Hofstadter Superconductivity

The advent of superconductivity in twisted bilayer graphene, now understood to have non-trivial topology in its non-superconducting state, has revived the question of the nature of superconductivity in doped topological bands. Motivated by this, in this talk we will consider the theory of superconductivity in the earliest-known topological bands: 2D Chern bands displaying the integer quantum Hall effect modeled by the Harper-Hofstadter Hamiltonian. Surprisingly, the symmetry of such Hofstadter superconductors (HSCs) had previously not been well-understood, leading us to classify the irreducible representations (irreps) of the magnetic translation group realized by HSCs. These irreps turn out to have dimensions that depend on the fraction of the flux quantum per unit cell of the lattice, resulting in potentially rich superconducting phase diagrams with different patterns of symmetry breaking. In particular, we apply the new representation theory in a renormalization group study of weak-coupling HSCs driven by repulsive interactions. We identify nodal and chiral topological HSC states possessing emergent self-similar symmetries, as well as a triplet pair density wave state for the special case of π-flux per unit cell with additional nearest-neighbor attraction.

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