Floquet product mode and eigenphase order

TL;DR

The paper analyzes the robustness of Floquet strong modes in the Floquet quantum Ising chain, focusing on the $0\pi$ phase where both Majorana zero modes and Majorana $\pi$ modes coexist. By examining eigenphase order and spectral quadruplets in the integrable limit, it explains how integrability-breaking perturbations split these quadruplets into doublets and why the Floquet product mode remains unusually robust. A combination of numerical simulations and Floquet perturbation theory reveals that inter-parity splittings are sensitive to small perturbations (scaling roughly as $J_x^2$), while intra-parity splittings are exponentially suppressed and much more stable, leading to Gaussian boundary correlation decay at small $J_x$ that crosses to exponential decay at larger $J_x$. The results illuminate a hierarchy of edge-mode stability driven by symmetry sectors and spectral statistics, with potential extensions to other driven systems and multi-edge scenarios such as Floquet clock models.

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.

Figures (9)

• Figure 1: Floquet dynamics in the integrable case for the phase exhibiting both Majorana zero and $\pi$ modes. (a) Phase diagram of the Floquet quantum Ising chain, with phases exhibiting Majorana zero modes (MZM) and Majoranas $\pi$ modes (MPM). We focus on the $0\pi$ phase where both modes are present (red lines). (b) Eigenphase order of the Floquet many-body spectrum in the $0\pi$ phase: Eigenphases form quadruplets, labeled by their boundary mode occupation $(n_{0},n_\pi)$ in the different parity sectors (colors). (c) Correlation functions $G_{X,Z}(t)$ of the boundary operators $X_1$ and $Z_1$ show period doubling. (d) Coherent long-time oscillations of $G^\pm_Z(t)$ (defined in the text) and $(-1)^tG_X(t)$ are induced by hybridization of boundary modes, with oscillation period determined by inverse Majorana-hybridization splittings [Fig. \ref{['figsplit']}(a)]. Parameters: $N=8$, $J_z=0.9$, $g=0.505$.
• Figure 2: Quadruplet splittings in the many-body Floquet spectrum (not to scale). (a) Non-interacting case. Quadruplets are labeled by their boundary mode occupation $(n_{0},n_\pi)$ in the different parity sectors (see colors). Spectral pairings are hybridization-split described through inter-parity splittings $\delta_{0,\pi}$ (left) or intra-parity splittings $\delta_\pm$ (right). Inter- and intra-parity splittings are comparable in magnitude and splittings are identical for all quadruplets. (b) Interacting case. Splittings can be distinguished spectrally between parity sectors and depend on the quadruplet $n$. Inter-parity splittings (left) are much larger than intra-parity splittings (right zoom).
• Figure 3: Floquet quadruplet dynamics (interacting case). (a) Correlation functions $G^+_Z(t)$ (solid) and $G^-_Z(t)$ (dashed). The dynamics relaxes quickly for small interactions. (b) Fourier transforms $G^+_Z(\omega)$ (solid) and $G^-_Z(\omega)$ (dashed). Peaks associated to the intra-parity splittings $\delta_{0,\pi}$ vanish with increasing coupling. (c) Inter-parity splitting distributions for zero (solid) and $\pi$-splittings (dashed). The behavior tracks $G^{\pm}_Z(\omega)$. (d) Correlation function $G_X(t)$. Long-time oscillations are visible for much larger interactions than for $G_Z(t)$. (e) Fourier transform $G_X(\omega)$. Peaks associated to the inter-parity splittings $\delta_{\pm}$ can still be identified in the presence of moderate interactions. (f) Intra-parity splittings. The behavior follows $G_X(\omega)$. Parameters: $N=12$, $J_z=0.9$, $g=0.52$.
• Figure 4: Temporal decay of correlation functions in the presence of coupling $J_x$ (data: colored lines). (a) $G^+_Z(t)$ decays as a Gaussian (black dashed) for intermediate couplings and as an exponential for larger couplings $J_x$ (black dotted). (b) $G_X(t)$ decays as a linear combination of terms with Gaussian envelope as detailed in Eq. \ref{['eq:fit double Gauss']} (black dashed) for intermediate couplings and as an exponential for larger interactions (black dotted). Parameters: $N=12$, $J_z=0.9$, $g=0.52$.
• Figure 5: Splittings in the presence of transverse couplings $J_x$. (a) Typical inter-parity splittings are lifted by small $J_x$ (double-logarithmic scale). Dash-dotted: exact-diagonalization data. Solid: perturbation theory. (b) Typical intra-parity splittings are robust against large $J_x$ (semi-logarithmic scale). Dots: exact-diagonalization data. Solid: perturbation theory. (c) Intra-parity splitting fluctuations for which $|\delta^\prime_{+}|>\delta_{t}$. The threshold $\delta_{t}=(\delta_++\delta_-)/2$ is chosen to separate the peaks in the intra-parity splitting distributions. Solid: Perturbation theory. Black dashed and black dotted lines indicate linear ($\propto J_x$) and quadratic ($\propto J_x^2$) scaling, respectively, and serve as guides to the eye for identifying the relevant perturbative order. Inset: Distributions at the corresponding $J_x$-couplings (see colors), with respective Gaussian and Lorentzian fits. (d) Typical intra-splittings vs. $J_z$-coupling. Solid black lines for bare splittings. Parameters: (a)-(d) $g=0.52$; (a)-(c) $J_z=0.9$.