# BCS theory of time-reversal-symmetric Hofstadter-Hubbard model

## TL;DR

This paper investigates how competing length scales in a Hofstadter-Hubbard model with time-reversal symmetry influence superfluid phases and phase transitions in a two-component Fermi gas on an optical lattice.

## Contribution

It introduces a detailed analysis of the many-body BCS pairing in a Hofstadter-Hubbard model with opposite synthetic magnetic fields, revealing complex phase diagrams with lobe structures.

## Key findings

- Identification of superfluid, semi-metal, and quantum spin-Hall phases.
- Discovery of lobe-shaped phase boundaries similar to Mott insulators.
- Analysis of phase transitions from various insulating and metallic states.

## Abstract

The competition between the length scales associated with the periodicity of a lattice potential and the cyclotron radius of a uniform magnetic field is known to have dramatic effects on the single-particle properties of a quantum particle, e.g., the fractal spectrum is known as the Hofstadter butterfly. Having this intricate competition in mind, we consider a two-component Fermi gas on a square optical lattice with opposite synthetic magnetic fields for the components, and study its effects on the many-body BCS-pairing phenomenon. By a careful addressing of the distinct superfluid transitions from the semi-metal, quantum spin-Hall insulator or normal phases, we explore the low-temperature phase diagrams of the model, displaying lobe structures that are reminiscent of the well-known Mott-insulator transitions of the Bose-Hubbard model.