Floquet–Bloch Transform in Periodic Systems

• Floquet–Bloch transform is a unitary mapping that re-expresses functions with discrete periodicity as parameterized problems over the Brillouin zone, enabling precise spectral analysis.
• The truncated Floquet–Bloch transform adapts this method to finite systems by incorporating DFT and section decomposition, facilitating efficient numerical simulations and error control.
• Applications include analysis of band structures in condensed matter, photonic crystals, and topological modes in defect-decorated periodic systems.

The Floquet–Bloch transform is a unitary mapping that re-expresses functions or operators displaying discrete periodicity (in space, time, or both) as direct integrals over a fundamental domain—the Brillouin zone—parametrized by quasimomentum or quasifrequency. By mapping unbounded and/or infinite-dimensional problems to families of parametrized problems on a finite cell with explicit quasiperiodic boundary conditions, the transform enables analytical, computational, and asymptotic studies of spectral, scattering, and dynamical aspects in periodic and defect-decorated structures. In its truncated (finite-size) version, the transform is adapted to large but finite systems and retains the essential features of its infinite-domain counterpart for applications ranging from condensed matter physics to photonic crystal analysis.

1. Mathematical Definitions and Foundations

The Floquet–Bloch transform generalizes the concept of Fourier analysis to problems with lattice periodicity. For a spatial lattice with period matrix $A\in\mathbb{R}^{d\times d}$ and its dual (the Brillouin zone) $W_A^*=2\pi A^{-\mathsf T}[ -\frac12, \frac12 )^d$, the transform for $u\in C_0^\infty(\mathbb{R}^d)$ is

$$(\mathcal{J}u)(k,x) = |{\det A}|^{1/2}\sum_{j\in\mathbb{Z}^d}u(x+Aj) e^{-ik\cdot (Aj)} / (2\pi)^{d/2},\quad (k,x)\in W_A^*\times W_A.$$

The inverse is

$$u(x+Aj) = |{\det A}|^{1/2}\int_{W_A^*} (\mathcal{J}u)(k,x) e^{ik\cdot (Aj)}\,dk / (2\pi)^{d/2}.$$

The transform is an isometry on $L^2(\mathbb{R}^d)$ and admits rigorous extension to Bochner–Sobolev spaces $$H^s_r(\mathbb{R}^d)\leftrightarrow H^r(W_A^*;H^s(W_A))$$ (Lechleiter, 2016).

For wave or Green's function analysis on twofold-periodic structures, the double Floquet–Bloch transform is employed: $$F(\xi) = \sum_{m\in\mathbb{Z}^n} u(r+\Lambda m) e^{+im\cdot \xi},\quad u(r+\Lambda m) = (2\pi)^{-n} \int_{[-\pi,\pi]^n} F(\xi) e^{-im\cdot \xi}\,d\xi.$$ Generalization to systems periodic in both space and time—spin, time-dependent, or driven—leads to the Floquet–Bloch–Sambe transform, where the Hilbert space is augmented by $k$ and a frequency (Floquet) index (Beule et al., 30 Jan 2024, Baba et al., 30 Aug 2024).

2. Truncated Floquet–Bloch Transform for Finite Lattices

The truncated Floquet–Bloch transform (TFBT) is adapted to large but finite systems. For $u\in\mathbb{C}^{mk}$ (vectorized data: $m$ unit cells, $k$ internal degrees of freedom/cell), TFBT is built in two steps:

• Section decomposition: $u$ is reshaped via a unitary map $\Phi:\mathbb{C}^{mk} \to (\mathbb{C}^m)^k$, extracting $k$ sequences of length $m$.
• DFT application: To each section (length $m$), apply the discrete Fourier transform $F$ with frequencies $\alpha_j=2\pi j/m$.

The TFBT is then

$$\mathcal T_m:\mathbb{C}^{mk} \xrightarrow{\Phi} (\mathbb{C}^m)^k \xrightarrow{F\oplus\cdots\oplus F} (\mathbb{C}^m)^k,$$

explicitly,

$$(\mathcal{T}_m u)_{p,j} = \frac{1}{\sqrt m} \sum_{s=0}^{m-1} u_{p+sk} e^{-i (2\pi j/m) s},\quad p=0,\dots,k-1,\; j=0,\dots,m-1.$$

This identification is unitary, directly diagonalizes circulant matrices, and maps delocalized eigenmodes to sharply peaked quasimomentum distributions (Ammari et al., 23 Oct 2024).

A phase-average is defined for $u\in\mathbb{C}^{mk}$ as

$$\mathcal Q_m(u) = \sum_{j=0}^{m-1} \alpha_j \left\|\left(\mathcal T_m u\right)_{:,j}\right\|^2,$$

where $\|\cdot\|$ is the Euclidean norm in $\mathbb{C}^k$, providing a quasimomentum label for eigenmodes.

3. Theoretical Properties: Spectral Convergence and Error Analysis

For truncated Toeplitz matrices $T_{mk}(f)$ derived from a matrix-valued, Hermitian symbol $f(e^{i\theta})$ with simple eigenvalues, the essential spectrum of the infinite system is

$$\sigma_\mathrm{ess}(T(f)) = \bigcup_\theta \sigma(f(e^{i\theta})).$$

As $m\to\infty$, discrete spectra of the truncated problem $\mathrm{spec}\,T_{mk}(f)$ cluster onto this set. For a sequence of delocalized eigenvectors $(\lambda_m, u_m)$ converging to $(\lambda_0, \alpha_0)$ in the interior of a simple band, $\mathcal{Q}_m(u_m)\to\pm\alpha_0$ (Ammari et al., 23 Oct 2024).

Error is controlled via difference to the circulant truncation $C_{mk}(f)$. If $a_n=O(|n|^{-s})$ ($s>1$) are the Fourier coefficients of $f$, then $\delta_m=O(m^{1-s})$ governs the control over

$$|\mathcal Q_m(u_m)-\alpha_0| = O(\delta_m) + O(\varepsilon_m),$$

where $\varepsilon_m$ is the eigenpair's residual norm. For long-range interactions, band-limitation followed by perturbative correction is employed (Ammari et al., 23 Oct 2024).

In full-space settings with locally perturbed periodic surfaces, the regularity of the $\alpha$-dependence of the Bloch transform governs spatial decay: higher $W^{1,p}$ regularity in $\alpha$ implies weighted $H^1$ decay in the physical space (Lechleiter, 2016).

4. Numerical Schemes and Algorithmic Implementation

Application of the (truncated) Floquet–Bloch transform in numerical analysis consists of the following steps (Ammari et al., 23 Oct 2024, Lechleiter et al., 2016):

Step	Operation	Output
1	Assemble Hamiltonian or system matrix $A$	$A\in\mathbb{C}^{mk\times mk}$
2	Solve eigenproblem $A u^{(\ell)} = \lambda^{(\ell)} u^{(\ell)}$	Eigenpairs $(\lambda^{(\ell)}, u^{(\ell)})$
3	Apply $\Phi$ and DFT to each $u^{(\ell)}$ (TFBT)	$(\mathcal T_m u^{(\ell)})_{p,j}$
4	Compute $\mathcal Q_m(u^{(\ell)})$	Quasimomentum labels
5	Plot $(\mathcal Q_m(u^{(\ell)}), \lambda^{(\ell)})$	Band diagram
6	Histogram eigenvalues, optionally weight by $|\partial_\theta\lambda_p(\theta)|^{-1}$	Approximate density of states (DOS)

Finite element discretizations and quadrature over quasiperiodicity parameters are used to solve continuum or PDE problems. The cost for the spectral decomposition is $O(N^3)$ for dense matrices, $O(N r^2)$ for banded (bandwidth $r$), and TFBT cost per eigenvector is $O(k m \log m)$ via FFTs.

Parallelization is trivial across modes, and TFBT provides natural Brillouin-zone discretization; edge effects are suppressed for $m$ several times larger than the interaction range.

5. Applications: Scattering, Spectral Theory, and Topological Modes

The Floquet–Bloch transform is the primary analytic tool in scattering from periodic and locally perturbed surfaces (Lechleiter, 2016, Lechleiter et al., 2016). It reduces the variational Helmholtz or Schrödinger equations on unbounded domains to a parametrized family of bounded-cell problems with Bloch or quasimomentum boundary conditions. After solving for each quasimomentum, the original solution is reconstructed by integration (continuous transforms) or summation (discrete/TFFT), yielding access to physically relevant quantities including scattering amplitudes and resonance modes.

TFBT discerns bulk bands from defect-induced in-gap states. In SSH dimer chains, it separates continuous band curves (delocalized modes) and singular in-gap eigenstates (localized/topological), which are characterized by flat TFBT images and ambiguous quasimomentum (Ammari et al., 23 Oct 2024). Similarly, for dislocated or defected chains, TFBT recovers periodic bulk bands, while defect states appear as outliers.

In renormalization group analysis for lattice models with sublattice periodicity, a modified Floquet–Bloch decomposition yields a direct sum of block matrices indexed by coset momentum, enabling uniform decay bounds and analytic continuation for operator norm control (Balaban et al., 2016).

For wave propagation and Green's function analysis, the Floquet–Bloch double-transform is central for explicit far-field asymptotics and handling of singularities (shrinking loops, conical points, hyperbolic crossings) via complex contour deformation (Shanin et al., 12 Feb 2024).

6. Extensions: Time-Dependence and Nonperturbative Dynamics

For systems with temporal periodicity, notably in strongly driven condensed matter/optical settings, the Floquet–Bloch–Sambe transform elevates the problem to a static eigenproblem in momentum–frequency space. For commensurate static fields, the evolution in time maps into an enlarged synthetic dimension with block-coupling structure, yielding the Wannier–Stark ladder and encompassing Zener tunneling (Beule et al., 30 Jan 2024).

For pulsed or non-strictly periodic drives (e.g., Gaussian-envelope applied to 2D Dirac models), the $t$–$t'$ formalism introduces a slow (envelope) and fast (periodic) time scale, tracked by an augmented operator acting in combined momentum–frequency–envelope space. The resulting evolution equations capture sideband formation and selection rules: for pure Dirac systems, linear polarization populates only intraband sidebands, while circular polarization induces interband transitions and hybridization (Baba et al., 30 Aug 2024).

7. Significance, Limitations, and Computational Considerations

The Floquet–Bloch framework underpins the spectral and dynamical theory of periodic structures, providing both theoretical guarantees (unitarity, isomorphism, regularity–decay correspondences) and robust algorithmic workflows for large finite and perturbed systems. TFBT retains the physical and computational advantages of the full transform, faithfully capturing bulk band structure and isolating defect-localized phenomena. Computational tractability requires that the truncation size exceed the interaction length scale, and careful treatment of defects, boundaries, and degeneracies may necessitate refined methods (complex deformation, weighted norms, windowing for edge suppression).

The approach subsumes a wide range of applications: from electronic and photonic band-structure computations and surface/scattering problems, to dynamical and nonequilibrium phenomena in strongly driven periodic media. The analytic mapping properties provide critical insights into the structure of solutions, their decay, and stability, which are directly inherited by practical numerical discretizations.