Virial Expansions, Exclusion Statistics, and the Fractional Quantum Hall Effect

The spectrum of the kinetic energy operator for a non-interacting electron in two dimensions in a perpendicular magnetic field consists of Landau levels with degeneracy and energy separation both proportional to the magnetic field strength B. In strong magnetic fields, the situation where all electrons can be accommodated within the lowest Landau level can be realized experimentally and affords an interesting archetype for strongly interacting electron systems. In such systems, the many-particle kinetic energy operator has a ground state degeneracy that grows exponentially with system size, frustrating attempts to describe the system by treating electron-electron interactions perturbatively. The physical properties of two-dimensional electron systems (2DES) have a corresponding richness, exhibiting a number of unusual properties including the fractional quantum Hall 1 effect (FQHE). The FQHE is related to discontinuities in the chemical potential 2 of a 2DES at zero temperature. Progress has been made in its theoretical understanding by a variety of approaches, including the use of variational many-electron wave functions, 3 the idenfication of model Hamiltonians 4,5 for which the ground state may be determined analytically, and the introduction of artificial ChernSimons magnetic flux quanta attached to each particle. 6–9 In this article we discuss an initial attempt to address finite-temperature properties of 2DES’s in the FQHE regime using virial expansions. Throughout this article we will restrict our attention to the situation where the energy separation between Landau levels (¯ hωc = ¯ ∗ c) is much larger than the thermal energy kBT and assume that the electronic system is completely spin polarized so that we can truncate the Hilbert space to the lowest spin-polarized Landau level. In Section II we introduce appropriate definitions for cluster and virial expansion coefficients in this FQHE regime. Here we discuss the non-interacting electron limit and some exact properties of the virial expansion that follow from particle-hole symmetry within the lowest Landau level. For interacting electron systems, the non-interacting results for the virial coefficients are recovered in the high-temperature limit where kBT is much larger than the typical interaction energy per particle. All interaction-dependent results discussed in this article are for the case of hardcore-model electron-electron interactions. This model can be regarded as the ideal model of the fractional quantum Hall effect since the seminal variational wave functions of Laughlin are exact in this case. 4,5 In Section III we derive exact results for the zero-temperature limit of the virial coefficients of the hard-core model. As we discuss there, the thermodynamics in this regime corresponds precisely to that of an exclusion statistics gas. 10 In order to study how the virial coefficients interpolate between their high-temperature and low-temperature limits, we have obtained numerically exact results for the temperature dependence of the five leading virial coefficients by calculating finitesize partition functions for small numbers of electrons. These results are presented and discussed in Section IV. In Section V we discuss the degree to which fractional quantum Hall behavior is reflected in the virial coefficients we have evaluated. Finally in Section VI we briefly summarize our results. II. VIRIAL EXPANSIONS IN THE LOWEST LANDAU LEVEL The virial expansion 11 can be derived from the expansion of the grand potential in powers of the fugacity, z ≡ exp(�/k BT). This expansion is motivated in part by the fact that only its leading order term is non-zero for the classical non-interacting gas. Higher order terms in the expansion are due to quantum effects and interactions. The grand potential is extensive and for the two-dimensional case, the dimensionless coefficients of the