Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators (1511.00755v3)

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.

Analysis of High-Frequency Expansion in Periodically Driven Systems Using Brillouin-Wigner Theory

The paper details a comprehensive paper of the Brillouin-Wigner (BW) theory's application in creating high-frequency expansions for periodically driven quantum systems. This approach generates an effective Hamiltonian within the Floquet framework, which reproduces the quasienergies and eigenstates of the original Floquet Hamiltonian for systems subjected to periodic driving. The primary advantage of the BW method lies in its ability to efficiently derive higher-order perturbative terms, offering, uniquely, the potential to write the whole infinite series expansion.

Key Contributions

1. BW Theory in Floquet Context:
   • The BW formalism overcomes limitations faced in other methodologies, such as the van Vleck degenerate perturbation theory and Floquet-Magnus expansions, by efficiently handling higher-order terms and eliminating spurious phase dependencies from driving phases.
2. Effective Hamiltonian Constructed:
   • The BW expansion allows the projection of infinite-dimensional Floquet matrices onto a zero-photon subspace, simplifying the treatment of quasienergy spectra.
   • BW expansion provides a recursive mechanism for computing perturbative series across $(1/\omega)^n$ orders ($n\geq0$) without needing special treatments of degenerate energies, as necessary in standard perturbation theories.
3. Applications in Non-Interacting and Interacting Electron Models:
   • The paper illustrates the BW theory’s application to periodic driving of non-interacting electrons in lattice models (honeycomb, Lieb, kagome lattices) delivering phases characterized by various Chern numbers and revealing intricate topological transitions.
   • For interacting systems, the paper demonstrates the emergence of Floquet topological to Mott insulator phase transitions within a driven Hubbard model, using the Floquet DMFT approach.

Numerical Insights

• Floquet Topological Transitions:
  • Application to notable lattice models under circularly polarized light reveals a series of topological-to-topological phase transitions, depicted through varying Chern numbers.
  • Significant findings include nearly fractal behavior in phase diagrams at low frequencies and novel transitions induced by strong ac driving fields.
• Impact of External Parameters:
  • Dissection of lattice models under varying amplitude and frequency of external periodic drive sheds light on how photonic interactions affect band structure and topological properties.
  • Analysis establishes predictive analytic conditions for these transitions, validated across several lattice structures.

Implications and Future Directions

Besides addressing immediate questions related to topological phases in periodically driven systems, the paper opens pathways for further exploration in several areas: - Development of formalism and computational tools extendable to broader classes of periodically driven quantum systems. - Inviting future work involves revisiting the assumptions underpinning high-frequency regimes to unravel effects in ultralow-frequency scenarios potentially leading to dynamic stabilization of novel phases. - Exploration of many-body interactions such as fractional Chern insulators under nonequilibrium settings could provide unprecedented insights into complex phenomena at fractional fillings.

This research provides a robust foundation for employing BW theory in dynamically tuning quantum states, offering a promising direction towards applications spanning quantum information processing, photonic device engineering, and condensed matter physics. The new insights into Floquet topological phases underscore a paradigm shift in understanding driven quantum systems.