Band mass anisotropy and the intrinsic metric of fractional quantum Hall systems

TL;DR

This paper investigates how band mass anisotropy affects the geometric and topological properties of fractional quantum Hall states, demonstrating their robustness or phase transitions under anisotropic conditions across multiple Landau levels.

Contribution

It introduces a variational approach to model the intrinsic metric of FQHE states under band mass anisotropy and explores the resulting phase transitions in different Landau levels.

Findings

FQHE states survive moderate band mass anisotropy in the lowest Landau level.

The intrinsic metric of Laughlin states adjusts with anisotropy while maintaining phase robustness.

Mass anisotropy induces phase transitions to charge density waves and stripe order in higher Landau levels.

Abstract

It was recently pointed out that topological liquid phases arising in the fractional quantum Hall effect (FQHE) are not required to be rotationally invariant, as most variational wavefunctions proposed to date have been. Instead, they possess a geometric degree of freedom corresponding to a shear deformation that acts like an intrinsic metric. We apply this idea to a system with an anisotropic band mass, as is intrinsically the case in many-valley semiconductors such as AlAs and Si, or in isotropic systems like GaAs in the presence of a tilted magnetic field, which breaks the rotational invariance. We perform exact diagonalization calculations with periodic boundary conditions (torus geometry) for various filling fractions in the lowest, first and second Landau levels. In the lowest Landau level, we demonstrate that FQHE states generally survive the breakdown of rotational invariance by moderate values of the band mass anisotropy. At 1/3 filling, we generate a variational family of Laughlin wavefunctions parametrized by the metric degree of freedom. We show that the intrinsic metric of the Laughlin state adjusts as the band mass anisotropy or the dielectric tensor are varied, while the phase remains robust. In the n=1 Landau level, mass anisotropy drives transitions between incompressible liquids and compressible states with charge density wave ordering. In n>=2 Landau levels, mass anisotropy selects and enhances stripe ordering with compatible wave vectors at partial 1/3 and 1/2 fillings.