Lattice Monte Carlo for Quantum Hall States on a Torus

TL;DR

This paper introduces a fast lattice Monte Carlo method for quantum Hall problems on a torus, enabling efficient computation of various physical quantities and confirming phase structures in larger systems.

Contribution

The paper presents a mathematically exact reformulation of continuum quantum Hall problems into a lattice Monte Carlo approach, significantly improving computational efficiency.

Findings

Monte Carlo results agree with phase structures on large systems

Computed Coulomb energy, structure factor, and particle-hole symmetry for various states

Enhanced algorithm boosts Monte Carlo efficiency by several orders

Abstract

Monte Carlo is one of the most useful methods to study the quantum Hall problems. In this paper, we introduce a fast lattice Monte Carlo method based on a mathematically exact reformulation of the torus quantum Hall problems from continuum to lattice. We first apply this new technique to study the Berry phase of transporting composite fermions along different closed paths enclosing or not enclosing the Fermi surface center in the half filled Landau level problem. The Monte Carlo result agrees with the phase structure we found on small systems and confirms it on much larger sizes. Several other quantities including the Coulomb energy in different Landau levels, structure factor, particle-hole symmetry are computed and discussed for various model states. In the end, based on certain knowledge of structure factor, we introduce a algorithm by which the lattice Monte Carlo efficiency is further boosted by several orders.