Stability of topological edge states under strong nonlinear effects

We examine the role of strong nonlinearity on the topologically robust edge state in a one-dimensional system. We consider a chain inspired from the Su-Schrieffer-Heeger model but with a finite-frequency edge state and the dynamics governed by second-order differential equations. We introduce a cubic onsite nonlinearity and study this nonlinear effect on the edge state's frequency and linear stability. Nonlinear continuation reveals that the edge state loses its typical shape enforced by the chiral symmetry and becomes generally unstable due to various types of instabilities that we analyze using a combination of spectral stability and Krein signature analysis. This results in an initially excited nonlinear-edge state shedding its energy into the bulk over a long time. However, the stability trends differ both qualitatively and quantitatively when softening and stiffening types of nonlinearity are considered. In the latter, we find a frequency regime where nonlinear edge states can be linearly stable. This enables high-amplitude edge states to remain spatially localized without shedding their energy, a feature that we have confirmed via long-time dynamical simulations. Finally, we examine the robustness of frequency and stability of nonlinear edge states against disorder, and find that those are more robust under a chiral disorder compared to a nonchiral disorder. Moreover, the frequency-regime where high-amplitude edge states were found to be linearly stable remains intact in the presence of a small amount of disorder of both types.

Figure 1

(a) A chain of masses that are interconnected with linear springs and grounded with nonlinear springs. Relations between the force (F) and deformation (dx) are specified for the springs in different colors. (b) Spectrum of the linearized chain ($\mathrm{\Gamma }=0$) as a function of $\gamma$. For $\gamma >0$ we see a topological edge state emerging at ${\omega }_{\mathrm{ed}}=\sqrt{2+{\gamma }_0}$ inside the band gap. (c) Profile of the topological state localized at the left end of the chain for $\gamma =0.4$. Inset shows the zoomed-in view of the edge state corresponding to the shaded area. Due to the chiral symmetry, the state has a zero displacement at its even sites.

Figure 2

Effect of softening nonlinearity (with ${\gamma }_0=1$, $\gamma =0.4$ and $\mathrm{\Gamma }=-0.8$). (a) Nonlinear continuation of the topological edge state. (b) Maximum amplitude of FMs of the periodic solutions deviating from unity (vertical dashed line) with the decrease in frequency. The horizontal dashed line in red marks the onset of high growth rate of instability. (c) Profile of the nonlinear state at $\omega =1.65$ as indicated by the circular marker in (a). Its even sites have a nonzero displacement. (d) Transient simulation with the nonlinear state as the initial condition. (e) STFT of the first particle to indicate the change in frequency over a long time. The inset shows the zoomed-in view corresponding to the box. [(f)–(h)] Same for the nonlinear state at $\omega =1.54$, which has a stronger instability.

Figure 3

Effect of stiffening nonlinearity (with ${\gamma }_0=1$, $\gamma =0.4$, and $\mathrm{\Gamma }=0.8$). (a) Nonlinear continuation of the topological edge state. (b) Maximum amplitude of FMs of the time-periodic solutions. Note the region above the horizontal dashed line in red supports large-amplitude nonlinear states that are linearly stable. (c) Profile of the nonlinear state at $\omega =1.92$ as indicated by the circular marker in (a). (d) Transient simulation with the nonlinear state as the initial condition. Due to linear stability, the state remains localized for a long time without shedding its energy into the bulk. (e) STFT of the first particle verifies that there is no energy transfer across different frequencies. [(f)–(h)] Same for the nonlinear state at $\omega =1.85$; however, in this case the instability of the state causes the delocalization and change in frequency of the edge state over a long time.

Figure 4

Variation of FMs with frequency. (a) FMs in the complex plane for the two cases discussed in Fig. 2 for softening nonlinearity. Color denotes the Krein signature of each FM. (b) Variation in the amplitude of FMs with frequency. Values away from $\left|\lambda \right|=1$ indicate the presence of instability. (c) Variation in the phase of FMs with frequency. Blue dots denote the unstable FMs with zero Krein signature. The circled region represents the onset of a different instability. [(d)–(f)] The same for stiffening nonlinearity.

Figure 5

Effect of disorder on the spectrum of the linearized system. (a) Chiral disorder preserves the frequency of the edge state (${\omega }_{\mathrm{ed}}^2=3$) inside the band gap. Also, the bands above and below remain symmetric about the edge-state frequency. Color map indicates the localization (IPR) of states. Below is a typical shape of the edge state where even sites have zero displacements at $\delta =0.4$. (b) Nonchiral disorder does not impose any such symmetry constraints on the spectrum and the frequency of the edge state changes with disorder. Moreover, the edge state develops a nonzero displacement at even sites in the presence of disorder ($\delta =0.4$).

Figure 6

Effect of disorder on the nonlinear continuation and instabilities of the edge state. [(a)–(d)] For the softening case when the system has chiral [(a) and (b)] and nonchiral [(c) and (d)] disorder. Colors indicate disorder strengths of 0 (pristine), 0.05, and 0.1. Standard deviations in $x$-axes are shown by the shaded areas. [(e)–(h)] The same for the stiffening case. Notice the region of stability for $5\%$ disorder. A slight reduction in its frequency span compared to the pristine case is marked with horizontal dashed lines.

Figure 7

Mapping between the dispersion and the corresponding FMs. (a) Dispersion for our linearized system with $\gamma =0.4$ and ${\gamma }_0=1$. The cutoff frequencies are marked for the acoustic (below) and optical (above) bands. (b) Corresponding FMs $\lambda =e^{i\omega \left(\beta \right)T}$ on the complex plane for $T=2\pi /{\omega }_{\mathrm{ed}}$. Edges of the arcs are marked with the associated cutoff frequency of dispersion band. (c) Corresponding FM $\lambda =e^{-i\omega \left(\beta \right)T}$ for the same $T$.

Figure 8

Effect of disorder on the nonlinear continuation and instabilities of the edge state for stiffening nonlinearity. The conditions are similar to the ones in Figs. 6–6 but for more steps in disorder strength and 100 realizations of each. A short chain of 10 particles in considered.