Nonlinear topological edge states: From dynamic delocalization to thermalization

Abstract

We consider a mechanical lattice inspired by the Su-Schrieffer-Heeger model along with cubic Klein-Gordontype nonlinearity. We investigate the long-time dynamics of the nonlinear edge states, which are obtained by nonlinear continuation of topological edge states of the linearized model. Linearly unstable edge states delocalize and lead to chaos and thermalization of the lattice. Linearly stable edge states also reach the same fate, but after a critical strength of perturbation is added to the initial edge state. We show that the thermalized lattice in all these cases shows an effective renormalization of the dispersion relation. Intriguingly, this renormalized dispersion relation displays a unique symmetry, i.e., its square is symmetric about a finite squared frequency, akin to the chiral symmetry of the linearized model.

Topological Edge Breathers in a Nonlinear Su-Schrieffer-Heeger Lattice

“…Similar instability mechanisms have been described also for mechanical analogs of the SSH model [3,12,13]. The long-term dynamics arising from the unstable edge modes in mechanical lattices has been seen to, after an initial oscillatory behavior, typically lead to a chaotic spreading into the lattice, while leaving only a small-amplitude linear edge mode at the edge [3,12]. On the other hand, for the Kerr nonlinear case the example shown in Fig.…”

“…In both cases, the instabilities are caused by resonances either between a localized internal soliton oscillation mode and extended modes from the continuous spectrum, or between two localized modes originating from the two different sub-bands [6,11]. Similar instability mechanisms have been described also for mechanical analogs of the SSH model [3,12,13]. The long-term dynamics arising from the unstable edge modes in mechanical lattices has been seen to, after an initial oscillatory behavior, typically lead to a chaotic spreading into the lattice, while leaving only a small-amplitude linear edge mode at the edge [3,12].…”

“…There has been a large recent interest in the study of nonlinear effects in lattices with topologically induced gaps in their linear spectra, particularly within nonlinear optics which has led to the emerging field of nonlinear topological photonics (see reviews [1,2]), but also more generally for, e.g., mechanical or electrical lattices (see, e.g., recent works [3,4] and references therein). To elucidate the most fundamental effects arising from the simultaneous presence of topology and nonlinearity, much attention has been directed to nonlinear generalizations of the one-dimensional Su-Schrieffer-Heeger (SSH) lattice [5].…”

Delocalization and higher-order topology in a nonlinear elastic lattice

“…A transition from edge to bulk in an elastic TI by increasing the excitation amplitude was experimentally demonstrated [39]. In addition, topological phase transition [40], self-tunability [41], edge solitons [42], discrete breathers [45], nonlinear harmonic generation [43,46], instability [44,47], and thermalization [48] were reported in nonlinear elastic TIs. However, these works are limited to the traditional first-order TIs, whilst nonlinear elastic HOTIs remain largely unexplored.…”

Emergence of purely nonlinear localized states with frequencies exited from spectral bands

“…More recently, there have been growing interests on the topological materials with nonlinearity [20,21,22,23]. Among others an important problem is the effects of nonlinearity on the existence of edge states (or topologicallyprotected states) and it mainly has two branches: One is whether linear edge states continue to be localized as the strength of nonlinearity grows; The other is what new edge states will appear in the nonlinear regime.…”