Floquet Modes in Periodic Dynamics

• Floquet modes are the eigenstates of the one-period propagator in time-periodic systems that encapsulate quasienergy and stroboscopic dynamics.
• They enable the analysis of topological phases and edge phenomena, bridging quantum and photonic contexts through tools like Sambe space and Fourier harmonics.
• Research on Floquet modes transforms complex time-dependent problems into static eigenvalue challenges, offering insights into stability, localization, and quantum control.

Floquet modes are the natural spectral objects of time-periodic dynamics. For a quantum system with $H(t+T)=H(t)$, the one-period propagator

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$

has eigenvectors $|\phi_n\rangle$ and eigenphases $\mu_n\in[-\pi,\pi]$ defined by

$U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$

so that $\mu_n/T$ are quasi-energies and the Floquet modes diagonalize stroboscopic evolution. Equivalent formulations use time-periodic states $\phi_n(t+T)=\phi_n(t)$ multiplying a phase factor $e^{-i\epsilon_n t}$, and the same structure extends to perturbations of periodic classical fields, photonic time crystals, and space-time modulated media (Möckel et al., 23 Feb 2026, Zhou et al., 2014, Evslin et al., 6 Nov 2025, Gupta et al., 2017).

1. Formal structure and spectral meaning

The basic content of Floquet theory is that periodic time dependence replaces energy conservation by quasienergy conservation modulo the drive frequency. In quantum settings one may write

$$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$

with $\epsilon_n$ defined modulo $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$0, $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$1. Equivalently one defines an effective Floquet Hamiltonian $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$2 through $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$3, although several studies emphasize that the logarithm is multivalued and that branch choices can matter for intermediate constructions even when physical stroboscopic evolution is fixed (Zhou et al., 2014, Park et al., 2022, Yates et al., 2021).

A complementary formulation uses the extended Sambe space. Expanding the periodic part in Fourier harmonics,

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$4

turns the time-dependent problem into a static matrix eigenvalue problem in the harmonic index. In periodically driven lattice systems this produces a tight-binding-like structure in the Floquet harmonic direction; in multi-frequency drives the same idea yields a higher-dimensional “Floquet lattice” whose geometry depends on commensurability relations among the drives (Zhou et al., 2014, Park et al., 2022).

The mode concept is not restricted to stationary quantum states. For perturbations $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$5 around a periodic field configuration $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$6, one seeks

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$7

so a Floquet mode is the linearized analogue of a normal mode around a periodic background. This is the setting in which universal oscillon and breather Floquet modes have been derived at leading nonrelativistic order (Evslin et al., 6 Nov 2025).

2. Zero, $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$8, and eigenphase order

Because quasienergies are defined only modulo $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$9, two values are spectrally distinguished in particle-hole- or chiral-symmetric problems: $|\phi_n\rangle$0 and $|\phi_n\rangle$1. The former gives conventional Floquet zero modes; the latter gives $|\phi_n\rangle$2 modes, which are genuinely Floquet and change sign after one period. In superconducting language one writes Hermitian Majorana operators $|\phi_n\rangle$3 and $|\phi_n\rangle$4 satisfying $|\phi_n\rangle$5 and $|\phi_n\rangle$6 (Bomantara et al., 2018, Wu et al., 2023).

A concrete realization appears in the periodically driven quantum Ising chain with open boundaries,

$|\phi_n\rangle$7

In the integrable limit $|\phi_n\rangle$8, Jordan–Wigner fermionization maps the problem to a free-fermion Floquet-Kitaev chain, with topological phases determined by

$|\phi_n\rangle$9

Finite open chains can then host Majorana zero modes $\mu_n\in[-\pi,\pi]$0 and Majorana $\mu_n\in[-\pi,\pi]$1 modes $\mu_n\in[-\pi,\pi]$2 at each edge (Möckel et al., 23 Feb 2026).

In the $\mu_n\in[-\pi,\pi]$3 phase, all many-body Floquet eigenstates form quadruplets labeled by occupations $\mu_n\in[-\pi,\pi]$4 of the edge fermions. Their splittings consist of exponentially small zero-mode and $\mu_n\in[-\pi,\pi]$5-mode contributions, with $\mu_n\in[-\pi,\pi]$6. This quadruplet structure separates into inter-parity and intra-parity doublets, and that distinction controls the robustness of edge observables under integrability-breaking perturbations (Möckel et al., 23 Feb 2026).

Related spin-chain work formulates strong zero and strong $\mu_n\in[-\pi,\pi]$7 modes directly as boundary operators. In the thermodynamic limit, a strong zero mode commutes with the Floquet unitary while a strong $\mu_n\in[-\pi,\pi]$8 mode anticommutes with it; both also anticommute with a global $\mu_n\in[-\pi,\pi]$9 symmetry. These algebraic relations enforce full-spectrum pairing, while weak interactions deform the exact modes into almost strong modes with long but finite lifetimes (Yates et al., 2021).

3. Topological Floquet modes and bulk–edge correspondence

Floquet modes are central to periodically driven topological matter because they populate all quasienergy gaps, including the zone-boundary or $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$0-gap. In a harmonically driven Hofstadter model at flux $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$1, the quasienergy phase diagram contains phases with counter-propagating edge modes in the $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$2-gap. In the weak-drive limit one of the two edge states can be viewed as a static edge mode shifted by $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$3, i.e. by absorption or emission of a “photon” of frequency $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$4. Static disorder does not couple the dominant $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$5 and $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$6 Fourier sectors to leading order, and exact diagonalization shows that the paired edge modes persist under strong on-site impurities or random hopping disorder (Zhou et al., 2014).

The same departure from static band topology appears in driven two-dimensional lattice models with flat bands. In the three-step protocol analyzed for a honeycomb-type system, the flat-band anomalous limit has perfectly flat bulk bands at $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$7 but still supports chiral edge modes. The Berry curvature vanishes in the strict flat-band limit, so the Chern number is zero even though edge transport remains. This is a standard example of an anomalous Floquet phase, where quasienergy winding rather than static Chern data controls edge propagation (Dag et al., 2022).

One-dimensional superconductors provide the most developed classification of zero and $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$8 Floquet edge modes. In planar Josephson junctions mapped to driven long-range Kitaev chains, bulk invariants $U(T)\,|\phi_n\rangle=e^{-i\mu_n}\,|\phi_n\rangle,$9 and $\mu_n/T$0 count Majorana zero and $\mu_n/T$1 modes. For realistic InAs/Al parameters the phase diagram includes regions with $\mu_n/T$2, and the modes appear as end-localized peaks in the time-averaged LDOS at $\mu_n/T$3 and $\mu_n/T$4, together with a subharmonic response when both zero and $\mu_n/T$5 modes coexist (Liu et al., 2018). Wu, Wu and Zhou further showed in periodically kicked Kitaev chains that the corresponding winding numbers $\mu_n/T$6 and $\mu_n/T$7 can become arbitrarily large, giving

$\mu_n/T$8

with the same bulk–edge correspondence reproduced by the Floquet entanglement spectrum (Wu et al., 2023).

Floquet bulk–edge ideas also extend to defects in three dimensions. For spatially modulated periodic driving with a line-defect geometry, the number of defect-bound chiral modes is counted by an integer invariant $\mu_n/T$9 defined on a five-dimensional torus parameterized by $\phi_n(t+T)=\phi_n(t)$0. In the weak-drive limit one finds $\phi_n(t+T)=\phi_n(t)$1, and direct diagonalization shows $\phi_n(t+T)=\phi_n(t)$2 chiral modes at quasienergy $\phi_n(t+T)=\phi_n(t)$3 propagating along the defect. Their existence does not require nonzero static second Chern numbers, which is why they are termed anomalous (Bi et al., 2016).

Coexisting zero and $\phi_n(t+T)=\phi_n(t)$4 Majorana edge modes have also been proposed as computational resources. In a periodically driven one-dimensional superconducting superlattice, three pairs of Majorana edge modes suffice to encode two logical qubits, realize quantum gate operations, and execute two simple quantum algorithms through adiabatic lattice deformation (Bomantara et al., 2018).

4. Composite, anomalous, and long-lived boundary modes

A particularly sharp refinement of the Floquet mode concept is the Floquet product mode. In the $\phi_n(t+T)=\phi_n(t)$5 phase of the driven quantum Ising chain, one defines

$\phi_n(t+T)=\phi_n(t)$6

This composite edge operator anticommutes with the Floquet unitary exactly,

$\phi_n(t+T)=\phi_n(t)$7

and at a solvable sweet spot such as $\phi_n(t+T)=\phi_n(t)$8, $\phi_n(t+T)=\phi_n(t)$9, it becomes exactly $e^{-i\epsilon_n t}$0. By contrast, the individual Majorana operators fail to commute with the integrability-breaking perturbation $e^{-i\epsilon_n t}$1, whereas $e^{-i\epsilon_n t}$2 at the sweet spot. The result is a stability hierarchy in which the product mode is substantially more robust than the separate zero and $e^{-i\epsilon_n t}$3 modes (Yeh et al., 2024, Möckel et al., 23 Feb 2026).

The spectral explanation is formulated in terms of eigenphase order. Boundary autocorrelations

$e^{-i\epsilon_n t}$4

probe different members of the quadruplet spectrum. For $e^{-i\epsilon_n t}$5, the Fourier spectrum $e^{-i\epsilon_n t}$6 is controlled by inter-parity splittings $e^{-i\epsilon_n t}$7, and these broaden rapidly even for tiny $e^{-i\epsilon_n t}$8, so $e^{-i\epsilon_n t}$9 decays extremely fast. For $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$0, the relevant peaks sit at the intra-parity splittings $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$1, which remain sharply defined up to $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$2; correspondingly $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$3 exhibits long-lived $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$4-period oscillations. Numerically and perturbatively, intra-parity splittings broaden only weakly with $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$5, while inter-parity levels cross freely under perturbation (Möckel et al., 23 Feb 2026).

The language of strong and almost strong modes makes this robustness quantitative. In Krylov-subspace constructions, Heisenberg evolution of an edge operator maps to a fictitious single-particle problem on a topological chain of SSH type. In the Arnoldi-based construction the decay rate of an almost strong $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$6 or $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$7 mode is

$$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$8

with $$\psi_n(t)=e^{-i\epsilon_n t/\hbar}\,\phi_n(t),\qquad \phi_n(t+T)=\phi_n(t),$$9 the edge-mode amplitude at the far end of the truncated Krylov chain. This yields exponential lifetimes when the effective dimerization localizes the mode near the boundary (Yates et al., 2021).

Not all boundary Floquet modes are pinned to $\epsilon_n$0 or $\epsilon_n$1. In a harmonically driven $\epsilon_n$2-wave superconducting wire, periodic modulation of the hopping magnitude or phase generates end modes whose Floquet eigenvalues can lie at $\epsilon_n$3 or, unusually, at other points on the unit circle in complex-conjugate pairs. These anomalous end modes still have equal particle and hole weight, and for small driving amplitude their Floquet eigenvalues and Fourier peaks align with extrema of the bulk Floquet bands (Saha et al., 2016).

Dissipation provides a further modification. In a periodically driven proximitized nanowire, integrating out the equilibrium $\epsilon_n$4-wave superconductor produces a frequency-dependent self-energy $\epsilon_n$5, so the effective Floquet Hamiltonian becomes non-Hermitian and the Majorana zero and $\epsilon_n$6 modes acquire finite lifetimes. An effective model gives $\epsilon_n$7, and the lifetime can be engineered by varying the drive amplitude $\epsilon_n$8 and frequency $\epsilon_n$9 (Yang et al., 2020).

5. Generalizations: field theory, multi-frequency drives, and spontaneous Floquet states

In relativistic scalar field theory, Floquet modes of breathers, quasi-breathers, and oscillons exhibit a strong universality. For small-amplitude configurations of width $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$00 in $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$01 dimensions, all leading nonrelativistic Floquet modes depend only on $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$02 and the mass $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$03, not on the detailed potential $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$04. The spectrum contains a continuum labeled by real momentum $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$05 with nonrelativistic dispersion

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$06

plus four discrete zero-frequency modes corresponding to spatial translation, time translation, Lorentz boost, and amplitude/width variation. There are no discrete shape modes at leading order (Evslin et al., 6 Nov 2025).

For commensurate multi-frequency drives, the Fourier-space representation itself acquires boundaries that can bind edge states. In the minimal two-level model with drives at $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$07 and $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$08, the Floquet problem maps to a quasi-one-dimensional lattice in the harmonic indices. In the strong-frequency regime $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$09, the effective model becomes a Rice–Mele or SSH chain with alternating hoppings $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$10 and $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$11, winding number

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$12

and edge states localized at a “Floquet boundary” set by commensurability (Park et al., 2022).

A different extension concerns spontaneous rather than externally imposed periodicity. In spontaneous Floquet states that simultaneously break several continuous symmetries, Goldstone’s theorem becomes a statement about zero-quasienergy Floquet–Nambu–Goldstone modes. When continuous time-translation symmetry itself is broken, there is a genuine temporal FNG mode whose amplitude realizes a time operator, with $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$13 for the Gibbs generator $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$14. The same formalism introduces a conserved Floquet charge $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$15 and the Floquet enthalpy $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$16, providing a thermodynamic description of spontaneous and conventional Floquet states (Nova et al., 2024).

Open time-periodic media motivate yet another generalization: Floquet quasinormal modes. In a dispersive photonic time crystal, one seeks complex quasifrequencies $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$17 at which an operator $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$18 has a nontrivial null vector, equivalently $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$19. These modes can exhibit exceptional points on symmetry axes $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$20, where phase locking to the drive occurs and non-perturbative gain or loss appears. In large cavities, families of Floquet quasinormal modes cluster toward both static limit points and exceptional points, with approach rates scaling as $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$21 in the exceptional case (Hooper et al., 14 Oct 2025).

6. Photonic and wave realizations

In space-time modulated photonics, Floquet modes are often directly measurable harmonic components of scattered or guided fields. For a zero-thickness Huygens metasurface treated with Generalized Sheet Transition Conditions, the electric and magnetic surface susceptibilities are periodically modulated in space and time, and the reflected and transmitted fields are expanded as Floquet sums over harmonics $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$22 and transverse momenta $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$23. Substitution into the Lorentz-oscillator equations and GSTCs yields a finite-dimensional linear system $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$24, whose solution determines the steady-state scattered fields and their Fourier-propagated refracted harmonics (Gupta et al., 2017).

Periodic longitudinal modulation in waveguide arrays produces several distinct localization mechanisms. In arrays with a disclination, longitudinal oscillation of the waveguides switches the structure between topological and trivial phases during a modulation cycle, yet the one-period operator $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$25 develops a genuine Floquet gap containing modes bound to the disclination core. These appear only above a threshold $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$26, with five localized modes for a pentagonal disclination ($$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$27) and seven for a heptagonal one ($$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$28); the localization length obeys $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$29. With focusing cubic nonlinearity, the corresponding Floquet solitons remain localized below $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$30 and $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$31 (Sabour et al., 8 Dec 2025).

In arrays of out-of-phase curved waveguides with a transverse Aubry–André modulation, the undriven Floquet bands undergo pseudocollapses at special bending amplitudes. For $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$32, the first four pseudocollapses occur at

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$33

while at $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$34 only one pseudocollapse in the same interval appears near $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$35. Near these points, the effective coupling is suppressed and localized Floquet modes emerge even for weak quasiperiodic modulation below the straight-array localization threshold; representative modes reproduce their profile to within $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$36 after each period (Kartashov et al., 13 Sep 2025).

Floquet mode engineering can also localize light without physical boundaries. In a Floquet octagon lattice on silicon-on-insulator, perturbing the driving sequence induces a Floquet Mode Resonance when a bulk mode satisfies

$$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$37

The resonant state forms a closed bulk-mode loop with $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$38 sites in the ideal limit, and experiments reported loaded quality factors $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$39, together with scattered-light images showing light trapped on the predicted three-site loop (Afzal et al., 2021).

Across these settings, Floquet modes are not a single phenomenological category but a unifying spectral language for periodic dynamics: they describe stroboscopic eigenstates, symmetry-protected edge operators, defect-bound channels, universal perturbations of nonlinear periodic backgrounds, harmonic scattering states, and even complex resonances of open time crystals. The recurring technical themes are quasienergy periodicity, the special role of $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$40 and $$U(T)=\mathcal{T}\exp\!\Bigl[-i\int_0^T H(t)\,dt\Bigr]$$41, and the possibility of topological or interference-based localization in dimensions that include real space, frequency space, or both.