Floquet Majorana Flat Edge Modes

• Floquet Majorana flat edge modes are boundary-localized quasiparticles in periodically driven superconductors, pinned at quasienergies 0 and π/T due to symmetry protection.
• In one-dimensional systems like driven Kitaev chains, exact pinning of discrete Majorana modes arises, while two-dimensional setups yield dispersionless edge bands via momentum-resolved topological invariants.
• These modes are robust against disorder and interactions, offering promising routes for quantum information processing and the realization of novel Floquet time-crystalline phases.

Searching arXiv for the cited works and related FMFEM literature to ground the article in current research. Floquet Majorana flat edge modes (FMFEMs) are boundary-localized Majorana quasiparticles of periodically driven superconductors whose quasienergies are pinned to special Floquet values—most prominently $0$ and $\pi/T$—and whose edge dispersion is flat or effectively flat. In two-dimensional settings this means dispersionless edge bands as a function of momentum parallel to the boundary; in strictly one-dimensional settings, where no transverse edge momentum exists, “flatness” reduces to exact pinning of discrete edge levels at $0$ or $\pi/T$. The terminology is not uniform across the literature: some works speak of Floquet Majorana flat bands or Floquet Majorana edge modes, whereas others discuss many $0$- and $\pi$-Majorana edge modes without using the acronym FMFEM, but the common structure is a Floquet topological superconductor with symmetry-protected Majorana boundary states in the $0$ and/or $\pi$ quasienergy gaps (Poudel et al., 2014, Wu et al., 2023, Bera et al., 2 Jun 2026).

1. Conceptual setting and early development

The natural kinematic object for FMFEMs is the Floquet operator

$$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$

whose eigenphases define quasienergies modulo $2\pi/T$. Because BdG particle–hole symmetry identifies $\pi/T$0 with $\pi/T$1, the quasienergies $\pi/T$2 and $\pi/T$3 are distinguished: at these values a Floquet mode can be self-conjugate and hence Majorana. This Floquet doubling of special gaps has no static analog; it permits simultaneous $\pi/T$4- and $\pi/T$5-Majorana edge sectors and underlies the basic distinction between equilibrium Majorana flat bands and Floquet Majorana flat edge modes (Bomantara et al., 2018).

A first systematic route to Floquet Majorana end modes in one dimension was the periodically driven Kitaev chain with time-dependent chemical potential, hopping, or pairing. In that setting, periodic driving was shown to generate Majorana end modes even when the static system was trivial, and for time-reversal-symmetric drives the Floquet eigenvalues of the end modes are exactly $\pi/T$6. The same work introduced two Floquet topological invariants for the kicked-$\pi/T$7 protocol and showed that the number of end modes can become very large as the driving frequency decreases (Thakurathi et al., 2013).

The extension from one-dimensional end modes to genuinely momentum-flat edge structures emerged in higher-dimensional driven systems. In the Kitaev honeycomb model with periodically kicked bond couplings, Floquet Majorana modes appear on edges and corners, including on edges that do not support equilibrium Majorana solutions. In driven $\pi/T$8-wave superconductors and later in two-dimensional magnet–superconductor heterostructures, periodic control was shown to generate dispersionless edge bands at quasienergy $\pi/T$9, $0$0, or $0$1, making the flat-band interpretation explicit (Thakurathi et al., 2013, Poudel et al., 2014).

2. One-dimensional realizations and the meaning of “flatness”

In one dimension, FMFEMs are realized most explicitly in periodically driven Kitaev-type chains. The periodically kicked Kitaev chain provides a canonical example. Two protocols were analyzed in detail: kicked pairing amplitude (PKKC1) and kicked hopping amplitude (PKKC2), both with period $0$2. In symmetric time frames the Floquet operators belong to class BDI, and the two integer winding numbers $0$3 and $0$4 combine into the physically relevant gap invariants

$0$5

Under open boundary conditions, the numbers of Majorana edge modes obey

$0$6

These driven chains support phases with large topological invariants and arbitrarily many Majorana edge modes, strictly pinned to quasienergy $0$7 or $0$8, and the number of $0$9- and $\pi/T$0-modes grows stepwise as drive parameters are increased (Wu et al., 2023).

This one-dimensional setting also fixes a common ambiguity in the phrase “flat edge mode.” In a chain there is no momentum along the edge, so flatness does not mean a dispersionless band $\pi/T$1. Rather, it means that all boundary modes at a given edge quasienergy are exactly pinned to $\pi/T$2 or $\pi/T$3, up to exponentially small finite-size splittings. Several works make this point explicitly, including driven Kitaev chains with multiple harmonic drivings and Floquet superconducting superlattices with coexisting $\pi/T$4- and $\pi/T$5-modes (Wang et al., 2019, Bomantara et al., 2018).

One-dimensional Floquet engineering also broadens the accessible parameter space. In the multiply driven Kitaev chain with

$\pi/T$6

the relative phase $\pi/T$7 changes which avoided crossings open in the Floquet band structure. A generalized Zak-phase construction was proposed to diagnose edge modes in the $\pi/T$8 and $\pi/T$9 gaps, and multiple driving creates new regions where Majorana zero modes and Majorana $0$0-modes coexist (Wang et al., 2019).

A particularly concrete one-dimensional realization is the periodically quenched superconducting superlattice with Floquet BDI invariant $0$1. Under open boundary conditions it hosts three Majorana modes per edge: two at quasienergy $0$2 and one at $0$3. In the ideal limit these modes are exactly localized on the first and last unit cells and satisfy

$0$4

which is the one-dimensional prototype of a flat, symmetry-protected Floquet Majorana edge multiplet (Bomantara et al., 2018).

3. Two-dimensional FMFEMs and flat-band geometries

True momentum-flat FMFEMs arise when the boundary itself is one-dimensional. In the periodically kicked Kitaev honeycomb model, a strip geometry maps each good edge momentum $0$5 to an effective one-dimensional Floquet Kitaev chain. Whenever the chain is topological for a continuum of $0$6, the edge spectrum contains a macroscopic set of modes with quasienergy exactly $0$7 or $0$8, independent of $0$9. In this sense the driven honeycomb model realizes Floquet Majorana flat edge bands, and periodic kicks can generate them on edges that are trivial in equilibrium (Thakurathi et al., 2013).

A distinct route uses two-band $\pi$0-wave superconductors. There, periodic modulation of the chemical potential or an in-plane magnetic field was shown to generate flat bands of symmetry-protected Majorana edge modes in the quasienergy spectrum, starting from equilibrium conditions that are either topologically trivial or only support individual Majorana pairs. In one mechanism, the drive activates a pre-existing chiral symmetry and induces Floquet Majorana flat bands at quasienergy $\pi$1 and $\pi$2. In a second mechanism, effective parity kicks dynamically generate a desired chiral symmetry by suppressing chirality-breaking terms in the static Hamiltonian (Poudel et al., 2014).

The kicked $\pi$3-wave superconducting Harper model provides a synthetic-dimension realization. Two periodic phase parameters $\pi$4 act as artificial dimensions, so open boundary conditions along the physical lattice direction expose a two-dimensional surface Brillouin zone. In the static problem, line nodes generate zero-energy surface Majorana flat bands; in the kicked problem, line nodes appear at quasienergy $\pi$5 and $\pi$6, and the surface hosts Floquet Majorana flat bands in both gaps. For certain values of the pairing order parameter, Floquet Su–Schrieffer–Heeger-like edge modes appear as arcs connecting different Floquet Majorana flat bands (1609.01865).

A recent two-dimensional realization is the noncolinear magnet–superconductor heterostructure, or 2D Shiba lattice, with harmonic chemical-potential driving. In a real-space tight-binding model, this system supports both regular $\pi$7- and anomalous $\pi$8-Floquet Majorana flat edge modes. The phase structure is frequency dependent: at low frequency only $\pi$9-FMFEMs appear, at intermediate frequency $0$0- and $0$1-FMFEMs coexist, and at high frequency only $0$2-FMFEMs remain. The $0$3-sector is anomalous in the precise sense that it has no static analog and is not captured by simple static effective-Hamiltonian reasoning (Bera et al., 2 Jun 2026).

4. Symmetry, topology, and bulk–edge correspondence

The protecting symmetry structure of FMFEMs is typically BDI-like, with particle–hole symmetry, time-reversal symmetry, and chiral symmetry. In the periodically kicked Kitaev chain, the symmetric-frame Floquet operators satisfy

$0$4

and the associated winding numbers

$0$5

generate the two Floquet invariants $0$6 and $0$7. The resulting bulk–edge correspondence

$0$8

is the basic counting principle for many one-dimensional FMFEM phases (Wu et al., 2023).

Floquet systems also admit invariants that separately count $0$9- and $\pi$0-modes when a single winding number is insufficient. For the kicked-$\pi$1 Kitaev chain, a second invariant based on the special momenta $\pi$2 was constructed from

$\pi$3

and it correctly predicts the numbers $\pi$4 and $\pi$5 of end modes with Floquet eigenvalues $\pi$6 and $\pi$7. This invariant is more powerful than the standard winding number in separating the $\pi$8 and $\pi$9 sectors (Thakurathi et al., 2013).

In two dimensions, flat-band FMFEMs are commonly tied to a momentum-resolved winding invariant. For the driven 2D Shiba lattice, chiral symmetry of the bulk evolution operator leads to

$$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$0

and nonzero $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$1 appears exactly in the momentum windows where edge spectra display $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$2- or $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$3-FMFEMs. In the kicked Harper model, the 1D Floquet chains at fixed $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$4 are characterized by a pair of $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$5 invariants $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$6, where $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$7 signals an odd number of Floquet Majorana zero modes and $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$8 an odd number of Floquet Majorana $$\mathcal{U}=\mathcal{T}\exp\left(-i\int_0^T H(t)\,dt\right),$$9-modes (Bera et al., 2 Jun 2026, 1609.01865).

An important limitation of static effective descriptions emerges in the anomalous $2\pi/T$0-sector. In the 2D Shiba lattice, Brillouin–Wigner and Floquet perturbation theory reproduce the $2\pi/T$1-quasienergy structure well in the higher-frequency and high-amplitude domain, particularly close to the $2\pi/T$2-quasi-energy modes, but they do not capture the $2\pi/T$3-FMFEMs. This makes the $2\pi/T$4-sector a genuinely dynamical topological phenomenon rather than a simple dressed version of an equilibrium band structure (Bera et al., 2 Jun 2026).

5. Entanglement structure, localization diagnostics, and dynamical signatures

The spectral bulk–edge picture has an entanglement counterpart. For periodically kicked Kitaev chains, the reduced density matrix in a symmetric time frame defines a Floquet entanglement Hamiltonian $2\pi/T$5, whose single-particle entanglement spectrum contains maximally entangled edge modes at $2\pi/T$6. A real-space entanglement winding number

$2\pi/T$7

was found numerically to satisfy

$2\pi/T$8

where $2\pi/T$9 is the number of entanglement edge modes at $\pi/T$00. Combined with the physical mode counts, this yields

$\pi/T$01

so the entanglement spectrum contains the full information about the multiplicities of $\pi/T$02- and $\pi/T$03-Majorana edge modes. The bipartite entanglement entropy shows sharp cusps at Floquet topological transitions, with $\pi/T$04 per entanglement mode appearing or disappearing (Wu et al., 2023).

Localization of FMFEMs is commonly diagnosed by inverse participation ratio and by real-space profiles. In driven Kitaev chains and kicked honeycomb models, edge modes are identified as Floquet eigenvectors with large IPR, quasienergy $\pi/T$05 or $\pi/T$06, and exponentially decaying weight away from the boundary. In the honeycomb case, finite-size numerics further distinguish edge-localized from corner-localized Floquet Majorana states (Thakurathi et al., 2013, Thakurathi et al., 2013).

Local density of states provides a complementary probe. In the dynamically generated Floquet Majorana flat bands of driven $\pi/T$07-wave superconductors, the quasienergy-resolved LDOS shows strong edge-localized peaks at $\pi/T$08 and $\pi/T$09. In the 2D Shiba lattice, LDOS snapshots over a single driving period track the appearance of $\pi/T$10- and $\pi/T$11-FMFEMs in real time and show that the relevant weight remains concentrated along the edges rather than the bulk (Poudel et al., 2014, Bera et al., 2 Jun 2026).

The $\pi/T$12-sector also yields a distinctive dynamical signature: subharmonic response. In a periodically driven 1D SPT chain equivalent to a driven Kitaev chain, Majorana $\pi/T$13-modes generate spin oscillations with twice the driving period, localized near the edges. Starting from highly nonequilibrium initial states, these edge oscillations can have exponentially long lifetimes in clean systems; disorder can be engineered to stabilize the subharmonic response in a Floquet many-body localized phase. This directly connects Floquet Majorana boundary modes to boundary time-crystalline behavior (Shtanko et al., 2019).

6. Quantum-information relevance, interacting extensions, and recurring clarifications

FMFEMs are of interest not only as boundary states but also as operational resources. One-dimensional Floquet superconducting superlattices with coexisting $\pi/T$14- and $\pi/T$15-Majorana modes realize a Floquet BDI SPT phase with $\pi/T$16, so that three pairs of Majorana edge modes suffice to encode two logical qubits, realize gate operations, and execute two simple quantum algorithms through adiabatic lattice deformation. A central advantage is that during even-period protocols both $\pi/T$17- and $\pi/T$18-modes acquire trivial dynamical phases, leaving a purely geometric holonomy (Bomantara et al., 2018).

The broader one-dimensional Floquet literature points in the same direction. Large-invariant phases with many $\pi/T$19- and $\pi/T$20-edge modes provide a boundary Hilbert space that is larger than in the static Kitaev chain. This enlarges the space of possible boundary time-crystal and Floquet quantum-computing schemes, and a plausible implication is that many-mode FMFEM phases furnish a multi-qubit resource whenever the corresponding edge modes can be individually addressed without closing the protecting quasienergy gaps (Wu et al., 2023).

A recurrent misconception is that Floquet Majorana flat modes are merely driven versions of pre-existing equilibrium edge states. Several counterexamples show otherwise. Periodic kicks in the honeycomb model generate Floquet Majorana modes on zigzag edges even when the corresponding equilibrium Hamiltonian has no Majorana mode solutions on those edges, and periodic driving in $\pi/T$21-wave superconductors can generate flat bands starting from equilibrium conditions that are topologically trivial (Thakurathi et al., 2013, Poudel et al., 2014).

A second recurring clarification concerns interaction effects. The parafermion chain with $\pi/T$22 symmetry subject to a periodic binary drive realizes edge modes at quasienergies $\pi/T$23 and $\pi/T$24, robust to weak disorder, in an intrinsically strongly interacting setting. Although these are parafermionic rather than Majorana modes, they show that the existence of Floquet edge modes at nontrivial quasienergy does not rely on a weakly interacting limit. This suggests a natural generalization of FMFEM ideas beyond $\pi/T$25 Majorana systems toward interacting Floquet anyonic boundary phases (Sreejith et al., 2016).

Taken together, the literature establishes FMFEMs as a family of Floquet boundary phenomena rather than a single model-specific effect. In one dimension they appear as quasienergy-flat multiplets at $\pi/T$26 and $\pi/T$27; in two dimensions they become genuinely dispersionless edge bands; in anomalous Floquet phases they may exist only in the $\pi/T$28-gap; and in interacting generalizations they extend beyond Majorana statistics. Across these settings, the defining ingredients remain periodic driving, BdG particle–hole structure, a protecting chiral symmetry or equivalent Floquet topological constraint, and a bulk–edge correspondence formulated directly in quasienergy.