Nonlinear topological phase transitions in the dimerized sine-Gordon model

The nonlinear system is a frontier of topological physics. As a prototype, we propose the dimerized sine-Gordon model by combining the sine-Gordon model and the Su-Schrieffer-Heeger model. It has rich varieties of phases depending on the nonlinearity and the dimerization. This system is realized by a simple system of coupled pendulums, which is an ideal playground to explore associated intriguing phenomena. They are observable by the quench dynamics starting from the left-end pendulum. The topological and trivial phases are well defined for the weak nonlinear regime, where a pendulum motion is well approximated by a harmonic oscillator. The emergence of the topological edge state is detected by the standing waves whose amplitude exponentially decays with stationary even-numbered pendulums. In addition, we find nonlinearity-induced phases, i.e., the trap phase with a strictly localized standing wave and the dimer phase with a few coupled standing waves trapped to the left end. We also study a nonlinear dimerized Duffing model, which is obtained by the third-order Taylor expansion of the dimerized sine-Gordon model.