Many-Body Chern Numbers of $\nu=1/3$ and $1/2$ States on Various Lattices

TL;DR

This paper investigates many-body Chern numbers for fractional quantum Hall states on various lattices, using pseudopotentials and numerical methods to analyze topological properties, energy gaps, and the effects of disorder.

Contribution

It introduces a scheme to compute many-body Chern numbers on different lattices and explores the nature of fractional quantum Hall states, including the Laughlin state and composite fermion picture.

Findings

The ground state at ν=1/3 is a lattice analogue of the Laughlin state with a unique topological degeneracy.

A simple scaling form of the energy gap is numerically obtained for ν=1/3.

The validity of the composite fermion picture at ν=1/2 is discussed, considering the Fermi surface and disorder effects.

Abstract

For various two dimensional lattices such as honeycomb, kagome, and square-octagon, gauge conventions (string gauge) realizing minimum magnetic fluxes that are consistent with the lattice periodicity are explicitly given. Then many-body interactions of lattice fermions are projected into the Hofstadter bands to form pseudopotentials. By using these pseudopotentials, degenerate many-body ground states are numerically obtained. We further formulate a scheme to calculate the Chern number of the ground state multiplet by the pseudopotentials. For the filling factor of the lowest Landau level, $\nu=1/3$, a simple scaling form of the energy gap are numerially obtained and the ground state is unique except the three-fold topological degeneracy. This is a quantum liquid, which can be lattice analogue of the Laughlin state. For the $\nu=1/2$ case, validity of the composite fermion picture is discussed in relation to the existence of the Fermi surface. Effects of disorder are also described.