Hofstadter Superconductivity

While the conventional wisdom is that the orbital effect is detrimental to superconductivity, it has long been predicted theoretically that re-entrant superconductivity should be possible at very high magnetic fields when only a few Landau levels are occupied. The orbital effect can be significant already at lower fields in systems with large unit cells, with the flux per unit cell Φ is on the order of the flux quantum Φ₀, resulting in the Hofstadter butterfly. Motivated by this, we investigate the problem of pairing within Hofstadter bands for a rational flux Φ=p/q Φ₀ by studying the symmetry properties of the order parameter under magnetic translation symmetries. Surprisingly, the irreducible representations realized by the superconducting order parameter are distinct from the well-known representations realized by the single-particle Bloch states. We show that as a result, Hofstadter superconductors necessarily break at least some magnetic translation symmetries, resulting in degenerate ground states that may preserve an order q symmetry, with implications for the topology of the states. With the recent advance of superconducting moiré systems in which the Hofstadter butterfly has already been seen, it may soon be possible to examine Hofstadter superconductors in experiment.