arXiv:2602.19795 · Physical review. B./Physical review. B

Floquet product mode and eigenphase order

Felix Möckel, Harald Schmid, Felix von Oppen

cond-mat.str-elquant-phSubmitted Feb 23, 2026• Announced Feb 24, 2026• 1 citation

Abstract

We study the robustness of the Floquet quantum Ising model against integrability-breaking perturbations, focusing on the phase hosting both Majorana zero and $π$ modes. A recent work [Phys. Rev. B 110, 075117, (2024)] observed that the Floquet product mode, a composite edge mode constructed from both Majorana operators, is considerably more robust than the individual Majorana edge modes. We analyze these strong modes from the point of view of the eigenphase order present in finite chains with open boundary conditions. As a result of the Majorana modes, all Floquet eigenstates come in quadruplets in the integrable limit. We show that the robustness of the various modes as well as the behavior of the boundary spin correlation functions can be understood in terms of the spectral statistics of these quadruplets in the presence of integrability-breaking perturbations.

Comments: 12 pages, 9 figures

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arXiv:2602.19795 · Physical review. B./Physical review. B

Floquet product mode and eigenphase order

Felix Möckel, Harald Schmid, Felix von Oppen

cond-mat.str-elquant-phSubmitted Feb 23, 2026• Announced Feb 24, 2026• 1 citation

arXiv HTML PDF

Abstract

We study the robustness of the Floquet quantum Ising model against integrability-breaking perturbations, focusing on the phase hosting both Majorana zero and

π

modes. A recent work [Phys. Rev. B 110, 075117, (2024)] observed that the Floquet product mode, a composite edge mode constructed from both Majorana operators, is considerably more robust than the individual Majorana edge modes. We analyze these strong modes from the point of view of the eigenphase order present in finite chains with open boundary conditions. As a result of the Majorana modes, all Floquet eigenstates come in quadruplets in the integrable limit. We show that the robustness of the various modes as well as the behavior of the boundary spin correlation functions can be understood in terms of the spectral statistics of these quadruplets in the presence of integrability-breaking perturbations.

Comments: 12 pages, 9 figures

Related papers

Decay rates of almost strong modes in Floquet spin chains beyond Fermi's Golden Rule

Hsiu-Chung Yeh, Achim Rosch, Aditi Mitra

The stability and dynamics of almost strong zero and $\ensuremath{\pi}$ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction ${J}_{z}$ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where $T$ is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to ${J}_{z}^{2n}$ . Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in ${J}_{z}$ , an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.The stability and dynamics of almost strong zero and $\ensuremath{\pi}$ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction ${J}_{z}$ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where $T$ is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to ${J}_{z}^{2n}$. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in ${J}_{z}$, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.

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The atomic dipoles in the Rabi lattice model exhibit quantum Ising dynamics in the limit of strong atom-photon interaction. It governs the paraelectric to ferroelectric phase transition in the ground state. On an open chain, it implies the existence of two Majorana-like, albeit topologically unprotected, edge modes in the ordered phase. The relation ${\ensuremath{\rho}}_{1L}^{x}={p}^{8}$ between the end-to-end dipole correlation ${\ensuremath{\rho}}_{1L}^{x}$ and the spontaneous polarization $p$ is proposed as an observable signature of these edge modes. The density-matrix renormalization-group calculations on the one-dimensional Rabi lattice support the strong-coupling quantum Ising behavior and correctly yield the proposed end-to-end dipole correlation. The conditions that protect the edge modes against the adverse perturbations are also identified.The atomic dipoles in the Rabi lattice model exhibit quantum Ising dynamics in the limit of strong atom-photon interaction. It governs the paraelectric to ferroelectric phase transition in the ground state. On an open chain, it implies the existence of two Majorana-like, albeit topologically unprotected, edge modes in the ordered phase. The relation ${\ensuremath{\rho}}_{1L}^{x}={p}^{8}$ between the end-to-end dipole correlation ${\ensuremath{\rho}}_{1L}^{x}$ and the spontaneous polarization $p$ is proposed as an observable signature of these edge modes. The density-matrix renormalization-group calculations on the one-dimensional Rabi lattice support the strong-coupling quantum Ising behavior and correctly yield the proposed end-to-end dipole correlation. The conditions that protect the edge modes against the adverse perturbations are also identified.

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Lei Su, Aashish A. Clerk, Ivar Martin

Inspired by current research on measurement-induced quantum phase transitions, we analyze the nonunitary Floquet transverse-field Ising model with complex nearest-neighbor couplings and complex transverse fields. Unlike its unitary counterpart, the model shows a number of steady phases, stable to integrability-breaking perturbations. Some phases have robust edge modes and/or spatiotemporal long-range orders in the bulk. The transitions between the phases have extensive entanglement entropy, whose scaling with the system size depends on the number of the real quasiparticle modes in the spectrum at the transition. In particular, the volume-law scaling appears on some critical lines, protected by pseudo-Hermiticity. Both the scaling of entanglement entropy in steady states and the evolution after a quench are compatible with the non-Hermitian generalization of the quasiparticle picture of Calabrese and Cardy at least qualitatively. Published by the American Physical Society 2024Inspired by current research on measurement-induced quantum phase transitions, we analyze the nonunitary Floquet transverse-field Ising model with complex nearest-neighbor couplings and complex transverse fields. Unlike its unitary counterpart, the model shows a number of steady phases, stable to integrability-breaking perturbations. Some phases have robust edge modes and/or spatiotemporal long-range orders in the bulk. The transitions between the phases have extensive entanglement entropy, whose scaling with the system size depends on the number of the real quasiparticle modes in the spectrum at the transition. In particular, the volume-law scaling appears on some critical lines, protected by pseudo-Hermiticity. Both the scaling of entanglement entropy in steady states and the evolution after a quench are compatible with the non-Hermitian generalization of the quasiparticle picture of Calabrese and Cardy at least qualitatively. Published by the American Physical Society 2024

arXiv:2306.07428Physical Review ResearchJan 31, 202412 citations

We study the robustness of the Floquet quantum Ising model against integrability-breaking perturbations, focusing on the phase hosting both Majorana zero and

π

{J}_{z}

is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where

T

is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to

{J}_{z}^{2n}

. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in

{J}_{z}

, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.

U>t

. We also derive the explicit form of the many-body Majorana edge wave function, revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point, as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems.

p

is proposed as an observable signature of these edge modes. The density-matrix renormalization-group calculations on the one-dimensional Rabi lattice support the strong-coupling quantum Ising behavior and correctly yield the proposed end-to-end dipole correlation. The conditions that protect the edge modes against the adverse perturbations are also identified.

Through the study of the Rep(

D_8

) non-invertible symmetry, we show how non-invertible symmetries manifest in dynamics. Results are presented for dynamics generated by Hamiltonians as well as Floquet unitaries. For both examples, the role of the non-invertible symmetry is studied through the appearance of non-invertible symmetry protected edge modes. In addition, the role of the non-invertible symmetry for the Hamiltonian is studied through eigenstate order. In particular, by considering the effect of symmetry preserving disorder, the non-invertible symmetry is shown to give rise to degeneracies in the spectra of the Hamiltonian that can only be completely lifted at orders of perturbation that scale with system size. The eigenstates of disordered Hamiltonians, whose ground states correspond to non-trivial symmetry protected topological (SPT) states, are shown to have either trivial or non-trivial SPT order that are detected as non-zero expectation value of string order-parameters. In contrast, non-trivial SPT order is absent in the eigenstates of trivial SPT Hamiltonians with disorder. The interface between two different SPT phases host edge modes whose dynamics is studied numerically and analytically. The edge mode is shown to oscillate at frequencies related to different effective chain lengths that are weighted by the temperature, becoming an exact zero mode in the limit of zero temperature. A Floquet model with the non-invertible symmetry is constructed whose edge mode is shown to exhibit period-doubled dynamics at low effective-temperatures. The zero and period-doubled edge modes differ from those in conventional SPTs by being symmetric under the invertible symmetry, while being charged under the non-invertible symmetry.