VOLUME 50, NUMBER 18 PHYSICAL REVIEW LETTERS 2MAY 198 Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations R.B.Laughlin Lawrence Livermore National Laboratory, University of California, Livermore, California 94550 (Received 22 February 1983) This Letter presents variational ground-state and excited-state wave functions which describe the condensation of twodimension- electron gas into new state of matter. PACS numbers: 71.45.Nt, 72.20.My,73.40.Lq The " effect, recently discovered by Tsu sistent with all the experimental facts and ex- Stormer, and Gossard, results from the conden-plai the effect. The ground state is a new state sation of the two-dimensional electron gas in a of matter, a quantum fluid the elementary exci- GaAs-Ga, Aly-As heterostructure into a new ty tations of which, the quasielectrons and quasi- of collective ground state. Important experimen- noles, are fractionally charged. I have verified tal facts are the following: (1)The electrons con- the correctness of these wave functions for the level. (2)They are capable onalization of the many-body Ham- current d have a iltonian is possible. I predict the existence of a Hall conductance o sequence of these ground states, decreasing in of the electron density do not affect either con- density and terminating in a Wigner crystal. ductivity, but large o Let us consider two-dimensional electron f 150 kgpe field of 150. ( The effect occurs in sometic agnetic gas in the x-y plane subjected to a magnetic field ample samples but not in others. T se of this vector potential=H xy-y] and write the Letter is to report variational ground-state and eigenstates of the ideal single-body Hamiltonian excited-state wave functions that I feel are con-Hsp()v-(e/c)]2 in the manner ,n)=(++ n ! -12 exp[(+ (1) with the cyclotron energy=h(e/mc)and the magnetic length a =(/mw )1/2 =(nc/ )/2 states of the lowest Landau level as set to 1. We have Hsplm, n)=(n+2)m, n). (2) m)=(2m+m)-1 2mexp2(-z2) (3) The manifold of states with energy n+ consti- tutes the nth Landau level. I abbreviate the where=x+iy. Im) is an elgenstate of angular momentum with eigenvalue m. The many-body Hamiltonian is H={(n/i)-(e/c)A,2+(2}+e2/12-2n, (4) > where j and run over the N particles and Visa potential generated by a uniform neutralizing m about the center of mass are of the form background. =(z1-z2)m(z1+z2)exp[-a(z12+z22).(5) could be understood in terms of the states in the My present theory generalizes this observation to lowest I andau level solely. With e2/a hwc, N particles. the situation in the experiment, quantization of I write the ground state as a product of Jastrow interelectronic spacing follows from quantization functions in the manner of angular momentum: The only wave fun ctions ={f(z-n)}exp(-2),(6) composed of states in the lowest Landau level mome which describe orbiting with angular momentum and minimize the energy with respect to f.We 1983 The American Physical Society 1395
VOLUME 50 NUMBER 18 PHYSICAL REVIEW LETTERS 2MAY1983 bserve that the condition that the electrons lie TABLE I. Projection of variational three-body wave in the lowest Landau level is that f(z) be poly functionsψ in the manner伸mlm)/({钟nlψm)(4n nomial in z. The antisymmetry of p requires that xom)2.4m is the lowest-energy eigenstate of angu f be odd. Conservation of angular momentum re nomial of degree M, where M is the total angular ctronic potential of either 1//,-In(r), or exp(-r2/ quires that Il f(2j-zkbe a homogeneous poly- momentum. We have, therefore, f(2)=2, with m odd. To determine which m minimizes the en- m 1/y exp ergy, I write m2={71 small m is more negative than that of a charge I generate the elementary excitations of *m by ity wave(CDw). It is given in terms of the piercing the fluid at zo with an infinitely thin ial distribution function g(r) of the ocp b solenoid and passing through it a flux quantum △φ=hc/ e adiabatically.. The effect of this opera Utot =fre[g(r)-1]rdr. tion on the single-body wave functions is (2-2)exp(-是|z12)-(z-20)m“exp(-划1212.(12) In the limit of large r, Utot is approximated Let us take as approximate representations of these excited states +磊0=Ag mxp(-An|212)江(x4-20)江<(2-2), …A(,江,-4列
VOLUME 50 NUMBER 18 PHYSICAL REVIEW LETTERS 2MAY1983 for the quasihole and quasielectron, respectively. For m=3, these estimates are 0.062e2/an and or four particles, I have projected these wave 0.038e2/an. This compares well with the value functions onto the analogous ones computed nu 0.033e2/an estimated from the numerical four merically. I obtain 0. 998 for ya o and 0. 994 for particle solution in the manner *s -. I obtain 0.982 for 73 +0=II (i-2)13, which is 43 to with the center-of-mass motion re- △E(3+E(J30)-2E(v3)}, moved here E(3) denotes the eigenvalue of the numer These excitations are particles of charge 1/m. ical analog of 43. This expression averages th To see this let us write 14+4012 as e"Ba,with electron and hole creation energies while sub tracting off the error due to the absence of v 中=中-21lnz1-zo I have performed two-component hypernetted (15) chain calculations for the energies of 43+ao and dp' describes an ocP interacting with a phantom 43-0. I obtain(0.022+0.002)e2/a, and(0.025 point charge at zo. The plasma will completely #0.005)e/ao. If we assume a value E=13for creen this phantom by accumulating an equal and the dielectric constant of GaAs, we obtain 0. 02e2/ opposite charge near zo. However, since the ∈ao-4 K when H=150kG plasma in reality consists of particles of charge The energy to make a particle does not depend 1 rather than charge m, the real accumulated on zo, so long as its distance from the boundary harge is 1 /m. Similar reasoning applies to "z0 is greater than its size. Thus, as in the single if we approximate it as Il, (2j-20)-"Pe Y3, where particle problem, the states are degenerate and P. is a projection operator removing all con- there is no kinetic energy. We can expand the gurations in which any electron is in the single- creation operator as a power series in a ody state(2-20)exp(-4 lz 12). The projection of this approximate wave function onto da "40 for four A=2A,(21…,2)20 (19) particles is 0. 922. More generally, one observes that far away from the solenoid, adiabatic addi g are the elementary symmetric poly tion of A moves the fluid rigidly by exactly one nomials, the algebra of which is known to span state, per Eq.(12). The charge of the particles the set of sy mmetric functions. Since every anti is thus 1/m by the Schrieffer counting argument, symmetric function can be written as a sym The size of these particles is the distance over metric function times y1, these operators and which the OCP screens. Were the plasma weakly their adjoints generate the entire state space. coupled(r 2)this would be the Debye length Ap= It is thus appropriate to consider them N lin ao/v2. For the strongly coupled plasma, a better early independent particle creation operator estimate is the ion-disk radius associated with a The state described by fm is incompressible charge of 1/: R=v2 an. From the size we can because compressing or expanding it is tanta estimate the energy required to make a particle. mount to injecting particles. If the area of the The charge accumulated around the phantom in system is reduced or increased by &a the en- the Debye-Huckel approximation i ergy rises by 6U=0m Al dA. Were this an elast ic solid characterized by a bulk modulus b, we op=2m Ko(r/aD), would have 6U=2B(6A)2/A. Incompressibility causes the longitudinal collective excitation here k. is a modified Bessel function of the roughly equivalent to a compressional sound second kind. The energy required to accumulate wave to be absent, or more precisely, to have it an energy -a in the long-wavelength limit. This ¨如p。T1e2 facilitates current conduction with no resistive (16) loss at zero temperature. Our prototype fc this behavior is full Landau level(m=1)for This estimate is an upper bound, since the plasma which this collective excitation occurs at hi is strongly coupled. To make a better estimate The response of this system to compressive let 6p=o inside the ion disk and zero outside, to stresses is analogous to the response of a type obtain II superconductor to the application of a magnetic field. The system first generates Hall currents △disk"2 without compressing, and then at a critica stress collapses by an area quantum m2a 2 1397
VOLUME 50 NUMBER 18 PHYSICAL REVIEW LETTERS 2MAY1983 and nucleates a particle. This, like a flux line, for helpful discussions. I also wish to thank is surrounded by a vortex of Hall current rotat P. A, Lee, D. Yoshioka, and B I. Halperin f ng in a sense opposite to that induced by the helpful criticism. This work was performed stress under the auspices of the U. S. Department of The role of sample impurities and inhomogenei- Energy by Lawrence Livermore National Labora ties in this theory is the same as that in my tory under Contract No W-7405-Eng-48 theory of the ordinary quantum Hall effect.The electron and hole bands, separated in the im purity-free case by a gap 24, are broadened into a continuum consisting of two bands of ex D. C. Tsui, H. L Stormer, and A. C. Gossard, Phys tended states separated by a band of localized Rev,Iet,48,1559(1982). ones. Small variations of the electron density 2R. B. Laughlin, Phys. Rev. B 27, 3383(1983) move the Fermi level within this localized state 3J. M. Caillol, D. Levesque, J.J. Weis, and J. P band as the extra quasiparticles become trapped Hansen, J. Stat. Phys. 28, 325 (1982) at impurity sites. The Hall conductance is (1/m) 47, 394(1979); D. Yoshioka and P.A. Lee, Phys. Rev X(e/h)because it is related by gauge invariance B 27, 4986(1983), and private communication to the charge of the quasiparticles e* by o hall B. Jancovici, Phys. Rev. Lett. 46, 386(1981).T=2 e*e/h, whenever the Fermi level lies in a corresponds to a full Landau level, for which the total localized state band. As in the ordinary quantum energy equals the Hartree-Fock energy -Vm/8 e/ap- Hall effect disorder sufficient to localize all the This correspondence may be viewed as the underlying states destroys the effect. This occurs when the 6W. P. Su and J. R. Schrieffer, Phys. Rev. Lett. 46 collision time T in the sample in the absence of a magnetic field becomes smaller than T<h/A Mass I wish to thank H. De witt for calling my atten 1965),p.13 tion to the monte carlo work and d, boercker R. B. Laughlin, Phys. Rev. B 23, 5632(1981)