Pair Density Waves and Superconductivity in the π-flux Hofstadter Model using Mean-Field and Renormalization Group Analyses*

The π-flux lattice, or equivalently the Hofstadter model with magnetic flux of half of the flux quantum per unit cell, has historically attracted attention in the context of the pseudogap phase of cuprate superconductors and the study of systems with Dirac fermions, located at half filling in the π-flux lattice. Recently it has become possible to realize the π-flux lattice in cold atomic systems and moiré materials. In our work we consider the model away from half-filling and study the possibility of weak coupling instabilities in the presence of extended Hubbard interactions occuring around the less-studied fillings corresponding to Van Hove singularities. Using both mean-field and renormalization group calculations, we find that paring instabilities, including s-wave and d-wave uniform superconductors (SC) and a triplet pair density wave (PDW), in general win over competing spin density wave instabilities. Of particular interest is the PDW phase, for which some evidence has been seen in cuprates. This phase is characterized by spontaneous formation of Cooper pairs with total non-zero momentum without an explicit breaking of time reversal symmetry, as opposed to the FFLO state. We find that the triplet PDW phase is realized in the π-flux lattice with repulsive on-site interactions together with a moderately small nearest-neighbor attractive interactions, making it the first model in which a PDW phase is realized at weak coupling as a symmetry-enforced logarithmic instability.