Abstract
Monte Carlo methods have been employed to evaluate trial wave-functions for quantized Hall states and excitations using up to 92 electrons on a sphere. Results are presented for the energy of quasiparticle and quasihole excitations at filling factorv=1/3. These are consistent with extrapolations of exact small system calculations and earlier Monte Carlo results. A trial wavefunction for neutral excitations atv=1/m (m odd integer) is proposed. Results for its excitation energy in the long wavelength limit, atv=1/3, are consistent with those based on the single mode approximation. We have also studied a trial wavefunction representing a quasiparticle excitation involving an electron with reversed spin. Results are presented for the effects on excitation energies of the finite extent of the electron wave function perpendicular to the surface. We have also evaluated the energy of a microscopic trial wave-function for the ground state atv=2/5.
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References
For a review see: The quantum Hall effect. Prange, R.E., Girvin, S.M. (eds.). Berlin, Heidelberg, New York: Springer 1987
For an up to date discussion of experimental results cf. the article by A.M. Chang. Prange, R.E., Girvin, S.M. (eds.). Berlin, Heidelberg, New York: Springer 1987 pp. 175–232
Laughlin, R.B.: Phys. Rev. Lett.50, 1395 (1983); cf. also the article by Laughlin, R. B. in [1], pp 233–301
Caillol, J.M., Levesque, D., Weis, J.J., Hansen, J.P.: J. Stat. Phys.28, 325 (1982)
Haldane, F.D.M.: Phys. Rev. Lett.51, 605 (1983)
Haldane, F.D.M.:. Prange, R.E., Girvin, S.M. (eds.) Berlin, Heidelberg, New York: Springer 1987 pp 303–352
Haldane, F.D.M.: Phys. Rev. Lett.55, 2095 (1985)
Haldane, F.D.M., Rezayi, E.H.: Phys. Rev. B31, 2529 (1985)
Haldane, F.D.M., Rezayi, E.H.: Phys. Rev. Lett.54, 237 (1985)
Fano, G., Ortolani, F., Colombo, E.: Phys. Rev. B34, 2670 (1986)
Morf, R., d'Ambrumenil, N., Halperin, B.I.: Phys. Rev. B34, 3037 (1986)
The energy per electronE/N of fluid like ground states (e.g. atv=1/3 orv=2/5) has a very smooth size dependence and can be extrapolated to the bulk limit with high precision, cf. [11]
Laughlin, R.B.: Surf. Sci.142, 163 (1984)
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Fertig, H., Halperin, B.I.: Phys. Rev. B (to be published)
Morf, R., Halperin, B.I.: Phys. Rev. B33, 2221 (1986)
Halperin, B.I.: Helv. Phys. Acta56, 75 (1983)
Yoshioka, D.: Phys. Rev. B29, 6833 (1984)
The derivative wavefunction in the spherical geometry was introduced by Haldane, cf. [5]: Phys. Rev. Lett.51, 605 (1983)
Chakraborty, T., Pietiläinen, P., Zhang, F.C.: Phys. Rev. Lett.57, 130 (1986)
In this case, too, the value 405-1 obtained from HNC approximation is significantly too small cf. T. Chakraborty: Phys. Rev. B34, 2926 (1986)
Halperin, B.I.: Phys. Rev. Lett.52, 1583 (1984); 2390(E) (1984)
Girvin, S.M., MacDonald, A.H., Platzman, P.M.: Phys. Rev. Lett.54, 581 (1985)
Boebinger, G.S., Chang, A.M., Stormer, H.L., Tsui, D.C.: Phys. Rev. Lett.55, 1606 (1985)
Yoshioka, D.: J. Phys. Soc. Jap.55, 885 (1986)
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The role of the impurity potential in experimental results for the energy gapE g is indicated in recent measurements on an exceptionally high mobility sample by R., Willett, H.L., Stormer, D.C., Tsui, A.C., Gossard, J.H., English, K.W., Baldwin: Proc., Electronic Properties of Two-Dimensional Systems VII, Santa Fé 1987 (to be published in Surf. Sci.). They find values ofE g that are substantially larger than in previous experiments. A complete theoretical treatment of the role of impurities in the low-temperature behaviour of ρ xx has not been given as far as we are aware. See, however, A. Gold.: Europhys. Lett.1, 241 (1986); 479E (1986); A.H. MacDonald, K.L. Liu, S.M. Girvin, P.M. Platzman: Phys. Rev. B33, 4014 (1986)
The value Δ≈0.075 is obtained on the basis of the single mode approximation, [23], and agrees very well with exact numerical results (cf. [6-10])
Girvin, S.M.,. Prange, R.E., Girvin, S.M. (eds.). Berlin, Heidelberg, New York: Springer 1987 pp 353–380
This point is discussed in [11]: Phys. Rev. B34, 3037 (1986)
For a review see: Binder, K. (ed.). Applications of the Monte Carlo method. In statistical physics. In: Topics in Current Physics. Lotsch, H.K.V. (ed.), Vol. 36. Berlin, Heidelberg, New York: Springer 1984
The calculations in the present work were carried out ona MAP-300 array processor. Efficient use was made by performingindependent MC computations on many electron systems in parallel. For a detailed description, cf. Appendix B of [16]: Phys. Rev. B33, 2221 (1986)
The methods employed in the present work are generalizations to the spherical geometry of those employed in [16]: Phys. Rev. B33, 2221 (1986)
Notice the smallness of the finite size corrections which is a consequence of quoting the energy values in units 406-3, cf. (2.17) and (3.7). In unit 406-4, the 0(1/N) correction is about 30 times larger, cf. also [11]: Phys. Rev. B34, 3037 (1986)
Notice that while the bulk limit of\(\tilde \varepsilon _ - \), must not depend on the distance definition used for computing the Coulomb energy, corrections of 0(N −1) to\(\tilde \varepsilon _ - \), representing the effects of curvature, are not expected to be the same
In the limitN→∞, (3.24) becomes equivalent (up to a normalization factor) with the corresponding expression in the disk geometry (cf. eq. (3.23) of [16])
Using as weighting functionw the more standard formw(z)=exp−{1/α2(2R(|z|−p))2} has the disadvantage that forp>0,D j w(z j) becomes singular at the north pole, which would tend to increase the statistical error of MC results
Results for the non-antisymmetrized pair-wavefunction\(\tilde \psi ^{( + )} \) are, respectively\(\tilde \varepsilon _ + \) (14), 0.1091 (14) and 0.112(2) forN=16, 22 and 32 electrons. The antisymmetrizerA thus leads to a reduction of the quasiparticle energy by 35 to 38 percent
ForN=32 electrons, the difference between great-circle and chord distance is about 5 percent atr=3
Results forn(r) in the antisymmetrized pair wavefunction are new. In [16], we have presented result for Gaussian weighted integrals of the densityn(r) (eq. (3.45, 46) and Table 3 of [16]) which are consistent with our new results, in particular with regard to the larger values ofn(r) atr<1 for the pair wavefunction compared with those for the derivative wavefunction
Strictly, it is not necessary to compute the full sum over 0(N 2) terms defining antisymmetrizerA since «O P1 l»=«O P2 m» =«O P13» and similarly «O P1 l P2 m»=«O P13 P24» and thus ΫOAλ «O P13»+1/2(N-2)(N-3) «O P13 P24». However, it turns out that unless 0(N 2) contributions are computed directly the statistical error due to the second order term «O P13 P24» becomes unacceptable
Ando, T., Fowler, A.B., Stern, F.: Rev. Mod. Phys.54, 437 (1982)
Cf. Fig. 4 of [25]. The reduction of the gap at β=1 varies from about 46 percent atn=4 to approximately 43 percent atN=7. Cf. also [20]
IfB is measured in Tesla, β≈0.525B 1/6, atv=1/3
The situation is very similar for the “spin polarized” gapE g for which the exact result atN=6 isE g≈0.061 (cf. Fig. 1 of [20]) vs our result for the bulk limit,E g≈0.092±0.004 (1.5).
Cf. the discussion in Sect. 8.8 of [6], and also [29]
In our MC calculation one electron is moved in each step
Members of a pair are connected by straight lines in Fig. 5 a and b
AtN=64 electrons, a state of broken symmetry, similar to the one displayed in Fig. 5a, appears and its energy isE/N≈−0.4174 (as high as the value forN=32). The phase transition can be eliminated by performing the MC sampling with weight\(|\bar \Phi _T |^2 = |\Phi _T |^2 \prod \exp - [(r_{2n} - r_{2n - 1} )^2 /R_c^2 ] = |\Phi _T |^2 G\) and calculating expectation values e.g. the energy <E> by\(\left\langle {\bar \Phi _T |} \right.EG^{ - 1} |\left. {\bar \Phi _T } \right\rangle /\left\langle {\bar \Phi _T |G^{ - 1} |\bar \Phi _T } \right\rangle \) The results are insensitive to the value of the cut-offR c and the calculation of antisymmetrization corrections becomes well behaved. We also tested this weighting procedure for the Laughlin state atv=1/3 and obtained results for the energy independent ofR c (forR c≳2R 0)
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Dedicated to Professor Harry Thomas on the occasion of his 60th birthday
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Morf, R., Halperin, B.I. Monte Carlo evaluation of trial wavefunctions for the fractional quantized Hall effect: Spherical geometry. Z. Physik B - Condensed Matter 68, 391–406 (1987). https://doi.org/10.1007/BF01304256
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DOI: https://doi.org/10.1007/BF01304256