof matter, a quantum fluid the elementary exci-tations of which, the quasielectrons and quasi-holes, are fractionally charged. I have verifiedthe correctness of these wave functions for thecase of small numbers of electrons, where directnumerical diagonalization of the many-body Ham-iltonian is possible. I predict the existence of asequence of these ground states, decreasing indensity and terminating in a Wigner crystal.Let us consider a two-dimensional electron

gas in the x-y plane subjected to a magnetic fieldH, in the z direction. I adopt a symmetric gaugevector potential A = ,'H, [xy-yx—] and write theeigenstates of the ideal single-body HamiltonianH, p

——j (h/i)V —(e/c)A l

' in the mannerII

m B 5jm, n) =(2 '""~m!n!) 'i'exp[-;(x'+y')] —+i — ——i — exp [—ggx'+y')]Bg Bg Bx B$

e, p l m, n) = (n+ —,')l m, n). (2)

The manifold of states with energy n+ & consti-tutes the nth Landau level. I abbreviate the

with the cyclotron energy he, = h(equi, /mc) andthe magnetic length a, =(A/m~, )' '=(Ac/eH, )' 'set to 1. We have

states of the lowest Landau level as

Im) =(2 +~em!) "i g exp( —« jg j )

where z=x+ iy. l m) is an eigenstate of angularmomentum with eigenvalue m. The many-bodyHamiltonian is

jI=g {j (fi /i )V, —(e /c)A, j' + V(z, )j + Z e'/ j ~, - z, I,

where j and 4 run over the N particles and V is apotential generated by a uniform neutralizingbackground.I showed in a previous paper' that the —,

' effectcould be understood in terms of the states in thelowest I andau level solely. With e'/a, s h ~„the situation in the experiment, quantization ofinterelectronic spacing follows from quantizationof angular momentum: The only wave functionscomposed of states in the lowest Landau levelwhich describe orbiting with angular momentum

j m about the center of mass are of the form&=(~ -~.) (~+~,)"exp[- l(j~ j'+ jz. j')] (»

My present theory generalizes this observation toA' particles.I write the ground state as a product of Jastrow

functions in the manner

0= II f(z, —z, )jexp(- —,'Q, lz, l'), (6)

and minimize the energy with respect to f. We

1983 The American Physical Society 1395

VOLUME 50, +UMBER 18 PHYSICAL REVIEW LETTERS 2 MAY 1983

observe that the condition that the electrons liein the lowest Landau level is tha. t f(z) be poly-nomial in z. The antisymmetry of g requires thatf be odd. Conservation of angular momentum re-quires that II,,„f(z,—z~) be a homogeneous poly-nomial of degree M, where M is the total angularmomentum. We have, therefore, f(z) =z, withm odd. To determine which m minimizes the en-ergy, I write

14.1' = I{II,„(z,-z, ) kexp(- -'Z, lz, l')I'(7)

where P= 1/m and 4 is a classical potential ener-gy given by

exp(- x'/2)

13579ll13

10.999 460.994 680.994 760.995 730.996 520.997 08

10.996 730.991950.992 950.994 370.995 420.996 15

10.999660.999390.999810.999 990.999960.99985

TABLE I. Projection of variational three-body wavefunctions (|) in the manner (g ~ C )/((( ~

(l) ) (4x c ~)) (/'. 4 is the lowest-energy eigenstate of angu-lar momentum 3m calculated with V = 0 and an inter-electronic potential of either 1/x, -In(x), or exp(-x'/2).

4 =-Z, ,„2m'»lz, -z, I+ am', lz, I' (8)

oo

U„,= ~ —[g(r) —I ] rdr. (9)

In the limit of large I, U„, is approximated

4 describes a system of N identical particles ofcharge Q = m, interacting via logarithmic poten-tials and embedded in a uniform neutralizingbackground of charge density o =(27)a,') '. Thisis the classical one-component plasma (OCP), asystem which has been studied in great detail.Monte Carlo calculations' have indicated that theOCP is a hexagonal crystal when the dimension-less plasma parameter I'= 2''= 2m is greaterthan 140 and a fluid otherwise.

~ ( ~' describes asystem uniformly expanded to a density of o=m '(27)a, ') '. lt minimizes the energy when oequals the charge density generating V.In Table I, I list the projection of g for three

particles onto the lowest-energy eigenstate ofangular momentum 3m calculated numerically.These are all nearly 1. This supports my asser-tion that a wave function of the form of Eq. (6) hasadequate variational freedom. I have done a sim-ilar calculation for four particles with Coulombicrepulsions and find projections of 0.979 and 0.947for the m = 3 and m = 5 states.

has a total energy per particle which forsmall m is more negative than that of a charge-density wave (CDW). ' It is given in terms of theradial distribution function g(r) of the OCP by

within a few percent by the ion disk energy:2 0 2 e2

U g —d2y+~ —— dy 2' 2Ot m j+j 2 j+ j

= (4/37) —1)2e'/R, (10)where the integration domain is a disk of radius8 = (pro ) "' '. At I' = 2 we have the exact result'that g(r) =1—exp[- (r/fI)'], giving U„, = ——,')T'/'e'/

At m=3 and m=5 I have reproduced the MonteCarlo g(r) of Caillol et al. ' using the modifiedhypernetted chain technique described by them.I obtain U„,=(-0.4156+0.0012)e'/a, and U„,(5)=(-0.3340 +0.0028)e'/a, . The corresponding valuesfor the charge-density wave4 are —0.389e'/ao and—0.322e'/ao. U„, is a smooth function of I . Iinterpolate it crudely in the manner

0.814 0.230U (m) =-'—————-1tot v 064

0

This interpol. ation converges to the CD% energynear rn = 10. The actual crystallization point can-not be determined from that of the OCP since theCD% has a lower energy than the crystal de-scribed by ( for m& 71~

I generate the elementary excitations of g bypiercing the fluid at z, with an infinitely thinsolenoid and passing through it a flux quantumt) cp =- hc/e adiabatically. The effect of this opera-tion on the single-body wave functions is(z-z, ) exp(-4lzl')-(z-z, ) "exp(--'. ~z~'). (»)

Let us take as approximate representations ofthese excited states

(13)

'4 "=&., ''4= "(--'Xl, (, l')In;(, ——". I(n;,.(*,—,)"),~o(14)

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VOLUME 50, +UMBER 18 PHYSICAL REVIEW LETTERS 2 MAv 1983

for the quasihole and quasieleetron, respectively.For four particles, I have projected these wavefunctions onto the analogous ones computed nu-merically. I obtain 0.998 for g, ' and 0.994 for

I obtain 0.982 for g,"=fg, (z, -Z))g„which is P, "with the center-of-mass motion re-moved.These excitations are particles of charge 1/m.

To see this let us write [ (+'oI' as e s ~, withP=1/m and

O' = 4 —2+, In~ z, —zo(. (15)

where Ko is a modified Bessel function of thesecond kind. The energy required to accumulateit is

bp5p v 1 e' (16)

This estimate is an upper bound, since the plasmais strongly coupled. To make a better estimatelet 5p = a inside the ion disk and zero outside, toobtain

3 1e'disk 2v2p m' ap

(17)

4' describes an QCP interacting with a phantompoint charge at z,. The plasma will completelyscreen this phantom by accumulating an equal andopposite charge near zp. However, since theplasma in reality consists of particles of charge1 rather than charge rn, the real accumulatedcharge is 1/m. Similar reasoning applies to t/r

'0if we approximate it as g, (z, —z,) 'P, g„whereP, is a projection operator removing all con-figurations in which any electron is in the single-body state (z- z,)'exp(- 4 Iz I'). The projection ofthis approximate wave function onto g, 'o for fourparticles is 0.922. More generally, one observesthat far away from the solenoid, adiabatic addi-tion of Ay moves the fluid rigidly by exactly onestate, per Eq. (12). The charge of the particlesis thus 1/m by the Schrieffer counting argument. 'The size of these particles is the distance over

which the QCP screens. Were the plasma weaklycoupled (I' & 2) this would be the Debye length AD=a,/N. For the strongly coupled plasma, a betterestimate is the ion-disk radius associated with acharge of 1/m: A=&2a, . From the size we canestimate the energy required to make a particle.The charge accumulated around the phantom inthe Debye-Huckel approximation is

For m--3, these estimates are 0.062e'/a, and0.038e'/a, . This compares well with the value0.033e'/ao estimated from the numerical four-partiele solution in the manner

~ =-.'(E(q. -') E(e.")—2E(y, )), (18)where E(g, ) denotes the eigenvalue of the numer-ical analog of P,. This expression averages theelectron and hole creation energies while sub-tracting off the error due to the absence of V.I have performed two-component hypernettedchain calculations for the energies of P, "o and

I obtain (0.022 a 0.002)e'/ao and (0.025+ 0.005)e'/a, . If we assume a value e = 13 forthe dielectric constant of GaAs, we obtain 0.02e'/cap ~4 K when Ho = 150 kG.The energy to make a particle does not depend

on z„so long as its distance from the boundaryis greater than its size. Thus, as in the single-partiele problem, the states are degenerate andthere is no kinetic energy. We can expand thecreation operator as a power series in z, :

E~, = gX, (, ~ ~, z,)z,'- .j=0

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These A. , are the elementary symmetric poly-nomials, ' the algebra of which is known to spanthe set of symmetric functions. Since every anti-symmetric function ean be written as a sym. -metric function times y„ these operators andtheir adjoints generate the entire state space.It is thus appropriate to consider them X lin-early independent particle creation operators.The state described by g is incompressible

because compressing or expanding it is tanta-mount to injecting particles. If the area of thesystem is reduced or increased by ~A the en-ergy rises by &U = o ss

~M ~. Were this an elast-

ic solid characterized by a bulk modulus B, wewould have 5U= —,'B(5A)'/A. Incompressibilitycauses the longitudinal collective excitationroughly equivalent to a compressional soundwave to be absent, or more precisely, to havean energy - 4 in the long-wavelength limit. Thisfacilitates current conduction with no resistiveloss at zero temperature. Our prototype forthis behavior is full Landau level (m = 1) forwhich this collective excitation occurs at R~, .The response of this system to compressivestresses is analogous to the response of a type-II superconductor to the application of a magneticfield. The system first generates Hall currentswithout compressing, and then at a criticalstress collapses by an area quantum ppl2~QO'

VOLUME 50, +UMBER 18 PHYSICAL REVIEW LETTERS 2 MA+ 198$

and nucleates a particle. This, like a flux line,is surrounded by a vortex of Hall current rotat-ing in a sense opposite to that induced by thestress.The role of sample impurities and inhomogenei-ties in this theory is the same as that in mytheory of the ordinary quantum Hall effect. ' Theelectron and hole bands, separated in the im-purity-free case by a gap 2A, are broadenedinto a continuum consisting of two bands of ex-tended states separated by a band of localizedones. Small variations of the electron densitymove the Fermi level within this localized stateband as the extra quasiparticles become trappedat impurity sites. The Hall conductance is (1/m)& (e'/h) because it is related by gauge invarianceto the charge of the quasiparticles e* by o H,z=e*e/h, whenever the Fermi level lies in alocalized state band. As in the ordinary quantumHall effect, disorder sufficient to localize all thestates destroys the effect. This occurs when thecollision time w in the sample in the absence ofa magnetic field becomes smaller than T &h/4.I wish to thank H. DeWitt for calling my atten-tion to the Monte Carlo work and D. Boercker

for helpful discussions. I also wish to thankP. A. Lee, D. Yoshioka, and B. I. Halperin forhelpful criticism. This work was performedunder the auspices of the U. S. Department ofEnergy by Lawrence Livermore National Labora-tory under Contract No. W-7405-Eng-48.

'D. C. Tsui, H. L. Stormer, and A. C. Qossard, Phys.Rev. Lett. 48, 1559 (1982).'R. B. Laughlin, Phys. Rev. B 27, 3383 (1983).3J. M. Caillol, D. Levesque, J. J. leis, and J. P.

Hansen, J. Stat. Phys. 28, 325 (1982).D. Yoshioka and H. Fukuyama, J. Phys. Soc. Jpn.

47, 394 (1979};D. Yoshioka and P. A. Lee, Phys. Rev.B 27, 4986 {1983),and private communication.~B. Jancovici, Phys. Rev. Lett. 46, 386 {1981). -I=-2

corresponds to a full Landau level, for which the totalenergy equals the Hartree-Fock energy P~/8 e2/ao. -This correspondence may be viewed as the underlyingreason an exact solution at 1 = 2 exists.6W. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46,

738 (1981).S. Lang, Algebra (Addison-Wesley, Reading, Mass. ,

1965},p. 132.'R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).

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