Symmetry-breaking bifurcation of coupled topological edge states

Abstract

We propose that the symmetry-breaking bifurcation of coupled topological edge states (CTESs) can be used as a general principle for achieving spontaneous symmetry breaking (SSB) in a nonlinear topological lattice. Using an optical resonator array composed of two Su-Schrieffer-Heeger (SSH) chains as an example, we find that as the nonlinearity strength increases, the symmetric CTESs undergo a supercritical bifurcation. Beyond the critical threshold, the originally stable symmetric state becomes unstable, leading to the formation of a pair of stable asymmetric states. Both sides of the symmetric CTESs exhibit sublattice polarization, while the side of the asymmetric CTESs that is predominantly occupied demonstrates stronger sublattice polarization. We further find that as interchain coupling increases, the frequency range for stable CTESs expands, while the frequency range for stable asymmetric CTESs decreases. Our work provides a universal mechanism for realizing SSB in nonlinear topological lattices.

Figures (3)

• Figure 1: Schematic of the resonator array and the symmetry-breaking bifurcation of CTESs. (a) The resonator array consists of two chains. The resonant frequency of the resonators is expressed as $\omega_{0} + g \left\vert \psi \right\vert^2$, where $\omega_{0}$ is a constant term, $g$ is the Kerr nonlinear coefficient, and $\psi$ represents the amplitude of the optical field at the respective resonator. The intracell and intercell couplings are denoted as $J$ and $J^{\prime}$, respectively, while the two chains are interconnected via a coupling $J_d$. (b) The frequencies $\omega$ of CTESs at different total powers $P$ are presented. The mode distributions of the symmetric and asymmetric CTESs are shown as insets, with their corresponding frequencies marked on the existence curves. The purple dashed lines in the insets indicate the interface between the two chains. (c) The asymmetry parameters $\Theta$ of the symmetric and asymmetric CTESs. (d) The sublattice polarizations $s_{\sigma}$ ($\sigma = \mathrm{L}, \mathrm{R}$) of the symmetric and asymmetric CTESs. In panels (b)-(d), the gray regions denote the frequency ranges of the bulk states in the original linear resonator array.
• Figure 2: Stability analysis of CTESs. (a) Maximum growth rates, $\mathrm{Max}\left(\lambda_{\mathrm{I}}\right)$, for symmetric and asymmetric CTESs at various frequencies. The critical frequency for symmetry-breaking bifurcation is indicated by the dashed line. (b) Temporal evolutions of the symmetric and asymmetric CTESs, with a relative perturbation amplitude of $\pm 2\%$ added to the initial excitations. The time $t$ is dimensionless and state intensities are normalized relative to their maximum values. The frequencies of the CTESs are marked on the curves in panel (a). The time-dependent asymmetry parameter $\Theta(t)$ is plotted alongside each intensity distribution.
• Figure 3: Relationship between the frequency ranges of linearly stable CTESs and the interchain coupling $J_{d}$. (a) Maximum growth rates for perturbed symmetric CTESs at various frequencies and different values of $J_{d}$. (b) Same for asymmetric CTESs. The dark regions in panels (a) and (b), corresponding to $\mathrm{Max}\left(\lambda_{\mathrm{I}}\right)=0$, denote the frequency ranges for linearly stable CTESs.