Stability of topologically protected edge states in nonlinear quantum walks: additional bifurcations unique to Floquet systems

Abstract

Quantum walk, a kind of systems with time-periodic driving (Floquet systems), is defined by a time-evolution operator, and can possess non-trivial topological phases. Recently, the stability of topologically protected edge states in a nonlinear quantum walk has been studied, in terms of an effective time-indepedent non-Hermitian Hamiltonian, by applying a continuum limit to the nonlinear quantum walk. In this paper, we study the stability of the edge states by treating a nonunitary time-evolution operator, which is derived from the time-evolution operator of nonlinear quantum walks without the continuum limit. As a result, we find additional bifurcations at which edge states change from stable attractors to unstable repellers with increasing the strength of nonlinearity. The additional bifurcation we shall show is unique to Floquet nonlinear systems, since the origin of the bifurcations is that a stable region for eigenvalues of nonunitary time-evolution operators is bounded, while that of effective non-Hermitian Hamiltonians is unbouneded.

Stability of topological edge states under strong nonlinear effects

“…In a driven-damped system, the chaotic dynamics have been shown to exhibit topological features [40]. Recently, stability of topological states in periodically driven systems such as nonlinear quantum walks has also been discussed [41][42][43].…”

Asymptotic stability of small bound state of nonlinear quantum walks

“…The above specific form is used in Lemma 3.2 to obtain the "orthogonality" of the linearized operator (see, Lemma 3.2 below). An important example which do not fall in the above framework is the model considered in [22,44], with the nonlinear coin given by N (u) = e ig( u,σ3u )σ2 u, where σ j 's are the Pauli matrices. Study of such wider class of nonlinearity may be a good direction for the future research.…”

“…Indeed, many papers studying nonlinear QWs numerically observe solitonic behavior of the solution and focus on the study of its dynamics [8,9,17,34,37,45,63]. For the stability analysis of bound states, related to the study of topological phases [3,4,10,11,28,29,40,60,61,62], Gerasimenko, Tarasinski, and Beenakker [22], followed by Mochizuki, Kawakami and Obuse [44] studied the linear stability of bound states bifurcating from linear bound states.…”

Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization

“…The presence of nonlinearity, can also lead to the formation of topologically-robust edge solitons [35], unique gap solitons [36], or "self-induced" edge solitons and domain walls [25,37,38]. Finally, nonlinear, periodically driven topological lattices have been also investigated both theoretically and experimentally [35,[39][40][41][42][43] All these works contribute to our understanding of how topological states behave or emerge in the presence of inter-particle interactions and nonlinearity. However, very little is known about the long time behavior of these nonlinear topological states [44].…”