Disordered Floquet-Chern Phase

• Disordered Floquet-Chern phases are two-dimensional, periodically driven systems in which disorder and micromotion jointly determine topological invariants, such as the Bott index.
• These phases encompass conventional Floquet-Chern insulators, disorder-induced topological Anderson insulators, and anomalous Floquet-Anderson insulators, each featuring distinct edge state behaviors.
• Varying disorder strengths drive transitions from robust topological edge modes to trivial localized states, with mobility gaps and real-space formulations capturing these effects even when clean-band indices vanish.

A disordered Floquet-Chern phase is a periodically driven two-dimensional topological phase in which disorder and micromotion jointly determine the topology of the quasienergy spectrum. In the clean limit, such phases are characterized by Chern numbers of Floquet bands or by gap invariants such as Rudner-type winding numbers; in the disordered setting, translationally invariant formulations are replaced by real-space invariants such as the Bott index, or by mobility-gap bulk indices and edge indices defined directly for the Floquet propagator. Depending on the model, weak disorder can preserve a Floquet topological insulator phase, moderate disorder can induce a Floquet topological Anderson insulator, and strong disorder can either trivialize the system or stabilize anomalous phases with chiral edge states and fully localized bulk states (Chen et al., 2018, Zheng et al., 2023, Shapiro et al., 2018).

1. Definition and scope

In the narrow sense used in model studies, a disordered Floquet-Chern phase is a driven phase whose relevant quasienergy gap carries a nonzero Chern-type topological invariant even after disorder breaks translational symmetry. In the Weyl-semimetal thin-film example, the clean resonant gap is characterized by a Chern number, while the disordered problem is characterized by a Floquet Bott index computed from the projector onto Floquet eigenstates below the resonant quasienergy gap near $\Omega/2$ (Chen et al., 2018). In periodically modulated honeycomb models, the disorder-averaged Bott index can reproduce either band Chern numbers or Floquet winding numbers, depending on whether the projector is taken onto a single band or onto all states below a chosen quasienergy such as $\epsilon=0$ or $\epsilon=\omega/2$ (Zheng et al., 2023). In rigorous treatments of strongly disordered class-A Floquet systems, the defining structure is a mobility gap of the one-period propagator $U(1)$ rather than a spectral gap, and the corresponding bulk and edge indices remain integer-valued and equal by bulk-edge correspondence (Shapiro et al., 2018).

The term also covers more than one phenomenology. One class consists of conventional Floquet-Chern insulators, in which driven bands carry nonzero Chern numbers and chiral edge modes cross the relevant gap. Another class consists of disorder-induced Floquet topological Anderson insulators, where a phase trivial at zero disorder becomes topological at intermediate disorder strength (Chen et al., 2018, Roy et al., 2016). A third class comprises anomalous Floquet-Anderson insulators, in which all bulk Floquet eigenstates are Anderson localized while chiral edge states exist at all quasienergies; in that case the appropriate invariant is a Floquet winding number rather than a band Chern number (Timms et al., 2020, Zhang et al., 2021).

A common misconception is that disorder is generically detrimental to Floquet topology. The cited models show a more differentiated picture: weak disorder can leave Floquet topology intact, moderate disorder can favor anomalous Floquet topology or induce a Floquet topological Anderson insulator, and only sufficiently strong disorder necessarily drives the system to a trivial Anderson-localized phase in the conventional Floquet-Chern cases studied (Chen et al., 2018, Zheng et al., 2023, Roy et al., 2016).

2. Topological characterization beyond the clean-band picture

For a time-periodic Hamiltonian $H(t+T)=H(t)$, Floquet theory replaces the Schrödinger problem by an eigenvalue problem for a static operator in Sambe space or, equivalently, by a quasienergy spectrum of the one-period propagator. In clean systems, the relevant topological data can be encoded in Berry-curvature integrals over quasienergy bands or in gap invariants. In the Weyl-semimetal thin film, the resonant Floquet gap is assigned the total Chern number of all quasienergy bands below $\epsilon=\Omega/2$ (Chen et al., 2018). In driven honeycomb models, the pair $(\mathcal C,\mathcal W_{1/2})$ or equivalently $(\mathcal W_0,\mathcal W_{1/2})$ distinguishes normal-insulating, Floquet-topological, anomalous, and staggered Floquet phases, with

$\mathcal W_0=\mathcal W_{1/2}+\mathcal C$

and $\mathcal W_0$, $\epsilon=0$0 counting net chiral edge modes in the $\epsilon=0$1 and $\epsilon=0$2 gaps, respectively (Zheng et al., 2023).

Once disorder destroys translational invariance, momentum-space Chern numbers are no longer directly available. The most widely used replacement in the cited numerical literature is the Bott index. For finite disordered Floquet systems with position operators $\epsilon=0$3 and projector $\epsilon=0$4 onto a chosen set of Floquet eigenstates, one defines projected unitaries $\epsilon=0$5 and $\epsilon=0$6, and then

$\epsilon=0$7

In the clean limit this matches the momentum-space Chern number, and with a projector onto all states below a chosen quasienergy it reproduces the corresponding Floquet winding number (Chen et al., 2018, Zheng et al., 2023).

A more general formulation uses mobility gaps. In the rigorous framework for strongly disordered two-dimensional Floquet systems, a mobility gap is an interval $\epsilon=0$8 of the Floquet propagator for which bounded Borel functions of $\epsilon=0$9 supported in $\epsilon=\omega/2$0 satisfy a weak-locality estimate, and all eigenvalues in $\epsilon=\omega/2$1 are of finite degeneracy (Shapiro et al., 2018). This allows the definition of an effective Hamiltonian

$\epsilon=\omega/2$2

a relative bulk invariant given by a three-dimensional noncommutative winding number, and an edge invariant defined from the half-space evolution; the two are equal by bulk-edge correspondence (Shapiro et al., 2018). In the completely localized case, alternative bulk and edge indices can be defined through quantized magnetization and pumping observables, again yielding the same integer (Shapiro et al., 2018).

A second misconception is that band Chern numbers are sufficient for Floquet topology. They are not. Anomalous Floquet phases can have vanishing Chern numbers for all Floquet bands while retaining robust chiral edge states in one or more quasienergy gaps, and in the AFAI the winding number rather than any band Chern number is the decisive bulk invariant (Zheng et al., 2023, Timms et al., 2020, Zhang et al., 2021).

3. Microscopic realizations

A concrete realization of a disordered Floquet-Chern phase is a Weyl-semimetal thin film subject to either a periodic Zeeman drive or elliptically polarized light. The starting point is a minimal two-band tight-binding model on a cubic lattice,

$\epsilon=\omega/2$3

with the parameter choice $\epsilon=\omega/2$4, $\epsilon=\omega/2$5, $\epsilon=\omega/2$6. Confinement along $\epsilon=\omega/2$7 produces a quasi-two-dimensional thin film with discrete subbands and a finite-size gap. In the quantum-well approximation each subband is described by a Bernevig-Hughes-Zhang-type $\epsilon=\omega/2$8 block

$\epsilon=\omega/2$9

with mass

$U(1)$0

For sufficiently thin films such as $U(1)$1, all subbands are trivial and the static film is a normal insulator with $U(1)$2 (Chen et al., 2018).

Two periodic drives generate Floquet topology in that system. A periodic magnetic field enters through a Zeeman term

$U(1)$3

while elliptically polarized light enters through Peierls substitution with

$U(1)$4

In the resonant window $U(1)$5, hybridization between different photon sectors opens a Floquet-engineered gap near $U(1)$6 (Chen et al., 2018).

A second family of realizations is provided by periodically modulated honeycomb lattices. In one model, nearest-neighbor hoppings are modulated as

$U(1)$7

with an added staggered sublattice potential $U(1)$8. The clean phase diagram contains a normal insulator, several Floquet topological insulators, an anomalous Floquet topological insulator with $U(1)$9, and a staggered Floquet topological insulator with $H(t+T)=H(t)$0 and edge states of alternating chirality in adjacent gaps (Zheng et al., 2023). In another honeycomb construction based on a four-step hopping sequence, the strong invariants $H(t+T)=H(t)$1 and $H(t+T)=H(t)$2 distinguish six phases, including an anomalous phase $H(t+T)=H(t)$3 with $H(t+T)=H(t)$4 and vanishing band Chern numbers (Bhargava et al., 2021).

A third realization is the two-step driven Chern insulator on a square lattice, where the mass alternates between $H(t+T)=H(t)$5 and $H(t+T)=H(t)$6 during the two half-periods. The phase is classified by the pair $H(t+T)=H(t)$7, counting net chiral edge modes in the $H(t+T)=H(t)$8 and $H(t+T)=H(t)$9 gaps, and the phase boundaries depend on the mean mass $\epsilon=\Omega/2$0 rather than on the drive amplitude $\epsilon=\Omega/2$1 (Roy et al., 2016).

Finally, the anomalous Floquet-Anderson insulator has been analyzed in a five-step square-lattice drive and in a driven, disordered quantum anomalous Hall film. In the former, the phase features chiral edge modes traversing the quasienergy Brillouin zone together with a fully localized bulk (Timms et al., 2020). In the latter, the proposed mechanism is resonant coupling and annihilation of two critical extended states carrying opposite Chern numbers in a disordered QAH system, producing a phase with $\epsilon=\Omega/2$2 for all quasienergies (Zhang et al., 2021).

4. Disorder-driven transitions and mechanisms

The simplest disorder effect is robustness of an already-topological Floquet-Chern phase. In the driven Weyl-semimetal thin film with on-site disorder $\epsilon=\Omega/2$3, the Floquet Bott index remains quantized for weak disorder and collapses only at stronger $\epsilon=\Omega/2$4. In the Zeeman-driven case, a clean $\epsilon=\Omega/2$5 Floquet topological insulator keeps $\epsilon=\Omega/2$6 over a finite disorder window before eventually becoming trivial; in the light-driven case, an analogous $\epsilon=\Omega/2$7 phase similarly survives weak disorder before collapsing at strong disorder (Chen et al., 2018). The two-step driven Chern insulator shows the same broad pattern: existing Floquet-Chern phases survive moderate disorder, but sufficiently strong disorder drives them to a trivial Anderson-insulating phase (Roy et al., 2016).

A more distinctive phenomenon is the Floquet topological Anderson insulator. In the Weyl-semimetal thin film, disorder can drive a trivial Floquet insulator into a quantized $\epsilon=\Omega/2$8 or $\epsilon=\Omega/2$9 phase, depending on the drive protocol, before stronger disorder again destroys the topology (Chen et al., 2018). In the two-step Floquet Chern insulator, intermediate disorder can similarly generate nonzero $(\mathcal C,\mathcal W_{1/2})$0 starting from a clean trivial phase, producing disorder-induced topology in either the $(\mathcal C,\mathcal W_{1/2})$1 or the $(\mathcal C,\mathcal W_{1/2})$2 gap (Roy et al., 2016).

Several mechanisms underlie such transitions. In the Weyl-semimetal thin film, a zero-order Born approximation yields a self-energy

$(\mathcal C,\mathcal W_{1/2})$3

evaluated at $(\mathcal C,\mathcal W_{1/2})$4, and the disorder-renormalized Floquet Hamiltonian $(\mathcal C,\mathcal W_{1/2})$5 reproduces the weak-disorder phase boundaries. The interpretation is that disorder renormalizes effective mass parameters and can induce a topological band inversion in the low-energy Floquet Hamiltonian (Chen et al., 2018).

In the modulated honeycomb model, weak disorder favors the anomalous Floquet topological insulator rather than a conventional Floquet-Chern phase. Nearby clean FTI phases with $(\mathcal C,\mathcal W_{1/2})$6 or $(\mathcal C,\mathcal W_{1/2})$7 flow under increasing disorder to the AFTI phase $(\mathcal C,\mathcal W_{1/2})$8, and short-range correlated disorder with $(\mathcal C,\mathcal W_{1/2})$9 induces this transition at weaker disorder than uncorrelated disorder. Within the Born approximation, uncorrelated disorder mainly renormalizes on-site terms such as $(\mathcal W_0,\mathcal W_{1/2})$0 and $(\mathcal W_0,\mathcal W_{1/2})$1, whereas correlated disorder also renormalizes hopping amplitudes $(\mathcal W_0,\mathcal W_{1/2})$2 and $(\mathcal W_0,\mathcal W_{1/2})$3 through off-site self-energy corrections (Zheng et al., 2023).

The mechanism is not universal. For point-like gap closings at $(\mathcal W_0,\mathcal W_{1/2})$4 or $(\mathcal W_0,\mathcal W_{1/2})$5, a low-energy two-level Dirac description plus Born approximation correctly captures the disorder-induced transition because disorder shifts an effective mass parameter $(\mathcal W_0,\mathcal W_{1/2})$6 across zero (Zheng et al., 2023). By contrast, for ring-shaped quasienergy gaps the same Born approach fails: weak disorder couples degenerate and near-degenerate states along an entire ring in momentum space, so the transition cannot be reduced to renormalization of a single mass at a high-symmetry point (Zheng et al., 2023). A related misconception is therefore that self-consistent or lowest-order Born theory always explains disorder-driven Floquet topology. The literature explicitly shows that it does so for point-gap transitions but fails for ring-shaped-gap transitions (Zheng et al., 2023).

In the two-step Floquet Chern insulator, disorder-driven transitions exhibit the levitation-annihilation mechanism familiar from static disordered Chern insulators. Mobility bands of delocalized quasienergy states drift toward the center of the relevant $(\mathcal W_0,\mathcal W_{1/2})$7 or $(\mathcal W_0,\mathcal W_{1/2})$8 gap and annihilate there at a critical disorder strength, at which the winding invariant changes and the phase becomes trivial (Roy et al., 2016). This mechanism operates separately in the $(\mathcal W_0,\mathcal W_{1/2})$9 and $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$0 gaps because Floquet spectra contain two inequivalent quasienergy gaps (Roy et al., 2016).

5. Edge physics, anomalous phases, and transport

Bulk-edge correspondence is the operational signature of a disordered Floquet-Chern phase. In the thin-film Weyl-semimetal model, clean calculations with open boundaries show that nonzero Chern number in the resonant gap is accompanied by chiral Floquet edge modes traversing that gap. In the disordered system, a quantized Bott index implies that boundary modes remain delocalized while bulk states at that quasienergy are localized (Chen et al., 2018). In the rigorous mobility-gap formalism, the bulk relative index equals the edge index, and in the completely localized regime the bulk magnetization index equals the long-time averaged edge pumping index (Shapiro et al., 2018).

Anomalous Floquet phases sharpen this distinction between edge transport and band topology. In the honeycomb model with modulated hoppings, the AFTI phase has $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$1, so all individual bands have zero Chern number but both the $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$2 and $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$3 gaps host robust chiral edge states with the same chirality (Zheng et al., 2023). The related staggered Floquet topological insulator has $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$4 and therefore edge states of alternating chirality in adjacent gaps; weak disorder does not easily destroy these counter-chiral modes because their quasienergy separation is about $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$5 (Zheng et al., 2023).

The anomalous Floquet-Anderson insulator is the most extreme disordered Floquet-Chern-like phase in the cited literature. In the five-step square-lattice drive, all bulk Floquet eigenstates can be Anderson localized, yet a chiral edge mode traverses the entire quasienergy Brillouin zone and pumps one unit of charge per cycle along a boundary (Timms et al., 2020). The same phase can be interpreted as a non-adiabatic topological charge pump, and a driven disordered QAH system has been proposed to realize it through resonant annihilation of oppositely charged critical Chern states (Zhang et al., 2021).

Temporal noise introduces a further distinction between strict Floquet topology and operationally quantized transport. In the noisy AFAI, the exact period-by-period Floquet symmetry is broken by random step durations, but the pumped charge exhibits a plateau $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$6 over long times for weak temporal noise and appropriate spatial disorder. The lifetime of that plateau scales as the Thouless time,

$\mathcal W_0=\mathcal W_{1/2}+\mathcal C$7

and the robustness is traced to diffusion of the density interface together with Pauli blocking of edge-state decay (Timms et al., 2020). The same work identifies a non-Hermitian winding number of the noise-averaged Floquet superoperator, which remains quantized below a finite temporal-noise threshold (Timms et al., 2020). This suggests that topological response can remain quantized even when exact discrete time-translation symmetry is absent, provided the relevant non-Hermitian gap and localization structure persist.

Not all disordered Floquet phases have the same transport constraints. In the two-step driven Chern insulator, nonzero band Chern number forces the presence of delocalized bulk states somewhere in quasienergy, precluding a true AFAI with all bulk states localized. By contrast, in anomalous phases with zero band Chern numbers, full bulk localization is compatible with robust chiral edge transport (Roy et al., 2016, Bhargava et al., 2021).

6. Conceptual structure, common pitfalls, and open directions

Three structural lessons recur across the literature. First, Floquet topology is gap-specific. The natural invariants are attached to quasienergy gaps at $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$8, $\mathcal W_0=\mathcal W_{1/2}+\mathcal C$9, or $\mathcal W_0$0, not merely to Floquet bands, and the same system can carry different topological data in different gaps (Zheng et al., 2023, Roy et al., 2016). Second, disorder must be characterized not only by strength but also by structure. Uncorrelated on-site disorder, short-range correlated disorder, bond disorder that preserves particle-hole symmetry, and temporal noise can lead to qualitatively different topological responses (Zheng et al., 2023, Bhargava et al., 2021, Timms et al., 2020). Third, mobility gaps rather than spectral gaps are sufficient for quantized Floquet topology, allowing fully rigorous bulk and edge indices in strongly disordered systems (Shapiro et al., 2018).

Several pitfalls follow from these points. One is to identify every disorder-stabilized driven phase with an AFAI. That is incorrect: conventional disordered Floquet-Chern phases with nonzero band Chern number, Floquet topological Anderson insulators induced by moderate disorder, and anomalous Floquet-Anderson insulators with fully localized bulk are distinct phenomena (Chen et al., 2018, Roy et al., 2016, Timms et al., 2020). Another is to assume that weak topological indices are robust to arbitrary disorder. In the four-step honeycomb model, weak phases survive hopping disorder that preserves particle-hole symmetry as statistical topological phases, but are destroyed by onsite disorder that breaks particle-hole symmetry (Bhargava et al., 2021). A third is to treat all disorder-driven transitions as simple mass inversions. That intuition is reliable for point-like gap closings but explicitly fails for ring-shaped gaps (Zheng et al., 2023).

Experimentally motivated platforms already appear in the literature. The driven honeycomb models are closely related to ultracold-atom realizations of anomalous Floquet topology, and spatially correlated disorder is naturally relevant there because optical speckle potentials are intrinsically correlated (Zheng et al., 2023). Photonic waveguide arrays are emphasized as a natural implementation of stepwise hopping drives and disorder engineering; in the anomalous phase, a deliberately disordered central region can function as a filter that localizes bulk components while allowing edge-state propagation (Bhargava et al., 2021). Driven magnetically doped topological-insulator films have been proposed as a solid-state route to the AFAI through Chern band annihilation (Zhang et al., 2021).

Open directions identified in the cited works include interactions, higher dimensions, symmetry classes beyond class A, quasiperiodic or noisy drives, and nonunitary extensions. The rigorous mobility-gap framework suggests that the essential topological content of a disordered Floquet-Chern phase is not tied to translation invariance or even to a spectral gap, but to the persistence of weak locality, quantized bulk indices, and matching edge transport (Shapiro et al., 2018). A plausible implication is that future classifications of nonequilibrium topological matter will continue to be organized around mobility-gap and real-space formulations rather than clean-band topology alone.