Fractional quantum Hall effect in the Hofstadter model of interacting fermions

TL;DR

This paper investigates the fractional quantum Hall effect within the Hofstadter model of interacting fermions, revealing how inhomogeneous effective fields and fine spectral structures lead to fractional states and edge modes.

Contribution

It introduces a unified approach to analyze IQHE and FQHE in the Hofstadter model, accounting for interaction effects and spectral fine structures.

Findings

Fractional quantum Hall states emerge from the Hofstadter spectrum's fine structure.

Edge chiral modes connect subbands and influence fractional conductance.

Topological Chern numbers remain conserved despite spectral fine structure.

Abstract

Applying a unified approach, we study integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interaction between fermions. An effective field, that takes into account the interaction, is determined by both the amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale (cell size corresponding to a unit quantum magnetic flux), the second scale (determines the inhomogeneity of the effective field) forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite sizes with open boundary conditions, the fine structure of the Hofstadter spectrum also includes the fine structure of the edge chiral modes. The subbands in a fine structure of the Hofstadter band (HB) are separated extremely small quasigaps. The Chern number of a topological HB is conserved during the formation of its fine structure. Edge modes are formed into HB, they connect the nearest-neighbor subbands and determine the fractional conductance for the fractional filling at the Fermi energies corresponding to these quasigaps.