Floquet engineering of point-gapped topological superconductors

TL;DR

Non-Hermitian systems harbor line-gap and point-gap topologies, but a universal construction for point-gap phases under open boundaries has been elusive. The authors develop a Floquet-engineered framework that, together with particle-hole symmetry, realizes point-gapped topological superconductors and Floquet Majorana edge modes; the overlap of Floquet bands with opposite winding numbers creates a point gap accompanied by a Floquet $Z_2$ skin effect and a double, size-dependent phase transition governed by non-Bloch PT symmetry breaking. Through non-Bloch GBZ analysis and biorthogonal fidelity susceptibility, they identify the transition boundaries, including $\omega = |\mu \pm 2t|$, and demonstrate the mechanism in both a driven Kitaev chain and a spinful $s$-wave model, confirming generality. The work highlights the role of critical skin effects in finite systems and suggests extensions to higher dimensions and geometry-dependent skin-topology phenomena, providing a versatile route to explore point-gap physics in non-Hermitian platforms. These Floquet point-gap phases promise new routes for control and manipulation of edge modes in photonic and quantum systems.

Abstract

Non-Hermitian systems exhibit two distinct topological classifications based on their gap structure: line-gap and point-gap topologies. Although point-gap topology is intrinsic to non-Hermitian systems, its systematic construction remains a challenge. Here, we present the Floquet engineering approach for realizing point-gapped topological superconductors. By combining Floquet theory with particle-hole symmetry (PHS), we show that a point gap hosting robust Majorana edge modes emerges at the overlap of Floquet bands with opposite winding numbers. In the thermodynamic limit, even weak non-Hermiticity opens a point gap from a gapless spectrum, driving a topological phase transition and breaking non-Bloch parity-time ($\mathcal{PT}$) symmetry. This transition is accompanied by the appearance of the Floquet $Z_2$ skin effect. Furthermore, the point-gapped topological phase and the non-Bloch $\mathcal{PT}$ symmetry exhibit size-dependent phenomena driven by the critical skin effect. Our work offers a new pathway for exploring the point-gapped topological phases in non-Hermitian systems.

Figures (5)

• Figure 1: (a) Schematic illustration of the mechanism of realizing the point-gapped topological phase in a periodically driven non-Hermitian systems. The regions with $+1$ and $-1$ PBC winding number are shadowed by blue and green, respectively. (b) Quasienergy spectrum as a function of the non-Hermitian strength $\delta$ for an open periodically driven non-Hermitian Kitaev chain with $\mu=9.6$. The Floquet Majorana zero modes are highlighted in red. (c) Complex quasienergy spectrum for $\delta=0.3$. The green (blue), and gray curves represent the OBC spectrum of particles (holes) and the PBC spectrum, respectively. (d) Density distribution of eigenstates exhibit Floquet $Z_{2}$ skin effect. (e) Complex quasieigenenergy proportion $P$ as a function of $\delta$ and lattice size $L$. Numerically, the quasienergies with $\Im(E)>10^{-8}$ are regarded as complex. The inset shows the complex quasienergy spectrum of the system with $\delta=0.01$ for different lattice sizes. Other parameters are $t=1$, $\Delta=0.8$, $\lambda=0.8$, and $\omega=10$.
• Figure 2: (a) The OBC energy spectra as functions of $\mu$ with $\delta=0.3$. (b)-(d) The GBZ bands on the surface $f(\beta,\varepsilon)=0$ for $\mu = 8.0$, $9.6$ and $12.0$, respectively. The loops of particles ($C^p_{\beta}$) and holes ($C^h_{\beta}$) are split by the Brillouin zone (dotted circle), corresponding to $Z_{2}$ skin effect. The loops of particles and holes intersect at the BPs.
• Figure 3: (a)-(c) Quasienergy spectra with $\mu=9.6$ for different lattice size. The green (blue) dots denote the OBC quasienergies of particles (holes), while the gray curves represent the PBC quasienergies. (d) The finite-size GBZ of particles ($\beta_p$) and holes ($\beta_h$) for systems with different lattice sizes. (e) The decay length $\kappa_i=-\log(\abs{\beta_i})$ as a function of lattice size for the points in finite-size GBZs, which are closest to the BP. The inset shows the corresponding density distribution.
• Figure 4: (a) Ground state fidelity susceptibility $\chi$ varies with $\mu$ with different lattice size. (b) Finit-size scaling of the minimum of the susceptibility $\chi_{m} \propto - L^{3}$ . The insert shows the finit-size scaling of critical chemical potential.
• Figure 5: (a) The OBC quasienergies as functions of $\mu$ with $\delta=0.3$. (b) The quasienergy gaps. (c) The OBC quasienergy spectrum for $\mu=11.5$. (d) The fidelity susceptibilities of the states whose quasienergies are closest to 0 (denoted by $\chi_0$) and $\omega/2$ (denoted by $\chi_\pi$), respectively. Other parameters are $t=1, h=2.5$, $\alpha=0.8$, $\Delta=1.2$, $\lambda=0.8$, $\omega=10$.