Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit

TL;DR

This paper explores the fractional quantum Hall effect in the thin-torus limit, revealing a fractal phase diagram that aligns with experimental observations and offers new insights into the phenomenon.

Contribution

It maps the thin-torus Hamiltonian to a one-dimensional lattice gas, providing a novel fractal interpretation of the Hall resistivity plateaux.

Findings

Fractal phase diagram of Hall resistivity

Qualitative agreement with experimental data

Mapping to a 1D lattice gas model

Abstract

After more than three decades the fractional quantum Hall effect still poses challenges to contemporary physics. Recent experiments point toward a fractal scenario for the Hall resistivity as a function of the magnetic field. Here, we consider the so-called thin-torus limit of the Hamiltonian describing interacting electrons in a strong magnetic field, restricted to the lowest Landau level, and we show that it can be mapped onto a one-dimensional lattice gas with repulsive interactions, with the magnetic field playing the role of a chemical potential. The statistical mechanics of such models leads to interpret the sequence of Hall plateaux as a fractal phase diagram, whose landscape shows a qualitative agreement with experiments.