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.
Highlights
Driven systems or equivalently Floquet systems have attracted much attention
We have studied Floquet stationary states in periodically driven nonlinear systems which have anomalous Floquet topological phases in the linear regime
By gradually altering the strength of nonlinearity, we have numerically obtained stationary states which are directly connected to anomalous topological edge states
Summary
Driven systems or equivalently Floquet systems have attracted much attention. In addition to the periodic driving mentioned above, various physical effects, which can make systems nonequilibrium, such as gain-loss and nonlinearity, can be present in many physical settings Even under such effects, topological edge states and related phenomena have been observed [10,12,25– 32]. We focus on periodically driven nonlinear systems described by the Gross-Pitaevskii equation and obtain Floquet stationary states, which originate from topological edge states characterized by the Rudner winding number [15]. While solitons and their relation to topological phases have been studied in similar settings [30–33,38], we obtain stationary edge states, which are directly related to anomalous Floquet topological phases in a manner different from solitons.
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