Abstract

The spectrum of electrons subject to a 2D periodic potential and a constant magnetic field is formed by Hofstadter bands characterized by fractal and topological properties that are traditionally associated with the quantum Hall effect. In recent years, the ability to realize such appealing band structures, both in AMO and in moiré superlattices, has renewed interest in exploring novel quantum orders in Hofstadter systems. In this talk, I will discuss our recent work [1], which investigates the properties of mean-field states resulting from the pairing of electrons in time-reversal broken Hofstadter bands in 2D lattices where the unit cell traps magnetic flux  = (p/q)0 comparable to the flux quantum 0 = h/e. It will be described how the dimension and degeneracy of the irreducible representations of the magnetic translation group (MTG) furnished by the charge 2e pairing fields have different properties from those furnished by single particle Bloch states, and in particular are shown to depend on the parity of the denominator q. This symmetry analysis will set the stage for a phenomenological Ginzburg-Landau theory describing the thermodynamic properties of Hofstadter superconductors in terms of a multicomponent order parameter that describes the finite momentum pairing of electrons across different Fermi surface patches. I will describe the characteristic properties of the phase diagram resulting from different symmetry breaking patterns of the MTG, and show that it can support several interesting phases such as topological superconductors with tunable Chern numbers as well as Bogoliubov Fermi surfaces protected by parity and MTG symmetries that are captured by a new topological invariant. Consequently, Hofstadter superconductors emerge as a new setting to explore the rich interplay of symmetry broken and topological orders.
[1] Theory of Hofstadter Superconductors, D. Shaffer, J. Wang and L.H. Santos, Phys. Rev. B 104, 184501 (2021)