Superconducting properties of the three-dimensional Hofstadter-Hubbard model below the critical flux for Weyl points

TL;DR

This study explores how magnetic flux influences superconductivity in a 3D Hofstadter model, revealing a critical flux that separates regimes with different superconducting behaviors and highlighting the role of topology.

Contribution

It provides the first detailed phase diagram of superconductivity in the 3D Hofstadter-Hubbard model across a critical flux, emphasizing the interplay between topology and pairing.

Findings

Superconductivity occurs at arbitrarily weak attraction below the critical flux.

A finite critical interaction strength is needed for superconductivity above the critical flux.

Near the transition, the system exhibits specific critical exponents and scaling behavior.

Abstract

The three-dimensional Hofstadter model exhibits a critical rational flux at which Weyl points emerge in the single-particle spectrum. We study the superconducting regime of the model in the presence of a Hubbard attractive interaction by tuning the magnetic flux across its critical value. We determine the phase diagram in the plane of the coprime pairs parametrizing the magnetic flux. We show that the system exhibits two distinct regimes separated by a critical flux $\Phi_c$: for $\Phi>\Phi_c$, a semimetal-to-superconductor quantum phase transition occurs at a finite interaction strength ($U_c\neq0$), while for $\Phi<\Phi_c$ superconductivity arises for arbitrarily weak attraction, with a BCS-like exponential scaling of the gap due to the finiteness of the density of states. Close to the transition, we study the scaling behavior and identify the critical exponents. Our results highlight the interplay between magnetic band topology and attractive pairing in three-dimensional Hofstadter systems.