Nonlinear topological edge states: From dynamic delocalization to thermalization

We consider a mechanical lattice inspired by the Su-Schrieffer-Heeger model along with cubic Klein-Gordon–type 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.

Figure 1

A semi-infinite chain consisting of equal masses (gray spheres) interconnected with two types of linear springs with elastic constants $1+\gamma$ and $1-\gamma$, denoted by the green and blue spiral coils, respectively. Each mass is grounded with a nonlinear spring (gray spiral coil) whose elastic and nonlinear constants are ${\gamma }_0$ and $\mathrm{\Gamma }$. The $dx$ denotes the spring deformation and $F$ the resulting external force applying to a single mass.

Figure 2

Nonlinearity-modified topological edge states. (a) Emergence of topological edge state inside the band gap. (b) Spatial profile $x_j$ of the edge state with ${\gamma }_0=1$ and $\gamma =0.4$ whose frequency is indicated by the red dot in (a). (c)–(e) Nonlinear continuations of the topological edge state for $(\mathcal{H}=3.779,\mathrm{\Gamma }=0.8), (\mathcal{H}=1.608,\mathrm{\Gamma }=-0.8)$, and $(\mathcal{H}=4.214,\mathrm{\Gamma }=-0.8)$, respectively. In all the cases, the chosen phase of the mode is such that the corresponding momentum profile is zero, $p_j=0$.

Figure 3

The normalized modal energy ${\nu }_k\left(t\right)$ [Eq. (6)] evolution profiles for representative realizations of setup [Eq. (17)] of three different nonlinear topological edge states at ($\mathcal{H}, \mathrm{\Gamma }$) values with ($\mathcal{H}=0.307, \mathrm{\Gamma }=-0.8$), ($\mathcal{H}=1.608, \mathrm{\Gamma }=-0.8$), and ($\mathcal{H}=4.214, \mathrm{\Gamma }=-0.8$) in (a), (b), and (c), respectively. The white vertical line in (a) shows the position of the cross section of ${\nu }_k\left(t\right)$ at $t\approx 6.3\times {10}^4$ depicted in Fig. 11 of Appendix pp2. The edge states associated with (b) and (c) are, respectively, shown in Figs. 2 and 2. Each point is colored according to the color scale at the right end of the figure.

Figure 4

Temporal evolution of the (a) rescaled spectral entropy $\eta \left(t\right)$ [Eq. (8)] and (b) the ftMLE $\lambda \left(t\right)$ [Eq. (15)] of the model [Eq. (1)] for representative realizations of the initial nonlinear topological edge states with $(\mathcal{H}=0.307,\mathrm{\Gamma }=-0.8)$ (blue), $(\mathcal{H}=1.608,\mathrm{\Gamma }=-0.8)$ (green), and $(\mathcal{H}=4.214,\mathrm{\Gamma }=-0.8)$ (red) of Fig. 3. The black dashed horizontal line in (a) measures ${\langle \eta \rangle }_G\approx 0.125$, the Gibbs average for the case $(\mathcal{H}=4.214,\mathrm{\Gamma }=-0.8)$ [red curves in (a) and (b)]. The shaded curves around the dark-colored curves' course represent the evolution of 30 other realizations of perturbation of the same topological edge states. The black solid line in (b) shows the relation $\lambda \left(t\right)\propto ln\left(t\right)/t$ observed in the case of regular motion.

Figure 5

(a) Spatiotemporal profile of the normalized energy distribution ${\xi }_j\left(t\right)$ [Eq. (11)] for the case $(\mathcal{H}=4.214,\mathrm{\Gamma }=-0.8)$ of Figs. 3 and 4 (red curves in that figure). Each lattice site is colored according to the magnitude of its ${\xi }_j$ value [see color scale at the right of (a)]. (b) Evolution of the associated participation number $P\left(t\right)$ [Eq. (14)]. We see that the saturation of the entropy $\eta \left(t\right)$ [red curve in Fig. 4] practically coincides with the one displayed by $P\left(t\right)$ in (b).

Figure 6

$\mathrm{\Omega }-k$ dependence of the frequency shift distribution $\langle |a_k\left(\mathrm{\Omega }\right){|}^2\rangle$ for (a) $(\mathcal{H}=2.417,\mathrm{\Gamma }=-0.8)$, (b) $(\mathcal{H}=4.409,\mathrm{\Gamma }=-0.8)$, and (c) $(\mathcal{H}=7.019,\mathrm{\Gamma }=-0.8)$. The red dots indicate the dispersion relation for the linearized system, whereas the black dots indicate the renormalized dispersion relation for the nonlinear system (see text for details).

Figure 7

Change in the squared frequency $\delta {\omega }_k^2={\omega }_k^2-{\tilde{\omega }}_k^2$ [Eq. (19)] of the dispersion relation at different energy levels: $\mathcal{H}=2.417$ (blue line-connected circles), $\mathcal{H}=4.214$ (green line-connected triangles), and $\mathcal{H}=7.019$ (red line-connected squares). For all cases, $\mathrm{\Gamma }=-0.8$. The horizontal solid lines of the same colors indicate the average values $\overline{\delta {\omega }^2}$ of the relevant data points they fit: $\overline{\delta {\omega }^2}\approx 0.025\pm 0.004, 0.047\pm 0.004$, and $0.074\pm 0.004$, respectively, for the blue, green, and red line-connected points.

Figure 8

Similar to Fig. 3 but for three different perturbation strengths $\varepsilon$ [Eq. (17)] of the nonlinear topological edge state at ($\mathcal{H}=3.779, \mathrm{\Gamma }=0.8$) with (a) $\varepsilon =0.01$, (b) $\varepsilon =0.25$, and (c) $\varepsilon =1.0$ (see text for details). Each mode's number is colored according to the intensity of its energy.

Figure 9

Fraction of thermalized $f_{th}$ [Eq. (22)] states of the system [Eq. (1)] at time $t={10}^8$ as function of the distance $d$ [Eq. (20)] from the linearly stable nonlinear topological state in the phase space with $(\mathcal{H}=3.779,\mathrm{\Gamma }=0.8)$ (green line-connected circles) and $(\mathcal{H}=8.367,\mathrm{\Gamma }=0.8)$ (cyan line-connected crosses). The inset panel shows the dependence of $d$ against the perturbation parameter $\varepsilon$ of Eq. (17). The dashed vertical line indicates $d_c\approx 0.13$ and the dotted one $d_c\approx 0.031$ (see text for details).

Figure 10

(a) Similar to Fig. 6 but for a perturbation of the stable topological state with $(\mathcal{H}=8.367,\mathrm{\Gamma }=0.8)$ at distance $d\approx 0.285$ (see text for details). The red dotted line represents the dispersion relation of the linearized system ${\omega }_k$, while the black dotted line is the calculated dispersion relation ${\tilde{\omega }}_k$ [Eq. (18)]. (b) Change in the squared frequency $|\delta {\omega }_k^2|$ [Eq. (19)] (blue line-connected points) of the dispersion relations in (a). The value of $|\delta {\omega }_k^2|$ is oscillating around the average $\overline{\delta {\omega }^2}=0.085\pm 0.005$ [blue bold line in (b)].

Figure 11

(a) Geometric representation of the FMs ${\lambda }_i$ (see text for details) in the complex plane for the topological edge state with parameter $(\mathcal{H}=0.307,\mathrm{\Gamma }=-0.8)$ of Fig. 3. We superimpose the unit circle to guide the eye. The upper canvas depicts a zoom into the region of the unit circle where the strongest divergence of the FMs from the unit circle takes place. The most unstable eigenvalues are denoted by arrows. (b) Normalized modal energy ${\nu }_k\left(t\right)$ [Eq. (6)] profile at time $t\simeq 6.3\times {10}^4$ time units for a representative realization of perturbation of the edge breather mode at $(\mathcal{H}=0.307,\mathrm{\Gamma }=-0.8)$. This figure corresponds to a vertical cross section of ${\nu }_k\left(t\right)$ at $(t\approx 6.3\times {10}^4,k)$ [see white vertical line of Fig. 3]. The vertical bold and dotted lines label the resonant modes.

Figure 12

Frequency shift ${\tilde{\omega }}_k=\tilde{\omega }\left(k\right)$ [Eq. (18)] [(a), (c), (e)] and width of the frequency shift distribution ${\chi }_k=\chi \left(k\right)$ [Eq. (C1)] [(b), (d), (f)] as functions of the total energy $\mathcal{H}$ [Eq. (1)] of the system: (a), (b) $k=5$, (c), (d) $k=51$, and (e), (f) $k=52$. The ground spring nonlinear coefficient $\mathrm{\Gamma }$ is fixed at $\mathrm{\Gamma }=-0.8$.

Figure 13

The MLE $\langle \mathrm{\Lambda }\rangle$ as function of the energy of the system $\mathcal{H}$ [Eq. (1)]. The value of $\langle \mathrm{\Lambda }\rangle$ grows almost linearly with $\mathcal{H}$. We numerically approximated $\langle \mathrm{\Lambda }\rangle$ by $\langle \lambda \left(T\right)\rangle$, the ftMLE at the end of the integration time $T$ for each set of parameters in the lattice. The parameters used in $\mathcal{H}$ [Eq. (1)] are similar to the ones of Fig. 12.