# Theory of interacting fermions in shaken square optical lattice

## TL;DR

This paper develops a theoretical framework for weakly interacting fermions in shaken square optical lattices, revealing complex Fermi surfaces and unconventional pairing symmetries induced by orbital hybridization and periodic driving.

## Contribution

It introduces a four-band effective Hamiltonian for shaken lattices, analytically derives momentum-dependent interactions, and predicts an $s+d$-wave pairing phase in driven fermionic systems.

## Key findings

- Hybridized $s$-band develops multiple minima.
- Effective interactions gain momentum dependence.
- Predicted $s+d$-wave pairing symmetry.

## Abstract

We develop a theory of weakly interacting fermionic atoms in shaken optical lattices based on the orbital mixing in the presence of time-periodic modulations. Specifically, we focus on fermionic atoms in circularly shaken square lattice with near resonance frequencies, i.e., tuned close to the energy separation between $s$-band and the $p$-bands. First, we derive a time-independent four-band effective Hamiltonian in the non-interacting limit. Diagonalization of the effective Hamiltonian yields a quasi-energy spectrum consistent with the full numerical Floquet solution that includes all higher bands. In particular, we find that the hybridized $s$-band develops multiple minima and therefore non-trivial Fermi surfaces at different fillings. We then obtain the effective interactions for atoms in the hybridized $s$-band analytically and show that they acquire momentum dependence on the Fermi surface even though the bare interaction is contact-like. We apply the theory to find the phase diagram of fermions with weak attractive interactions and demonstrate that the pairing symmetry is $s+d$-wave. Our theory is valid for a range of shaking frequencies near resonance, and it can be generalized to other phases of interacting fermions in shaken lattices.