First-principles Floquet analysis from real-time propagation

Technical Summary: First-Principles Floquet Analysis from Real-Time Propagation

Problem Statement
The practical application of Floquet theory to realistic materials within first-principles simulations faces significant computational and interpretational hurdles. Traditional approaches require the explicit construction of an enlarged Floquet Hamiltonian, which combines electronic and photonic degrees of freedom into a massive Hilbert space. This construction incurs substantial computational overhead and necessitates arbitrary truncation of the Fourier series (choosing a maximum number of Floquet components, $n_{max}$). Furthermore, standard first-principles methods like Time-Dependent Density Functional Theory (TDDFT) naturally output real-time propagated wavefunctions rather than Floquet eigenstates. Additionally, experimental scenarios often involve finite pulses and transient regimes that lack strict time periodicity, conditions under which the formal Floquet framework is not strictly applicable, yet where "Floquet-like" electronic structures are frequently observed in pump-probe experiments.

Methodology
The authors propose a real-time Floquet analysis framework that extracts quasi-energies and Floquet states directly from propagated wavefunctions, bypassing the construction of the enlarged Floquet Hamiltonian. The methodology consists of three primary components:

1. Reconstruction of the One-Period Evolution Operator:
   Instead of diagonalizing a static Floquet Hamiltonian, the method reconstructs the one-period time-evolution operator, $U(t+T, t)$, using overlaps between time-evolved states. Given a set of orthonormal states $\{|\psi_i(t)\rangle\}$ (e.g., Kohn-Sham orbitals) at time $t$, the matrix elements of the evolution operator are computed as:
   $$U_{ij}(t+T, t) = \langle \psi_i(t) | \psi_j(t+T) \rangle$$
   Diagonalizing this matrix yields the Floquet quasi-energies ($E_\alpha = -\arg(\lambda_\alpha)/T$) and the periodic parts of the Floquet states ($|\phi_\alpha(t)\rangle$) within the subspace of the propagated states. This approach adds negligible computational cost to existing time-propagation schemes.

2. Floquet Band Unfolding via Harmonic Decomposition:
   To resolve the ambiguity of the reduced-zone representation (where quasi-energies are defined modulo the driving frequency $\Omega$), the authors introduce an unfolding procedure. They define a weight $w_\alpha$ based on the norm of the zeroth-harmonic component ($n=0$) of the Floquet state:
   $$w_\alpha = |u^{(0)}_\alpha|$$
   where $|u^{(0)}_\alpha\rangle$ is obtained by projecting the periodic Floquet state $|\phi_\alpha(t)\rangle$ onto the zeroth harmonic. This weight quantifies how strongly a Floquet state retains the character of the underlying equilibrium band. By unfolding the spectrum into an extended zone scheme and weighting the bands by $w_\alpha$, physically relevant branches (those with large $w_\alpha$) are distinguished from higher-order photon replicas (which have small $w_\alpha$).

3. Symmetry Analysis of Sidebands:
   The reconstructed real-space Floquet wavefunctions allow for the direct analysis of the symmetry properties of individual light-induced sidebands. By examining the spatial distribution of the wavefunctions, one can determine the symmetry character (e.g., glide-mirror symmetry) of specific Floquet sidebands ($n=0, \pm 1, \dots$) and verify optical selection rules.

Key Results
The framework was validated through first-principles TDDFT simulations on four distinct material systems:

• Graphene (2D): Under circularly polarized light (CPL), the method successfully identified the topological gap opening at the Dirac point. The unfolding procedure was shown to be essential; without it, the spectrum in the first Floquet Brillouin zone (FBZ) appeared as a dense, unintelligible set of folded bands. The weighted unfolding clearly revealed the hybridization gaps and the topological gap.
• Monolayer Black Phosphorus (2D): The method demonstrated the ability to resolve the symmetry character of light-induced sidebands. For armchair-polarized driving, the analysis confirmed that the $n=0$ valence and conduction bands retained their equilibrium glide-mirror symmetries (odd and even, respectively), while the $n=\pm 1$ sidebands exhibited reversed symmetries. This allowed for a direct explanation of observed band repulsion via symmetry-allowed hybridization.
• Monolayer Hexagonal Boron Nitride (2D): The approach was applied to a finite CPL pulse, demonstrating its utility beyond strict time periodicity. By analyzing the system at different time windows within the pulse (second, fifth, and eighth cycles), the authors captured transient Floquet features. The results showed that valley-dependent band renormalization (stronger at the $-K$ valley for left-CPL) emerged and faded in accordance with the pulse envelope, validating the method for transient non-equilibrium regimes.
• Bulk Tin Sulfide (3D): The method was extended to a fully periodic 3D bulk semiconductor. The simulations revealed polarization-dependent band renormalization, where armchair-polarized driving affected the $\Gamma X$ valley and zigzag-polarized driving affected the $\Gamma Y$ valley, consistent with previous symmetry-based analyses.

Significance and Claims
The paper claims to provide a practical and computationally efficient framework for connecting real-time simulations with Floquet observables. Its primary contributions are:

1. Computational Efficiency: It avoids the expensive construction and diagonalization of the enlarged Floquet Hamiltonian, utilizing only the overlaps of states already computed during standard time-propagation.
2. Interpretability: The unfolding scheme based on harmonic weights resolves the ambiguity of folded photon replicas, making the identification of physical band edges and hybridization gaps straightforward.
3. Generality: The method is applicable to both 2D monolayers and 3D bulk materials and extends to finite-pulse regimes where strict time periodicity is absent.
4. Analytical Depth: It enables the direct extraction of symmetry properties and selection rules for individual light-induced sidebands from real-space wavefunctions.

The authors conclude that Floquet theory serves not only as a tool for constructing periodically driven steady states but also as a powerful analysis framework for transient quasiparticle dynamics and non-equilibrium electronic structures in realistic materials.