Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators

TL;DR

This paper develops a systematic high-frequency expansion method based on Brillouin-Wigner perturbation theory for periodically driven quantum systems, enabling accurate effective Hamiltonians and phase transition analysis in Floquet topological insulators.

Contribution

It introduces a Brillouin-Wigner based high-frequency expansion that efficiently derives effective Hamiltonians and captures phase transitions in Floquet systems, surpassing previous methods like van Vleck theory.

Findings

Successfully explains Floquet topological phase boundaries at high frequencies.

Reveals intricate Floquet topological phases at lower frequencies.

Shows phase transitions from Floquet topological states to Mott insulators.

Abstract

We construct a systematic high-frequency expansion for periodically driven quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which generates an effective Hamiltonian on the projected zero-photon subspace in the Floquet theory, reproducing the quasienergies and eigenstates of the original Floquet Hamiltonian up to desired order in $1/\omega$, with $\omega$ being the frequency of the drive. The advantage of the BW method is that it is not only efficient in deriving higher-order terms, but even enables us to write down the whole infinite series expansion, as compared to the van Vleck degenerate perturbation theory. The expansion is also free from a spurious dependence on the driving phase, which has been an obstacle in the Floquet-Magnus expansion. We apply the BW expansion to various models of noninteracting electrons driven by circularly polarized light. As the amplitude of the light is increased, the system undergoes a series of Floquet topological-to-topological phase transitions, whose phase boundary in the high-frequency regime is well explained by the BW expansion. As the frequency is lowered, the high-frequency expansion breaks down at some point due to band touching with nonzero-photon sectors, where we find numerically even more intricate and richer Floquet topological phases spring out. We have then analyzed, with the Floquet dynamical mean-field theory, the effects of electron-electron interaction and energy dissipation. We have specifically revealed that phase transitions from Floquet-topological to Mott insulators emerge, where the phase boundaries can again be captured with the high-frequency expansion.