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arXiv:cond-mat/9701178 · Physical review. B, Condensed matter

Virial expansions, exclusion statistics, and the fractional quantum Hall effect

K. Tevosyan, Allan H. MacDonald

Published Sep 15, 1997• 10 citations
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Abstract

We report on a study of virial expansions for interacting electrons in the lowest Landau level of a two-dimensional electron gas. For hard-core-model interactions, we derive analytic results valid at low temperatures and filling factors smaller than $1/3$, and comment on their relationship with virial expansions for exclusion statistics models. In the high-temperature limit the virial coefficients reduce to those for a noninteracting electron system. For the first five virial coefficients, the crossover between low- and high-temperature limits has been studied numerically by using partition functions obtained from small system exact diagonalization calculations. Our results show that the exclusion statistics description of fractional Hall thermodynamics breaks down when the temperature exceeds a small fraction of the gap temperature.

DOI
10.1103/physrevb.56.7517

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We report on a study of virial expansions for interacting electrons in the lowest Landau level of a two-dimensional electron gas. For hard-core-model interactions, we derive analytic results valid at low temperatures and filling factors smaller than 1/31/31/3, and comment on their relationship with virial expansions for exclusion statistics models. In the high-temperature limit the virial coefficients reduce to those for a noninteracting electron system. For the first five virial coefficients, the crossover between low- and high-temperature limits has been studied numerically by using partition functions obtained from small system exact diagonalization calculations. Our results show that the exclusion statistics description of fractional Hall thermodynamics breaks down when the temperature exceeds a small fraction of the gap temperature.
We study two-dimensional bosons and fermions in a magnetic field with vanishing range interactions. This model was previously proposed for fermions to illustrate the exactness and uniqueness of Laughlin's wave function as a ground state within the lowest Landau level. We show that the restriction to the lowest Landau level is not a valid approximation of these models for arbitrarily high (but finite) cyclotron frequency, since it predicts fractional quantum Hall effect (FQHE) at an incorrect filling factor. In particular, in contrast to the lowest-Landau-level theory, hard-core bosons are shown not to exhibit any FQHE. However, the set of filling factors for the FQHE are identical in the two cases, thus reaffirming the universality of the FQHE.
The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of $\ensuremath{\nu}$ (the lowest-Landau-level filling factor) equal to certain multiples of 15\frac{1}{5}51​, 17\frac{1}{7}71​, 19\frac{1}{9}91​, and 111\frac{1}{11}111​. These $\ensuremath{\nu}'\mathrm{s}$ are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.
The fractional quantum Hall (FQH) effect arises from strong electron correlations in a quantising magnetic field, and features exotic emergent phenomena such as electron fractionalisation. Using the diagrammatic Monte Carlo approach with the combinatorial summation (CoS) algorithm, we obtain results with controlled accuracy for the microscopic model of interacting electrons in the lowest Landau level (LLL) in the thermodynamic limit. Starting from the macroscopically degenerate LLL at finite temperature, including interactions order by order, and applying a controlled resummation to the resulting series, we observe the emergence of the incompressible 1/3-filled state as the temperature is lowered. By analysing the long-time decay of the Green's function, we find spectral properties consistent with an energy gap at 1/3-filling, whereas at 1/2-filling our results are consistent with the pseudogapped behaviour previously observed experimentally and suggested theoretically. Our work provides the first demonstration that fractionalised phases of matter can be reliably described with Feynman's diagrammatic technique in terms of the fundamental electronic degrees of freedom, while also showing applicability of expansions in the bare Coulomb potential for precision calculations.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
It is argued that a short-range interaction induces a quasi-Landau-level structure in the filling-factor dependence of the ground-state energy of electrons in the lowest Landau level. The discussion is based on exact numerical results obtained from small-system calculations, and the adiabatic principle of Greiter and Wilczek. It is shown that by attaching a single negative flux quantum to each electron adiabatically, the noninteracting electron ground states occupying higher Landau levels are smoothly transformed into the ground states of a hard-core boson gas in the lowest Landau level. This implies a close correspondence between the noninteracting fermions and the hard-core bosons in the lowest Landau level. Numerical results suggest the possibility of a smooth transition from hard-core bosons to fermions with a short-range interaction, by continuously varying the interaction potential.