VOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10JULY1989 gels in order to obtain some of the observed fractions. occupied single-particle states in each LL is N/p.Since It is also worth mentioning that the present scheme natu- the largest power of a z; in Zm is 2m(N-1), y has rally produces the experimentally observed sequences 2m(N-1)+Np single-particle states occupied in of fractions converging to (for m-1), to (for each LL, which immediately yields a filling factor 2), to 4(hole analog of 4) P/(2mp+1)in the thermodynamic limit. Thus the state In the following I will construct explicit trial wave yi(unlike ptp) satisfies the fundamental requirement functions, analogous to the Laughlin wave functions, that the filling factor obtained by counting the total which have the correlations discussed above. The Ham- number of states agrees with that obtained from the iltonian for N noninteracting electrons (N-oo)at Alux-counting argument (i. e the number of flux quanta +eA, /c), where A; is chosen so as to produce a un? piercing the sample is equal to the number of single- form magnetic field in the +z direction of strength such partially projects the single-particle states of the higher that there is an average flux of p po per electron. The LLs into the lowest LL. Write corresponding ground-state wave functions are y+p with y-p=Y+p. We first consider the fractions p/(2mp+1) Az2m∏∏5-1(z1+) hich are obtained by starting from y+p. Gauge flux tubes 2moo are attached to each electron by adding to the vector potential A a singular gauge potential 0, II (2n2+11)-12-112.-n2, 2 VOjk, where A is the anitsymmetrization operator, Sl,s are the single-particle states, 7=0, 1 is the LL ind where Ojk is defined by (zj -z)-1z-Ik lexp(ie/k), and s-O,., N/p-1 is the angular momentum quan and zj=;+iy denotes the position (xj, yj) of the jth tum number. Zm is a sum of terms of type Ilf-1z7 particle as a complex number. The new ground-state with 2i t -mN(N-1), where t, is typically a large wave function is! power (in the thermodynamic limit) of order mN. Thus, in each term of y 4p the coordinate zi of a particle ap- 0-12-x1my+ pears as the product z, S,s(z,). For ty of order mN, this product lies almost entirely in the lowest LL. Expanding Clearly this is not an appropriate wave function for it as a sum of single-particle states, describing the FQHE of electrons. Following the analo- gy of the Laughlin wave functions, and for the reasor z/5, ak5+1+k-1, related (unnormalized trial wave function one can show that the ratio ak +ilak is of order 1/vN e,, the amplitude of z, 5i, s (zy is smaller by a factor of order I/vN in each successively higher LL. Thus in here Z=ll <k(z -zk)a This wave function has the yi p the amplitude is expected to be in general much same topological structure as dm, and also describes larger for the terms which have a greater number of the electrons carrying gauge flux tubes of strength 2moo. I lowest LL states occupied. Furthermore, there are mani this state addition of flux tubes is accompanied by a festly terms, with extremely large amplitudes, which change in the size of the system in such a way as to keep have only the lowest LL occupied. This implies that, un he total flux per unit area (i. e, the magnetic field)con- less there are some very strange cancellations, the state w2 lies predominantly in the lowest LL in the thermo This state has the following properties: (i)For p=1, dynamic limit. (vi)Last, y i p is expected to be a good w2? is identical with the corresponding Laughlin state. variational state in the presence of repulsive interactions, (ii)Since V+p is determined almost completely by the because both the factors Zm and Y+p are very efficient Pauli principle, and has little dependence on interelec- in keeping the electrons apart. This is a direct sense in tron interactions, y*p is also largely insensitive to the in- which the correlations of the higher LL are utilized to teractions. This is explicitly the case for the Laughlin obtain a low-energy state. Thus we believe that the states which have been found to be very accurate for a states y p possess all the necessary properties of a variety of interelectron interactions. (i) It describes an reasonable trial state. At present we are working to- electron gas of uniform density. This follows straightfo wards a quantitative test, which is complicated due to wardly from the fact that y+p describes an electron gas the complex structure and the inherent thermodynamic of uniform density. o an eigenstate of the angu- nature of these states. The form of the incompressible ar momentum. (iv) One can read off the filling factor state at v-p/(2mp-1), which is obtained starting from from the wave function. Take y +p with p LL's com- y-p, is not as obvious. pletely occupied in a disk-shaped region; the number of Normally the ground state is expected to be completeVOLUME 63, NUMBER 2 PHYSICAL REVIEW LETTERS 10 JULY 1989 levels in order to obtain some of the observed fractions. It is also worth mentioning that the present scheme natu￾rally produces the experimentally observed sequences ' of fractions converging to —, ' (for m=1), to —, ' (for m =2), to —, ' (hole analog of —, ' ), etc. In the following I will construct explicit trial wave functions, analogous to the Laughlin wave functions, which have the correlations discussed above. The Ham￾iltonian for N noninteracting electrons (N ~ ~) at filling factor p is given by Ho =Pj.=i (2m, ) ' (pj +eAj/c), where Aj is chosen so as to produce a uni￾form magnetic field in the T- z direction of strength such that there is an average flux of p '&0 per electron. The corresponding ground-state wave functions are W ~~ with We first consider the fractions p/(2mp+1) which are obtained by starting from ++~. Gauge flux tubes 2m&0 are attached to each electron by adding to the vector potential Ai a singular gauge potential' " i gi, =(2ir2'+'lis!) ' e!' 2 s —iz i'/2 occupied single-particle states in each LL is N/p. Since the largest power of a zj in Z is 2m(N —1), +~~ has 2m (N —1)+Np ' single-particle states occupied in each LL, which immediately yields a filling factor p/(2mp+1) in the thermodynamic limit. Thus the state ++p (unlike @+„)satisfies the fundamental requirement that the filling factor obtained by counting the total number of states agrees with that obtained from the flux-counting argument (i.e., the number of flux quanta piercing the sample is equal to the number of single￾particle states in each LL). (v) The factor Z in ++~ partially projects the single-particle states of the higher LL's into the lowest LL. Write A, = —2m 2z p(I ~, ) where Oik is defined by (zj —zk) = izj —zk i exp(i8jk), and zj =xj+iyj denotes the position (xj,yj) of the jth particle as a complex number. The new ground-state wave function is ' ' (zj —zk)' Clearly this is not an appropriate wave function for describing the FQHE of electrons. Following the analo￾gy of the Laughlin wave functions, and for the reasons mentioned below, we write instead the following closely related (unnormalized) trial wave function @2m+p Z2m@ +p where Z' =Qj & k (zi —zk ) '. This wave function has the same topological structure as ++~, and also describes electrons carrying gauge flux tubes of strength 2m&0. In this state addition of flux tubes is accompanied by a change in the size of the system in such a way as to keep the total flux per unit area (i.e., the magnetic field) con￾stant. This state has the following properties: (i) For p =1, ++i is identical with the corresponding Laughlin state. (ii) Since ++~ is determined almost completely by the Pauli principle, and has little dependence on interelec￾tron interactions, ++~ is also largely insensitive to the in￾teractions. This is explicitly the case for the Laughlin states which have been found to be very accurate for a variety of interelectron interactions. (iii) It describes an electron gas of uniform density. This follows straightfor￾wardly from the fact that ++~ describes an electron gas of uniform density. It is also an eigenstate of the angu￾lar momentum. (iv) One can read off the filling factor from the wave function. Take ++~ with p LL's com￾pletely occupied in a disk-shaped region; the number of where A is the anitsymmetrization operator, gi, are the single-particle states, l =0, . . . ,p —1 is the LL index, and s=O, . . . ,N/p —1 is the angular momentum quan￾tum number. Z2 is a sum of terms of type Qj ——izji' with Pitj =mN(N 1), where—tj is typically a large power (in the thermodynamic limit) of order mN. Thus, in each term of I+p the coordinate zj of a particle ap￾pears as the product zPgi, (zj). For tj of order rnN, this product lies almost entirely in the lowest LL. Expanding it as a sum of single-particle states, fj Zj gi, s X ak4k, + is+kS—i i k=0 one can show that the ratio akil/ak is of order I/JN; i.e., the amplitude of zji'gi, (zj) is smaller by a factor of order I/JN in each successively higher LL. Thus in +~~ the amplitude is expected to be in general much larger for the terms which have a greater number of the lowest LL states occupied. Furthermore, there are mani￾festly terms, with extremely large amplitudes, which have only the lowest LL occupied. This implies that, un￾less there are some very strange cancellations, the state ++~ lies predominantly in the lowest LL in the thermo￾dynamic limit. (vi) Last, ++~ is expected to be a good variational state in the presence of repulsive interactions, because both the factors Z and ++~ are very efficient in keeping the electrons apart. This is a direct sense in which the correlations of the higher LL are utilized to obtain a low-energy state. Thus we believe that the states ++~ possess all the necessary properties of a reasonable trial state. At present we are working to￾wards a quantitative test, which is complicated due to the complex structure and the inherent thermodynamic nature of these states. The form of the incompressible state at v p/(2mp —1), which is obtained starting from +—~, is not as obvious. Normally the ground state is expected to be complete- 201