Exact results for model wave functions of anisotropic composite fermions in the fractional quantum Hall effect

TL;DR

This paper derives exact analytical relations for the energies of anisotropic fractional quantum Hall states using composite fermion wave functions, explaining their robustness to anisotropy and proposing improved variational wave functions.

Contribution

It provides a universal scale factor relating anisotropic and isotropic wave function energies, enabling estimation of observables and insights into the robustness of fractional quantum Hall states.

Findings

Energy relations are analytically derived for anisotropic states.

Fractional quantum Hall states show weak dependence on electron mass anisotropy.

Variational wave functions can be improved but are computationally complex.

Abstract

The microscopic wave functions of the composite fermion theory can incorporate electron mass anisotropy by a trivial rescaling of the coordinates. These wave functions are very likely adiabatically connected to the actual wave functions of the anisotropic fractional quantum Hall states. We show in this article that they possess the nice property that their energies can be analytically related to the previously calculated energies for the isotropic states through a universal scale factor, thus allowing an estimation of several observables in the thermodynamic limit for all fractional quantum Hall states. The rather weak dependence of the scale factor on the anisotropy provides insight into why fractional quantum Hall effect and composite fermions are quite robust to electron mass anisotropy. We discuss how better, though still approximate, wave functions can be obtained by introducing a variational parameter, following Haldane [Phys. Rev. Lett. {\bf 107}, 116801 (2011)], but the resulting wave functions are not readily amenable to calculations. Our considerations are also applicable, with minimal modification, to the case where the dielectric function of the background material is anisotropic.