Unconventional Superconductivity in Hofstadter Electronic Bands

The interplay between correlations and topology in fractal Hofstadter electronic bands has been traditionally associated with the quantum Hall effect. Two-dimensional heterostructures with large unit cells, such as moiré quantum materials, have recently emerged as platforms realizing the Hofstadter regime when the magnetic flux per unit cell is comparable to the flux quantum. Motivated by such developments, I will present recent theoretical progress in understanding a new class of quantum states that result when Hofstadter electrons form Cooper pairs. A general classification of Hofstadter superconductors based on the representations of the magnetic translation symmetries will be presented, uncovering a rich phase diagram of symmetry-broken phases characterized by multi-component order parameters. Further, I will present a weak-coupling renormalization group analysis revealing how repulsive electronic interactions projected on Van Hove singularities can promote unconventional superconductivity, bringing to light microscopic mechanisms for chiral topological superconductors and pair-density wave states. This approach suggests a new venue for realizing long-sought unconventional symmetry breaking and topological orders using quantum Hofstadter materials.