Floquet perturbation theory for periodically driven weakly interacting fermions

We compute the Floquet Hamiltonian ${H}_F$ for weakly interacting fermions subjected to a continuous periodic drive using a Floquet perturbation theory (FPT) with the interaction amplitude being the perturbation parameter. This allows us to address the dynamics of the system at intermediate drive frequencies $\ensuremath{\hbar}{\ensuremath{\omega}}_D\ensuremath{\ge}{V}₀\ensuremath{\ll}{\mathcal{J}}₀$, where ${\mathcal{J}}₀$ is the amplitude of the kinetic term, ${\ensuremath{\omega}}_D$ is the drive frequency, and ${V}₀$ is the typical interaction strength between the fermions. We compute, for random initial states, the fidelity $F$ between wave functions after a drive cycle obtained using ${H}_F$ and that obtained using exact diagonalization (ED). We find that FPT yields a substantially larger value of $F$ compared to its Magnus counterpart for ${V}₀\ensuremath{\le}\ensuremath{\hbar}{\ensuremath{\omega}}_D$ and ${V}₀\ensuremath{\ll}{\mathcal{J}}₀$. We use the ${H}_F$ obtained to study the nature of the steady state of an weakly interacting fermion chain; we find a wide range of ${\ensuremath{\omega}}_D$ which leads to subthermal or superthermal steady states for finite chains. The driven fermionic chain displays perfect dynamical localization for ${V}₀=0$; we address the fate of this dynamical localization in the steady state of a finite interacting chain and show that there is a crossover between localized and delocalized steady states. We discuss the implication of our results for thermodynamically large chains and chart out experiments which can test our theory.