In the 3D Hofstadter-Hubbard model, superconductivity occurs for arbitrarily weak attraction below the critical flux for Weyl points with BCS-like gap scaling, but requires finite interaction strength above it.
The necessary condition for this step is that the Jacobian matrix must be invertible
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Superconducting properties of the three-dimensional Hofstadter-Hubbard model below the critical flux for Weyl points
In the 3D Hofstadter-Hubbard model, superconductivity occurs for arbitrarily weak attraction below the critical flux for Weyl points with BCS-like gap scaling, but requires finite interaction strength above it.