Fate of Topological Edge States in Disordered Periodically-driven Nonlinear Systems

TL;DR

This work investigates the fate of topological edge states in disordered, periodically driven nonlinear systems by constructing Floquet stationary states through a self-consistent method and analyzing their linear stability. It reveals a nonlinear-induced transition between long- and short-lived edge states, characterized by collisions of edge-dominant eigenstates and Krein-signature changes, linked to pseudo-Hermiticity breaking. Adding random potentials further homogenizes lifetimes across edge states via edge–bulk state mixing, with randomness prolonging lifetimes in the short-lived regime and shortening them in the long-lived regime. The findings highlight a nonlinear Floquet platform for exploring pseudo-Hermiticity breaking and propose observable signatures in experiments through edge-state intensity dynamics.

Abstract

We explore topological edge states in periodically driven nonlinear systems. Based on a self-consistency method adjusted to periodically driven systems, we obtain stationary states associated with topological phases unique to Floquet systems. In addition, we study the linear stability of these edge states and reveal that Floquet stationary edge states experience a sort of transition between two regions I and II, in which lifetimes of these edge states are extremely long and short, respectively. We characterize the transitions in lifetimes by Krein signatures or equivalently the pseudo-Hermiticity breaking, and clarify that the transitions between regions I and II are signified by collisions of edge-dominant eigenstates of Floquet operators for fluctuations. We also analyze the effects of random potentials and clarify that lifetimes of various stationary edge states are equalized due to the randomness-induced mixing of edge- and bulk-dominant eigenstates. This intriguing phenomenon originating from a competition between the nonlinearity and randomness results in that random potentials prolong lifetimes in the region II and vice versa in the region I. These changes of lifetimes induced by nonlinear and/or random effects should be detectable in experiments by preparing initial states akin to the edge states.

Figures (5)

• Figure 1: (a) A schematic picture of the hopping terms $H_m$ in four time frames $m$ in Eq. (\ref{['eq:Hamiltonian']}). Solid filled, solid empty, broken filled, and broken empty arrows are respectively couplings for $m=1,2,3,4$. (b) Eigenvalues of $\mathbb{T}\exp[-i\int_0^TdtH(t)\,]$ in the linear regime ($g=0$) under OBC, with $h=2.2\pi/T,\,T=1,\,L=24$, and $V=0$. Green empty circles correspond to topological edge states inside the gap of the bulk spectrum composed of navy filled squares. A few topological edge states in the bulk spectrum are included in navy filled squares. (c) Fidelities between Floquet stationary edge states (left axis) and quasienergies (green circles, right axis) as functions of $g$ with $V(\bm r)=0$. Gray asterisks and pink empty squares respectively represent $\Delta[\phi_{g-\delta g}(0),\phi_g(0)]$ and $\Delta[\phi_{g=0}(0),\phi_g(0)]$. We choose one stationary state $\ket{\phi_g(t)}$ that is continuously connected to the topological edge state with $\varepsilon T/\pi\simeq0.85$ in the linear regime $g=0$. (d) Fidelities between Floquet stationary edge states $\ket{\phi_v(t)}$ (left axis) and quasienergies (green circles, right axis) as functions of $v$ at $g=-3$. Gray asterisks and pink empty squares show $\Delta[\phi_{v-\delta v}(0),\phi_v(0)]$ and $\Delta[\phi_{v=0 }(0),\phi_v(0)]$ respectively. (e) Intensity profiles of an edge state $|\phi({\bm r},t)|^2$ corresponding to the state in (c), with $g=4,\,h=2.2\pi/T,\,T=1$, and $V=0$ at (e-0) $t=0$ (or $T$), (e-1) $t=T/4$, (e-2) $t=T/2$, and (e-3) $t=3T/4$.
• Figure 2: (a) Dependence of $\max (|\lambda|)$ on the strength of nonlinearity $g$ with $h=2.2\pi/T$ and $L=24$ for Floquet stationary states originating from topological edge states whose quasienergies at $g=0$ are $\varepsilon T/\pi\simeq0.98$ (green filled circles), $0.85$ (black filled squares), and $0.43$ (orange asterisks). Each stationary state has two distinct regions, the long-lifetime region I and the short-lifetime region II. The inset figure shows the results for states with negative quasienergies at $g=0$; $\varepsilon T/\pi\simeq-0.98$ (green empty circles), $-0.85$ (black empty squares), and $-0.43$ (orange crossess). (b)-(e) Eigenvalues of $G[\{\phi_\text{F}(t)\}]$ with various $g$ for $\ket{\phi_\text{F}(t)}$ that has the largest region I, corresponding to green circles in (a). Green triangles, red squares, and blue circles represent eigenvalues whose Krein signatures are $0$, $+1$, and $-1$, respectively. The dashed curves show the unit circle. (b) Eigenvalues at $g=1$, where $P_\text{edge}^n<0.5$ in (b-1) and $P_\text{edge}^n\geq0.5$ in (b-2). (c),(d) Eigenvalues with $g=5$ and $g=-3$ respectively , where the corresponding eigenstates satisfy $P_\text{edge}^n\geq0.5$. (e) Trajectories of two eigenvalues with $P_\text{edge}^n\geq0.5$ in (e-1) $0<g<2$ and (e-2) $2<g<4$. Symbols with light (deep) color correspond to smaller (larger) $g$ and eigenvalues move in the directions of arrows as $g$ is increased.
• Figure 3: (a) The fidelities between Floquet states $\ket{\phi_\text{F}(0)}$ and $\ket{\psi(t=mT)}$ with $m$ being integers when initial states are chosen as in Eq. (\ref{['eq:initial_state']}) with $g=4$ and $w=10^{-3}$. The meanings of symbols are the same as in Fig. \ref{['fig:max_eigenvalue']} (a). (b) The intensity profiles at (b-1) $t=0$ and (b-2) $t=500T$ corresponding to the dynamics shown by green filled circles in (a). (c) The same figure as (b) except that the dynamics corresponds to orange asterisks in (a).
• Figure 4: (a) The largest values of $|\lambda_n|$ and (b) mean edge weights for eigenstates with $|\lambda_n|\neq1$ as functions of $v$, averaged over $100$ ensembles with $h=2.2\pi/T,\,g=-3$, and $L=12$. Dark green, light blue, yellow, black, purple, pink, red, light green, and blue symbols respectively correspond to various Floquet stationary edge states whose origins are different topological edge states with $\varepsilon T/\pi\simeq-0.53,\,-0.61,\,-0.70,\,-0.78,\,0.70,\,0.61,\,0.96,\,-0.87$, and $-0.96$. Under the nonlinearlity $g=-3$ in the clean limit $v=0$, the blue one represents the stationary edge state in the region I, while the other states reside in the region II.
• Figure 5: (a-1,b-1) The intensities of Floquet stationary edge states at $t=0$, (a-2,b-2) corresponding eigenvalues of $G[\{\phi_\text{F}(t)\}]$, and (a-3,b-3) fidelities during the dynamics for initial states in Eq. (\ref{['eq:initial_state']}) with $w=10^{-3}$, under specific realizations of $V(\bm r)$. Stationary states in (a) and (b) respectively correspond to blue squares in the region I and yellow triangles in the region II in Fig. \ref{['fig:max_edge-ratio']}. In (a), green filled circles and orange empty squares correspond to $\ket{\phi_\text{F}(t)}$ respectively in random ($v=6.7$) and clean ($v=0$) systems, where (a-1) shows the intensity profiles in the former case. In (b), the meanings of symbols are the same as in (a) except that the strength of $V(\bm r)$ is $v=6.1$ in the random system.