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Calculations of the fractional quantized Hall effect
Surface Science 170 (1986) 115-124 North-Holland, Amsterdam
115
C A L C U L A T I O N S OF T H E F R A C T I O N A L Q U A N T I Z E D H A L L EFFECT Bertrand I. H A L P E R I N Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA Received 17 July 1985; accepted for publication 13 September 1985
1. Introduction The purpose of this paper is to review several aspects of the current theory of the fractional quantized Hall effect, and also to present the results of some recent calculations by Morf and myself [1]. I shall confine my review to the study of the ideal two-dimensional electron system, at zero temperature, in a uniform positive background. The results of such studies enable one to make predictions also about the behavior at very low temperatures and very small impurity concentrations, but they do not answer many questions of experimental interest, which involve larger concentrations of impurities, a n d / o r higher temperatures [2].
2. Current theoretical picture As is well known, the key dimensionless parameter for two-dimensional electrons in a strong magnetic field is the Landau-level filling factor -=
(1)
where n is the two-dimensional electron density, and l o = ( h c / e B ) 1/2 is the magnetic length. In this review, I shall discuss the situation of v ~< 1, and I shall assume that the electron spins are fully aligned in the direction of the applied magnetic field. In order to understand the observations of the fractional quantized Hall effect, it is necessary to assume that the energy per electron E / N , when plotted against v, has a series of downward pointing cusps, occurring at certain rational filling factors with odd denominators [3-5]. A basic question in any theory of the fractional quantized Hall effect is the nature of the stable ground state at these special values of the filling factor v. According to our current understanding, the stable ground states are in each case a translationally invariant "liquid" state [4,5]. In particular, we believe 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
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B.L Halperin / Calculations of fractional QHE
that the structure factor S(k) is a smooth function of the wavevector k, and is a finite, bounded function, for all k ~ 0 [6]. In 1983, Laughlin [5] proposed an approximate trial wavefunction for the "elementary" fractional quantized Hall states, I, = 1/m, where m is an odd integer. This trial wavefunction appears to be very close to the true ground state of the system (at least for u = 1/3), and can also be used to explain the Hall plateaus at i, = 1 - ( l / m ) , because of the particle-hole symmetry in the problem [7]. In order to explain the observed Hall plateaus at other rational values of v, various generalizations of Laughlin's ideas have been proposed [8-10]. As was first noted by Laughlin [5], the elementary charged excitations in the quantized hall states should be quasiparticles with fractional charge. (Specifically, the charge is +_qe, where q ~ is the denominator of the fraction u.) The energy cost Eg to produce a separated pair of quasiparticles of opposite charge may then be related to the discontinuity in slope of the energy curve by
Eg
=q(-~ d E ~x
dE
T ~ ,, j"
(2)
Analyses of the neutral excitations, such as the quasiexciton constructed of a bound pair of oppositely charged quasiparticles, support the idea that there is also an energy gap for production of neutral excitations in the quantized Hall states [11-14]. The properties of the quantized Hall states mentioned above, for the ideal case of no impurities, are sufficient to explain the central features of the quantized Hall effect, when a small density of impurities is added - viz., the absence of dissipation at zero temperature (a,, = 0) and the plateau in the Hall conductance (o,~ = v~e2/h), which exist over a range of filling factors near the stable state u~ [4,5,10]. In order to fill in the details of the above outline, however, it is necessary to perform quantitative calculations, or at least to develop a method of estimating the various quantities of interest. One basic quantity is the energy per particle in the quantized Hall ground state. Particularly, it is of interest to compare this energy with the energy of other possible ground states - for example a Wigner crystal [15], which would not display the quantized Hall effect. Of even greater interest is the energy gap Eg. In the limit of small impurity potential, the thermal activation energy for electrical resistance, at the filling factor corresponding to the quantized Hall value, should be simply equal to Eg/2. Other quantities which are of interest (at least to theorists) include the frequency spectrum for neutral excitations, the structure factor S(k), the matrix elements associated with various external perturbations, etc. The most direct method of carrying out quantitative calculations is an exact diagonalization of the Hamiltonian matrix for a small number of electrons, in the lowest Landau level. The most useful geometries for such calculations have been the rectangle (or parallelogram) with periodic boundary conditions, first
B.L Halperin / Calculationsof fractional QHE
117
exploited by Yoshioka, Halperin and Lee [4], and the spherical geometry, first proposed by Haldane 9. Because the size of the Hamiltonian matrix goes up very rapidly with the number of electrons considered, it seems that these methods are restricted to particle number N ~< 10. Fortunately, the dependence on particle number has proved to be remarkably weak, so that very useful results have been obtained even from such small systems [4,14,16]. Very recently, Chui [6] has employed a renormalization group method to obtain an approximate diagonalization of a system of 16 electrons at u = 1/3. It remains to be seen, however, whether this type of approximation is more reliable than the exact diagonalization of a small system. Another method of calculation is the evaluation of the energy of specified trial wavefunctions, using molecular dynamics or Monte Carlo techniques [1,5,17]. The number of particles employed in these calculations can be much larger than for the exact diagonalization methods - for example several hundred particles were used for the ground states in ref. [17], and 72 particles were used in a recent evaluation of the quasiparticle energy at u = 1/3, by Morf and Halperin [1]. On the other hand, evaluation by simulation methods can be used only for a restricted class of trial wavefunctions, and the accuracy is also limited, in practice, by statistical errors. Finally there exist a variety of approximate methods of calculation. One method, exploited by Laughlin [5,10,12], uses the hypernetted chain approximation to estimate the energy of various trial wavefunctions. Girvin, MacDonald and Platzman [13] employed a variational method to estimate the quasiexciton dispersion relation, using as input the structure function S ( k ) obtained from a molecular dynamics evaluation of Laughlin's wavefunction for the ground state at i, = 1/3. The present author has proposed an approximate energy formula, based on the hierarchical construction [9] of the quantized Hall states, in which the quasiparticles at various stages are treated as point particles, with fractional charge and obeying fractional statistics [18]. One application of the approximate energy formula of ref. [18] is to provide an estimate of the energy gap Eg. The gap at a stable filling factor uS has the approximate form
Eg ( us) = q~/2f( qs_,/qs),
(3)
where q51 is the denominator of the fraction us, and qs--~l is the corresponding quantity for the parent state in the hierarchy. The form of the function f must be determined by fitting to numerical calculations of the energy gap at u = 1 / 3 and other fractions with small denominator. It is expected, however, that f will be a relatively slowly varying function of its argument, so that for fractions with large denominator, the gap should vary roughly as the inverse 5 / 2 power of the denominator. In the limit of very weak impurity scattering, the width of a quantized Hall plateau should be determined by the melting density of the Wigner crystal of
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B.L Halperin / Calculationsoffractional QHE
quasiparticles, which is formed to accommodate the charge deviation from the stable filling factor us [8]. If one treats the quasiparticles as point charges obeying fractional statistics, one finds that the Hall conductance is constant over the range us - 3u < u < us + 3u, where 2 &' --- Cqs,
(4)
and c is a constant approximately equal to 1 / 6 [15,17]. In practice, however, the width of an observed Hall plateau is likely to be considerably affected by the finite impurity potential [19].
3. Computer simulations 3.1. Definitions In this section, I shall discuss some recent results from molecular dynamics simulations of trial wavefunctions and shall compare these results with other calculations. All result cited are for electrons in a uniform positive background, interacting with the pure Coulomb potential, V ( ~ -- rj) = e 2 / c I ~ -- t) ],
where ~ is the background dielectric potential, and r, is the position of the ith particle in the x - y plane. For p ~ 1, and h % >> e2/~.lo, we may write E/N
-
u(p)e2/,lo,
(5)
where we have subtracted off the constant kinetic energy, ½h%, and the Zeeman energy - l g ~ B B , and we are neglecting corrections which are higher order in the ratio e2/~loh~c. According to eqs. (2) and (5), the energy gap Eg at a stable filling factor us may be expressed, in units of e2/~.lo, as Eg = quS[ u'(u + ) - u'( u~-)],
(6)
where the prime indicates the derivative. There are several possible definitions of the separate energies for the quasielectron and quasihole which are all compatible with the above energy gap. We may define the "gross energy" of the quasielectron and quasihole, respectively, as Cqe ( U s ) =
q[us u,(u s+ ) + u(us)] =qd-NdE ~:'
,qh(lVs) =
_q[UsU,(U; ) + U(Us)] = - q d E , .
(7a) (7b)
s
These are equal, respectively, to q times the energy to add or subtract one electron from the system, keeping the magnetic field fixed, and assuming that
B.L Halperin / Calculations of fractional QHE
119
the excess electronic charge breaks up into 1/q quasiparticles of charge + qe, widely separated from each other. In other to preserve charge neutrality, one should adjust the uniform background charge at the same time as one adds or subtracts an electron. A n alternate definition of the quasiparticle or quasihole energy is given by q/v times the energy to remove or to add one q u a n t u m of magnetic flux, keeping the area and the n u m b e r of particles fixed. These definitions may be expressed as (qe -,°ux, = Cqe __ 3 P s U ( P s ) ,
Cq h-(flux)__ £qh
+
sU(ps).
(8)
A third definition, called the " p r o p e r energy" of a quasiparticle, and denoted g-+, was employed in ref. [18], in the context of the hierarchical construction [9] of the quantized Hall states. The proper energy reduces to the energy to remove or add a flux q u a n t u m for the first level of the hierarchy, where q = v = l/m, with m an odd integer. The quasiparticle energies used by Laughlin [5,10] are also equivalent to the quantities ~(nu~) and ~'(fl"") defined ~qe ~qh above. We note that (9)
-(flux) -]'- ~qh -(flux)_ Eg = Cqe -- ~qe At- £qp"
The energy gap Eg is also equal to the sum of the proper energies g + + g- in all cases.
3.2. Trial wavefunctions and results Laughlin's trial wavefunction for the stable state at v =
~'~[zJ]---I-I(zs-zt)"~exp(s
-
'zk[2)4l~ '
1/m
is [5] (10)
where zj ~- xj - iyj. This wavefunction is a legitimate trial wavefunction for a collection of fermions in the lowest L a n d a u level, if m is an odd integer. It is a valid wavefunction for bosons in the lowest L a n d a u level if m is an even integer, and by extension, it m a y be considered to be a wavefunction for particles obeying fractional sffttistics, if m is a non-integer rational [16,18]. In any case, the probability distribution [akm[Zj][ 2 m a y be simulated by the Monte Carlo algorithm or the molecular dynamics method, appropriate to a two-dimensional o n e - c o m p o n e n t plasma. The plasma pair-correlation function gpl(r) may be evaluated, and integrated against the potential v ( r ) = e2/~r, to obtain the potential energy per particle in the wavefunction q'm, which we shall write as Upl(m), in units of e2/clo . Clearly Upl is a smooth function of its argument m. However, Upj(m) is only a valid u p p e r b o u n d for the electron potential-energy function u(v), at v = 1/m, in the special case where m is an odd integer.
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120
The function Upl(m) has been evaluated accurately by Levesque, Weis and MacDonald [17] for a series of values of m, using as many as 256 particles. Their results were fit, in the range 1 ~< m ~ 20, by the empirical formula 0.782133(0.211 ml/2 1 m 0"74
U p l ( m ) --
0.012) -k- m 1"7
(11) .
At the point m = 3, corresponding to v = 3, the actual calculated value of Up~ was Upl ( 3 ) = - 0 . 4 1 0 0 _+ 0 . 0 0 0 1 .
(12)
This gives the energy per particle of Laughlin's wavefunction, eq. (10), for the v = 1 / 3 state. By comparison, some results of exact diagonalization of the Hamiltonian for small systems are the values u ( 1 / 3 ) = -0.4152, -0.4127, and -0.4128 for N = 4, 5, and 6 particles, respectively, in a periodic rectangular geometry, calculated by Yoshioka et al. [4,16]. More significantly, Haldane and Rezayi [14] have found that the difference in energy between the exact ground state and Laughlin's wavefunction for the same finite system (six particles on a sphere) is smaller than one part in 2000. Laughlin proposed a trial wavefunction for a quasihole, in the state v = 1 / m , which can also be evaluated by standard simulation methods [5]. The wavefunction, for a quasihole at origin, is given by N
+ ( - ) [ z j ] = ~bm[zj] I-I zj. j=l
(13)
Morf and Halperin [1] have computed the energy of this trial function, at v = 1/3, and have obtained the following result, for a system of 72 particles in a disk geometry: g~ux)(1/3) = 0.0268 _+ 0.003.
(14)
This result is in excellent agreement with the result 0.026, obtained by Laughlin, using a modified hypernetted chain approximation [10]. Morf and Halperin have also evaluated the energy of two different trial wavefunctions for the quasiparticle, in the v = 1 / 3 state. One of these is the "derivative wavefunction" proposed by Laughlin [5] ~ktd+n)v--- f i
exp
4l~
~zj
j=l
I--I(zt-zk) m .
(15)
l
The second trial wavefunction is the " p a i r form", proposed by Halperin, in ref. [8]. This may be written d~+ ) - e a /
..
.
.
.
.
~b,,, j=
(z,
---
--
zj)(z2-z,)
where ~¢ is the antisymmetrlzing operator.
.
(16)
B.L Halperin / Calculationsof fractional QHE
121
Neither (15) nor (16) can be simulated directly by the usual Monte Carlo or molecular dynamics techniques. However, in both cases evaluation of the energy is possible using some mathematical manipulation [1]. I present here only the results of these calculations. Using Laughlin's derivative wavefunction, for N = 20, 30, 42, and 72 particles in a disk geometry, Morf and Halperin have obtained the extrapolated result, for N = oo, of -(flux) 'qe ( 1 / 3 ) = 0.073 + 0.008.
(17)
The quoted uncertainty is a subjective estimate of the uncertainty of extrapolation to N = oo, combined with the statistical error of the simulations. The largest system employed for the pair wavefunction (16) was N = 42, where the quasiparticle energy was found to be 0.066 + 0.006. The energy of Laughlin's wavefunction, for the same size system, was found to be identical, within the statistical uncertainty. It seems likely that there is a high degree of overlap between the two wavefunctions, despite the difference in their written forms. We note that our results for the quasielectron energy at v = 1 / 3 are significantly higher than the estimate 0.030 given by Laughlin in ref. [10], or the estimate 0.025 obtained recently by Chakraborty [20]. The estimates of Laughlin and Chakraborty were based on the same trial wavefunction as we have used to obtain the results (17), but involved the use of further approximations to obtain the energy. The present results imply that these further approximations were inaccurate by approximately a factor of two. Results for the quasielectron and quasihole energies may be combined to give an estimate for the energy gap, at v = 1/3: Eg = 0.099 + 0.009.
(18)
This result is substantially larger than the estimate Eg = 0.056 obtained by Laughlin, in ref. [10], but it is somewhat smaller than a recent estimate Eg = 0.11 given by Girvin, MacDonald and Platzman [13]. The result is quite close to the estimate by Haldane and Rezayi [14] of Eg -- 0.105 + 0.005 based on computations of the exact spectrum of up to seven electrons on a sphere. It must be remarked that experimental values of Eg, extracted from the thermal activation of the electrical resistance at the center of a quantized Hall plateau, have been consistently ~< 0.03e2/~1o [21], significantly smaller than any of the theoretical calculations. The precise reason for this discrepancy is not currently understood. Corrections to the theoretical value arising from admixture of the higher Landau levels (i.e., higher order in e2/~loh~c), as well as corrections due to the finite thickness of the electron layer, will certainly lower the energy gap somewhat. It does not appear likely, however, that these corrections are large enough to explain the observed discrepancies [22]. It seems more likely that the disagreement between theory and experiment is a
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B.L Halperin / Calculations of fractional QHE
measure of the complications in interpretation in the presence of the disordered potential due to the ionized impurities in the sample. Morf and Halperin have also evaluated the energy of a trial wavefunction for the 2 / 5 state, whose form is
412o
(Zk--ZI)2U(Z2i--Z2i-I) i
1H(zi--Zj)2 i
,
(19) where k and l run from 1 to N, while i and j run from 1 to ½N and Zi = ½(z2, + z2,-1) is the center of gravity of the ith pair. The potential energy per electron of this trial function was found to be - 0 . 4 1 4 + 0.002. This energy is much higher than the value u ( 2 / 5 ) = -0.435, obtained by Yoshioka [16] from exact diagonalization of systems with up to eight particles, in a rectangular cell with periodic boundary conditions. (In fact it is comparable in energy to the Wigner crystal state.) Thus it seems that eq. (19) is not a very good approximation to the true ground state at v = 2/5. Currently under way are simulations of an alternate trial function, in which the final factor (Z, - Zj) 2 in eq. (19) is replaced by the factor (Z2;Z2,_l "4- Z2jZ2j_ 1 --
2ZiZj).
Preliminary results indicate a significant improvement in the energy, as a result of this modification. Specifically, we find u(2/5) = - 0.430 _+ 0.002 [23]. According to the analysis of ref. [18], the energy per particle in the v = 2 / 5 state should be approximately related to the quasielectron energy of the parent u = 1/3, by [1] u ( 2 / 5 ) ~ - A + ~ qte-(flux)/1/q~ ~'/-'J,
(20)
where A = ~,3u o l ( 3 ) + ~v/3-uo,(5/3) -~ - 0 . 4 6 1 .
(21)
If we substitute the value quoted in eq. (17), for the quasiparticle energy, we obtain the estimate u ( 2 / 5 ) - - -0.424, which we consider a reasonable approximation of Yoshioka's result.
Acknowledgements The author is grateful for conversations with many colleagues, including especially R. Morf, R.B. Laughlin, P.M. Platzman, M. Rasolt, F.D.M. Haldane, D. Yoshioka, and P.A. Lee. This work was supported in part by NSF grant DMR-82-07431.
B.L Halperin / Calculationsof fractional QHE
123
References [1] R. Morf and B.I. Halperin, Phys. Rev. B33 (1986) (Feb. 15 issue, in press). [2] For experiments on the fractional quantized Hall effect, see for example, works cited as refs. [3] and [21] below; also see, A.M. Chang, P. Berglund, D.C.Tsui, H.L. St~rmer and J.C.M. Hwang, Phys. Rev. Letters 53 (1984) 997; E.E. Mendez, L.L. Chang, M. Heiblum, L. Esaki, M. Naughton, K. Martin and J. Brooks, Phys. Rev. B30 (1984) 1087; and experimental papers elsewhere in this volume. [3] D.C. Tsui, H.L. St~rmer and A.C. Gossard, Phys. Rev. Letters 48 (1982) 1559. [4] D. Yoshioka, B.I. Halperin and P.A. Lee, Phys. Rev. Letters 50 (1983) 1219; Surface Sci. 142 (1984) 155. [5] R.B. Laughlin, Phys. Rev. Letters 50 (1983) 1395. [6] Recently, S.T. Chui, T. Hakim and K.B. Ma (preprint) and S.T. Chui (preprint) have proposed that S(k) has a power law divergence at some finite value of k, in the quantized Hall ground state. The evidence in favor of this does not appear to be very strong, however. We may remark that the analysis of Girvin et al. (ref. [13], below) implies rigorously that there can be no energy gap for neutral excitations, if S(k) diverges at some value of k. [7] See, for example: H. Fukuyama, P.M. Platzman and P.W. Anderson, Phys. Rev. B19 (1979) 5211; D. Yoshioka, Phys. Rev. B29 (1984) 6833; S.M. Girvin, Phys. Rev. B29 (1984) 6012. [8] B.I. Halperin, Helv. Phys. Acta 56 (1983) 75. [9] F.D.M. Haldane, Phys. Rev. Letters 51 (1983) 605. [10] R.B. Laughlin, Surface Sci. 142 (1984) 163. [11] C. Kallin and B.I. Halperin, Phys. Rev. B30 (1984) 5655. [12] R.B. Laughlin, Physica 126B (1984) 254. [13] S.M. Girvin, A.H. MacDonald and P.M. Platzman, Phys. Rev. Letters 54 (1985) 581. [14] F.D.M. Haldane and E.H. Rezayi, Phys. Rev. Letters 54 (1985) 237. [15] See, P.K. Lam and S.M. Girvin, Phys. Rev. B30 (1984) 473; B31 (1985) 613E. [16] D. Yoshioka, Phys. Rev. B29 (1984) 6833; D. Yoshioka, private communication. [17] D. Levesque, J.J. Weiss and A.H. MacDonald, Phys. Rev. B30 (1984) 1056. [18] B.I. Halperin, Phys. Rev. Letters 52 (1984) 1583; Phys. Rev. Letters 52 (1984) 2390E. [19] R.B. Laughlin, M.L. Cohen, J.M. Kosterlitz, H. Levine, S.B. Libby and A.M.M. Pruisken, preprint. [20] T. Chakraborty, Phys. Rev. B31 (1985) 4026. [21] See, for example: A.M. Chang, M.A. Paalanen, D.C. Tsui, H.L. St~rmer and J.C.M. Hwang, Phys. Rev. B28 (1984) 6133; S. Kawaji, J. Wakabayashi, J. Yoshino and H. Sasaki, J. Phys. Soc. Japan 53 (1984) 1915; J. Wakabayashi, S. Kawaji, J. Yoshino and H. Sasaki, in: Proc. 17th Intern. Conf. on Physics of Semiconductors, San Francisco, 1984, Eds. D.J. Chadi and W.A. Harrison (Springer, Berlin, 1984); G. Ebert, K. von Klitzing, J.C. Maan, G. Remenyi, C. Probst, G. Weimann and W. Schlapp, J. Phys. C17 (1984) L775; J. Ihm and J.C. Phillips, J. Phys. Soc. Japan 54 (1985) 1506. [22] See, for example: A.H. MacDonald and G.C. Aers, Phys. Rev. B29 (1984) 5976; D. Yoshioka, J. Phys. Soc. Japan 53 (1984) 3740.
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B.I. Halperin / Calculationsoffractional QHE
[23] Note added in proof: Further Monte Carlo evaluations of this improved trial wavefunction have given u ( 2 / 5 ) = -0.4305 + 0.0005. By comparison, extrapolations of exact calculations of N~<10 electrons on a sphere give u ( 2 / 5 ) = - 0 . 4 3 3 1 . These results are from N. d ' A m brumenil, B.I. Halperin and R. Morf, unpublished work, and G. Fano, private communication.
115
C A L C U L A T I O N S OF T H E F R A C T I O N A L Q U A N T I Z E D H A L L EFFECT Bertrand I. H A L P E R I N Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA Received 17 July 1985; accepted for publication 13 September 1985
1. Introduction The purpose of this paper is to review several aspects of the current theory of the fractional quantized Hall effect, and also to present the results of some recent calculations by Morf and myself [1]. I shall confine my review to the study of the ideal two-dimensional electron system, at zero temperature, in a uniform positive background. The results of such studies enable one to make predictions also about the behavior at very low temperatures and very small impurity concentrations, but they do not answer many questions of experimental interest, which involve larger concentrations of impurities, a n d / o r higher temperatures [2].
2. Current theoretical picture As is well known, the key dimensionless parameter for two-dimensional electrons in a strong magnetic field is the Landau-level filling factor -=
(1)
where n is the two-dimensional electron density, and l o = ( h c / e B ) 1/2 is the magnetic length. In this review, I shall discuss the situation of v ~< 1, and I shall assume that the electron spins are fully aligned in the direction of the applied magnetic field. In order to understand the observations of the fractional quantized Hall effect, it is necessary to assume that the energy per electron E / N , when plotted against v, has a series of downward pointing cusps, occurring at certain rational filling factors with odd denominators [3-5]. A basic question in any theory of the fractional quantized Hall effect is the nature of the stable ground state at these special values of the filling factor v. According to our current understanding, the stable ground states are in each case a translationally invariant "liquid" state [4,5]. In particular, we believe 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation
116
B.L Halperin / Calculations of fractional QHE
that the structure factor S(k) is a smooth function of the wavevector k, and is a finite, bounded function, for all k ~ 0 [6]. In 1983, Laughlin [5] proposed an approximate trial wavefunction for the "elementary" fractional quantized Hall states, I, = 1/m, where m is an odd integer. This trial wavefunction appears to be very close to the true ground state of the system (at least for u = 1/3), and can also be used to explain the Hall plateaus at i, = 1 - ( l / m ) , because of the particle-hole symmetry in the problem [7]. In order to explain the observed Hall plateaus at other rational values of v, various generalizations of Laughlin's ideas have been proposed [8-10]. As was first noted by Laughlin [5], the elementary charged excitations in the quantized hall states should be quasiparticles with fractional charge. (Specifically, the charge is +_qe, where q ~ is the denominator of the fraction u.) The energy cost Eg to produce a separated pair of quasiparticles of opposite charge may then be related to the discontinuity in slope of the energy curve by
Eg
=q(-~ d E ~x
dE
T ~ ,, j"
(2)
Analyses of the neutral excitations, such as the quasiexciton constructed of a bound pair of oppositely charged quasiparticles, support the idea that there is also an energy gap for production of neutral excitations in the quantized Hall states [11-14]. The properties of the quantized Hall states mentioned above, for the ideal case of no impurities, are sufficient to explain the central features of the quantized Hall effect, when a small density of impurities is added - viz., the absence of dissipation at zero temperature (a,, = 0) and the plateau in the Hall conductance (o,~ = v~e2/h), which exist over a range of filling factors near the stable state u~ [4,5,10]. In order to fill in the details of the above outline, however, it is necessary to perform quantitative calculations, or at least to develop a method of estimating the various quantities of interest. One basic quantity is the energy per particle in the quantized Hall ground state. Particularly, it is of interest to compare this energy with the energy of other possible ground states - for example a Wigner crystal [15], which would not display the quantized Hall effect. Of even greater interest is the energy gap Eg. In the limit of small impurity potential, the thermal activation energy for electrical resistance, at the filling factor corresponding to the quantized Hall value, should be simply equal to Eg/2. Other quantities which are of interest (at least to theorists) include the frequency spectrum for neutral excitations, the structure factor S(k), the matrix elements associated with various external perturbations, etc. The most direct method of carrying out quantitative calculations is an exact diagonalization of the Hamiltonian matrix for a small number of electrons, in the lowest Landau level. The most useful geometries for such calculations have been the rectangle (or parallelogram) with periodic boundary conditions, first
B.L Halperin / Calculationsof fractional QHE
117
exploited by Yoshioka, Halperin and Lee [4], and the spherical geometry, first proposed by Haldane 9. Because the size of the Hamiltonian matrix goes up very rapidly with the number of electrons considered, it seems that these methods are restricted to particle number N ~< 10. Fortunately, the dependence on particle number has proved to be remarkably weak, so that very useful results have been obtained even from such small systems [4,14,16]. Very recently, Chui [6] has employed a renormalization group method to obtain an approximate diagonalization of a system of 16 electrons at u = 1/3. It remains to be seen, however, whether this type of approximation is more reliable than the exact diagonalization of a small system. Another method of calculation is the evaluation of the energy of specified trial wavefunctions, using molecular dynamics or Monte Carlo techniques [1,5,17]. The number of particles employed in these calculations can be much larger than for the exact diagonalization methods - for example several hundred particles were used for the ground states in ref. [17], and 72 particles were used in a recent evaluation of the quasiparticle energy at u = 1/3, by Morf and Halperin [1]. On the other hand, evaluation by simulation methods can be used only for a restricted class of trial wavefunctions, and the accuracy is also limited, in practice, by statistical errors. Finally there exist a variety of approximate methods of calculation. One method, exploited by Laughlin [5,10,12], uses the hypernetted chain approximation to estimate the energy of various trial wavefunctions. Girvin, MacDonald and Platzman [13] employed a variational method to estimate the quasiexciton dispersion relation, using as input the structure function S ( k ) obtained from a molecular dynamics evaluation of Laughlin's wavefunction for the ground state at i, = 1/3. The present author has proposed an approximate energy formula, based on the hierarchical construction [9] of the quantized Hall states, in which the quasiparticles at various stages are treated as point particles, with fractional charge and obeying fractional statistics [18]. One application of the approximate energy formula of ref. [18] is to provide an estimate of the energy gap Eg. The gap at a stable filling factor uS has the approximate form
Eg ( us) = q~/2f( qs_,/qs),
(3)
where q51 is the denominator of the fraction us, and qs--~l is the corresponding quantity for the parent state in the hierarchy. The form of the function f must be determined by fitting to numerical calculations of the energy gap at u = 1 / 3 and other fractions with small denominator. It is expected, however, that f will be a relatively slowly varying function of its argument, so that for fractions with large denominator, the gap should vary roughly as the inverse 5 / 2 power of the denominator. In the limit of very weak impurity scattering, the width of a quantized Hall plateau should be determined by the melting density of the Wigner crystal of
118
B.L Halperin / Calculationsoffractional QHE
quasiparticles, which is formed to accommodate the charge deviation from the stable filling factor us [8]. If one treats the quasiparticles as point charges obeying fractional statistics, one finds that the Hall conductance is constant over the range us - 3u < u < us + 3u, where 2 &' --- Cqs,
(4)
and c is a constant approximately equal to 1 / 6 [15,17]. In practice, however, the width of an observed Hall plateau is likely to be considerably affected by the finite impurity potential [19].
3. Computer simulations 3.1. Definitions In this section, I shall discuss some recent results from molecular dynamics simulations of trial wavefunctions and shall compare these results with other calculations. All result cited are for electrons in a uniform positive background, interacting with the pure Coulomb potential, V ( ~ -- rj) = e 2 / c I ~ -- t) ],
where ~ is the background dielectric potential, and r, is the position of the ith particle in the x - y plane. For p ~ 1, and h % >> e2/~.lo, we may write E/N
-
u(p)e2/,lo,
(5)
where we have subtracted off the constant kinetic energy, ½h%, and the Zeeman energy - l g ~ B B , and we are neglecting corrections which are higher order in the ratio e2/~loh~c. According to eqs. (2) and (5), the energy gap Eg at a stable filling factor us may be expressed, in units of e2/~.lo, as Eg = quS[ u'(u + ) - u'( u~-)],
(6)
where the prime indicates the derivative. There are several possible definitions of the separate energies for the quasielectron and quasihole which are all compatible with the above energy gap. We may define the "gross energy" of the quasielectron and quasihole, respectively, as Cqe ( U s ) =
q[us u,(u s+ ) + u(us)] =qd-NdE ~:'
,qh(lVs) =
_q[UsU,(U; ) + U(Us)] = - q d E , .
(7a) (7b)
s
These are equal, respectively, to q times the energy to add or subtract one electron from the system, keeping the magnetic field fixed, and assuming that
B.L Halperin / Calculations of fractional QHE
119
the excess electronic charge breaks up into 1/q quasiparticles of charge + qe, widely separated from each other. In other to preserve charge neutrality, one should adjust the uniform background charge at the same time as one adds or subtracts an electron. A n alternate definition of the quasiparticle or quasihole energy is given by q/v times the energy to remove or to add one q u a n t u m of magnetic flux, keeping the area and the n u m b e r of particles fixed. These definitions may be expressed as (qe -,°ux, = Cqe __ 3 P s U ( P s ) ,
Cq h-(flux)__ £qh
+
sU(ps).
(8)
A third definition, called the " p r o p e r energy" of a quasiparticle, and denoted g-+, was employed in ref. [18], in the context of the hierarchical construction [9] of the quantized Hall states. The proper energy reduces to the energy to remove or add a flux q u a n t u m for the first level of the hierarchy, where q = v = l/m, with m an odd integer. The quasiparticle energies used by Laughlin [5,10] are also equivalent to the quantities ~(nu~) and ~'(fl"") defined ~qe ~qh above. We note that (9)
-(flux) -]'- ~qh -(flux)_ Eg = Cqe -- ~qe At- £qp"
The energy gap Eg is also equal to the sum of the proper energies g + + g- in all cases.
3.2. Trial wavefunctions and results Laughlin's trial wavefunction for the stable state at v =
~'~[zJ]---I-I(zs-zt)"~exp(s
-
'zk[2)4l~ '
1/m
is [5] (10)
where zj ~- xj - iyj. This wavefunction is a legitimate trial wavefunction for a collection of fermions in the lowest L a n d a u level, if m is an odd integer. It is a valid wavefunction for bosons in the lowest L a n d a u level if m is an even integer, and by extension, it m a y be considered to be a wavefunction for particles obeying fractional sffttistics, if m is a non-integer rational [16,18]. In any case, the probability distribution [akm[Zj][ 2 m a y be simulated by the Monte Carlo algorithm or the molecular dynamics method, appropriate to a two-dimensional o n e - c o m p o n e n t plasma. The plasma pair-correlation function gpl(r) may be evaluated, and integrated against the potential v ( r ) = e2/~r, to obtain the potential energy per particle in the wavefunction q'm, which we shall write as Upl(m), in units of e2/clo . Clearly Upl is a smooth function of its argument m. However, Upj(m) is only a valid u p p e r b o u n d for the electron potential-energy function u(v), at v = 1/m, in the special case where m is an odd integer.
B.I. Halperin / Calculationsoffractional QHE
120
The function Upl(m) has been evaluated accurately by Levesque, Weis and MacDonald [17] for a series of values of m, using as many as 256 particles. Their results were fit, in the range 1 ~< m ~ 20, by the empirical formula 0.782133(0.211 ml/2 1 m 0"74
U p l ( m ) --
0.012) -k- m 1"7
(11) .
At the point m = 3, corresponding to v = 3, the actual calculated value of Up~ was Upl ( 3 ) = - 0 . 4 1 0 0 _+ 0 . 0 0 0 1 .
(12)
This gives the energy per particle of Laughlin's wavefunction, eq. (10), for the v = 1 / 3 state. By comparison, some results of exact diagonalization of the Hamiltonian for small systems are the values u ( 1 / 3 ) = -0.4152, -0.4127, and -0.4128 for N = 4, 5, and 6 particles, respectively, in a periodic rectangular geometry, calculated by Yoshioka et al. [4,16]. More significantly, Haldane and Rezayi [14] have found that the difference in energy between the exact ground state and Laughlin's wavefunction for the same finite system (six particles on a sphere) is smaller than one part in 2000. Laughlin proposed a trial wavefunction for a quasihole, in the state v = 1 / m , which can also be evaluated by standard simulation methods [5]. The wavefunction, for a quasihole at origin, is given by N
+ ( - ) [ z j ] = ~bm[zj] I-I zj. j=l
(13)
Morf and Halperin [1] have computed the energy of this trial function, at v = 1/3, and have obtained the following result, for a system of 72 particles in a disk geometry: g~ux)(1/3) = 0.0268 _+ 0.003.
(14)
This result is in excellent agreement with the result 0.026, obtained by Laughlin, using a modified hypernetted chain approximation [10]. Morf and Halperin have also evaluated the energy of two different trial wavefunctions for the quasiparticle, in the v = 1 / 3 state. One of these is the "derivative wavefunction" proposed by Laughlin [5] ~ktd+n)v--- f i
exp
4l~
~zj
j=l
I--I(zt-zk) m .
(15)
l
The second trial wavefunction is the " p a i r form", proposed by Halperin, in ref. [8]. This may be written d~+ ) - e a /
..
.
.
.
.
~b,,, j=
(z,
---
--
zj)(z2-z,)
where ~¢ is the antisymmetrlzing operator.
.
(16)
B.L Halperin / Calculationsof fractional QHE
121
Neither (15) nor (16) can be simulated directly by the usual Monte Carlo or molecular dynamics techniques. However, in both cases evaluation of the energy is possible using some mathematical manipulation [1]. I present here only the results of these calculations. Using Laughlin's derivative wavefunction, for N = 20, 30, 42, and 72 particles in a disk geometry, Morf and Halperin have obtained the extrapolated result, for N = oo, of -(flux) 'qe ( 1 / 3 ) = 0.073 + 0.008.
(17)
The quoted uncertainty is a subjective estimate of the uncertainty of extrapolation to N = oo, combined with the statistical error of the simulations. The largest system employed for the pair wavefunction (16) was N = 42, where the quasiparticle energy was found to be 0.066 + 0.006. The energy of Laughlin's wavefunction, for the same size system, was found to be identical, within the statistical uncertainty. It seems likely that there is a high degree of overlap between the two wavefunctions, despite the difference in their written forms. We note that our results for the quasielectron energy at v = 1 / 3 are significantly higher than the estimate 0.030 given by Laughlin in ref. [10], or the estimate 0.025 obtained recently by Chakraborty [20]. The estimates of Laughlin and Chakraborty were based on the same trial wavefunction as we have used to obtain the results (17), but involved the use of further approximations to obtain the energy. The present results imply that these further approximations were inaccurate by approximately a factor of two. Results for the quasielectron and quasihole energies may be combined to give an estimate for the energy gap, at v = 1/3: Eg = 0.099 + 0.009.
(18)
This result is substantially larger than the estimate Eg = 0.056 obtained by Laughlin, in ref. [10], but it is somewhat smaller than a recent estimate Eg = 0.11 given by Girvin, MacDonald and Platzman [13]. The result is quite close to the estimate by Haldane and Rezayi [14] of Eg -- 0.105 + 0.005 based on computations of the exact spectrum of up to seven electrons on a sphere. It must be remarked that experimental values of Eg, extracted from the thermal activation of the electrical resistance at the center of a quantized Hall plateau, have been consistently ~< 0.03e2/~1o [21], significantly smaller than any of the theoretical calculations. The precise reason for this discrepancy is not currently understood. Corrections to the theoretical value arising from admixture of the higher Landau levels (i.e., higher order in e2/~loh~c), as well as corrections due to the finite thickness of the electron layer, will certainly lower the energy gap somewhat. It does not appear likely, however, that these corrections are large enough to explain the observed discrepancies [22]. It seems more likely that the disagreement between theory and experiment is a
122
B.L Halperin / Calculations of fractional QHE
measure of the complications in interpretation in the presence of the disordered potential due to the ionized impurities in the sample. Morf and Halperin have also evaluated the energy of a trial wavefunction for the 2 / 5 state, whose form is
412o
(Zk--ZI)2U(Z2i--Z2i-I) i
1H(zi--Zj)2 i
,
(19) where k and l run from 1 to N, while i and j run from 1 to ½N and Zi = ½(z2, + z2,-1) is the center of gravity of the ith pair. The potential energy per electron of this trial function was found to be - 0 . 4 1 4 + 0.002. This energy is much higher than the value u ( 2 / 5 ) = -0.435, obtained by Yoshioka [16] from exact diagonalization of systems with up to eight particles, in a rectangular cell with periodic boundary conditions. (In fact it is comparable in energy to the Wigner crystal state.) Thus it seems that eq. (19) is not a very good approximation to the true ground state at v = 2/5. Currently under way are simulations of an alternate trial function, in which the final factor (Z, - Zj) 2 in eq. (19) is replaced by the factor (Z2;Z2,_l "4- Z2jZ2j_ 1 --
2ZiZj).
Preliminary results indicate a significant improvement in the energy, as a result of this modification. Specifically, we find u(2/5) = - 0.430 _+ 0.002 [23]. According to the analysis of ref. [18], the energy per particle in the v = 2 / 5 state should be approximately related to the quasielectron energy of the parent u = 1/3, by [1] u ( 2 / 5 ) ~ - A + ~ qte-(flux)/1/q~ ~'/-'J,
(20)
where A = ~,3u o l ( 3 ) + ~v/3-uo,(5/3) -~ - 0 . 4 6 1 .
(21)
If we substitute the value quoted in eq. (17), for the quasiparticle energy, we obtain the estimate u ( 2 / 5 ) - - -0.424, which we consider a reasonable approximation of Yoshioka's result.
Acknowledgements The author is grateful for conversations with many colleagues, including especially R. Morf, R.B. Laughlin, P.M. Platzman, M. Rasolt, F.D.M. Haldane, D. Yoshioka, and P.A. Lee. This work was supported in part by NSF grant DMR-82-07431.
B.L Halperin / Calculationsof fractional QHE
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References [1] R. Morf and B.I. Halperin, Phys. Rev. B33 (1986) (Feb. 15 issue, in press). [2] For experiments on the fractional quantized Hall effect, see for example, works cited as refs. [3] and [21] below; also see, A.M. Chang, P. Berglund, D.C.Tsui, H.L. St~rmer and J.C.M. Hwang, Phys. Rev. Letters 53 (1984) 997; E.E. Mendez, L.L. Chang, M. Heiblum, L. Esaki, M. Naughton, K. Martin and J. Brooks, Phys. Rev. B30 (1984) 1087; and experimental papers elsewhere in this volume. [3] D.C. Tsui, H.L. St~rmer and A.C. Gossard, Phys. Rev. Letters 48 (1982) 1559. [4] D. Yoshioka, B.I. Halperin and P.A. Lee, Phys. Rev. Letters 50 (1983) 1219; Surface Sci. 142 (1984) 155. [5] R.B. Laughlin, Phys. Rev. Letters 50 (1983) 1395. [6] Recently, S.T. Chui, T. Hakim and K.B. Ma (preprint) and S.T. Chui (preprint) have proposed that S(k) has a power law divergence at some finite value of k, in the quantized Hall ground state. The evidence in favor of this does not appear to be very strong, however. We may remark that the analysis of Girvin et al. (ref. [13], below) implies rigorously that there can be no energy gap for neutral excitations, if S(k) diverges at some value of k. [7] See, for example: H. Fukuyama, P.M. Platzman and P.W. Anderson, Phys. Rev. B19 (1979) 5211; D. Yoshioka, Phys. Rev. B29 (1984) 6833; S.M. Girvin, Phys. Rev. B29 (1984) 6012. [8] B.I. Halperin, Helv. Phys. Acta 56 (1983) 75. [9] F.D.M. Haldane, Phys. Rev. Letters 51 (1983) 605. [10] R.B. Laughlin, Surface Sci. 142 (1984) 163. [11] C. Kallin and B.I. Halperin, Phys. Rev. B30 (1984) 5655. [12] R.B. Laughlin, Physica 126B (1984) 254. [13] S.M. Girvin, A.H. MacDonald and P.M. Platzman, Phys. Rev. Letters 54 (1985) 581. [14] F.D.M. Haldane and E.H. Rezayi, Phys. Rev. Letters 54 (1985) 237. [15] See, P.K. Lam and S.M. Girvin, Phys. Rev. B30 (1984) 473; B31 (1985) 613E. [16] D. Yoshioka, Phys. Rev. B29 (1984) 6833; D. Yoshioka, private communication. [17] D. Levesque, J.J. Weiss and A.H. MacDonald, Phys. Rev. B30 (1984) 1056. [18] B.I. Halperin, Phys. Rev. Letters 52 (1984) 1583; Phys. Rev. Letters 52 (1984) 2390E. [19] R.B. Laughlin, M.L. Cohen, J.M. Kosterlitz, H. Levine, S.B. Libby and A.M.M. Pruisken, preprint. [20] T. Chakraborty, Phys. Rev. B31 (1985) 4026. [21] See, for example: A.M. Chang, M.A. Paalanen, D.C. Tsui, H.L. St~rmer and J.C.M. Hwang, Phys. Rev. B28 (1984) 6133; S. Kawaji, J. Wakabayashi, J. Yoshino and H. Sasaki, J. Phys. Soc. Japan 53 (1984) 1915; J. Wakabayashi, S. Kawaji, J. Yoshino and H. Sasaki, in: Proc. 17th Intern. Conf. on Physics of Semiconductors, San Francisco, 1984, Eds. D.J. Chadi and W.A. Harrison (Springer, Berlin, 1984); G. Ebert, K. von Klitzing, J.C. Maan, G. Remenyi, C. Probst, G. Weimann and W. Schlapp, J. Phys. C17 (1984) L775; J. Ihm and J.C. Phillips, J. Phys. Soc. Japan 54 (1985) 1506. [22] See, for example: A.H. MacDonald and G.C. Aers, Phys. Rev. B29 (1984) 5976; D. Yoshioka, J. Phys. Soc. Japan 53 (1984) 3740.
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B.I. Halperin / Calculationsoffractional QHE
[23] Note added in proof: Further Monte Carlo evaluations of this improved trial wavefunction have given u ( 2 / 5 ) = -0.4305 + 0.0005. By comparison, extrapolations of exact calculations of N~<10 electrons on a sphere give u ( 2 / 5 ) = - 0 . 4 3 3 1 . These results are from N. d ' A m brumenil, B.I. Halperin and R. Morf, unpublished work, and G. Fano, private communication.