A Floquet perturbation theory for periodically driven weakly-interacting fermions

TL;DR

This paper develops a Floquet perturbation theory to analyze weakly interacting fermions under periodic driving, providing improved accuracy over Magnus expansion and exploring steady-state behaviors including localization and thermalization.

Contribution

Introduces a Floquet perturbation theory for weakly interacting fermions that improves upon Magnus expansion and investigates steady states and localization phenomena.

Findings

FPT yields higher fidelity than Magnus expansion for certain parameters.

Identifies parameter regimes with subthermal and superthermal steady states.

Shows crossover between localized and delocalized steady states in interacting chains.

Abstract

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 $\hbar \omega_D \ge V_0 \ll {\mathcal J}_0$, where ${\mathcal J}_0$ is the amplitude of the kinetic term, $\omega_D$ 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 wavefunctions 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 $V_0\le \hbar \omega_D$ and $V_0\ll {\mathcal J}_0$. 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 $\omega_D$ 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.