Nonlinear topological phase diagram in dimerized sine-Gordon model

We investigate the topological physics and the nonlinearity-induced trap phenomenon in a coupled system of pendulums. It is described by the dimerized sine-Gordon model, which is a combination of the sine-Gordon model and the Su-Schrieffer-Heeger model. The initial swing angle of the left-end pendulum may be regarded as the nonlinearity parameter. The topological number is well defined as far as the pendulum is approximated by a harmonic oscillator. The emergence of the topological edge state is clearly observable in the topological phase by solving the quench dynamics starting from the left-end pendulum. A phase diagram is constructed in the space of the swing angle $\xi \pi $ with $|\xi |\leq 1$ and the dimerization parameter $\lambda $ with $|\lambda |\leq 1$. It is found that the topological phase boundary is rather insensitive to the swing angle for $% |\xi |\lesssim 1/2$. On the other hand, the nonlinearity effect becomes dominant for $|\xi |\gtrsim 1/2$, and eventually the system turns into the nonlinearity-induced trap phase. Furthermore, when the system is almost dimerized ($\lambda \simeq 1$), coupled standing waves appear and are trapped to a few pendulums at the left-end, forming the dimer phase. Its dynamical origin is the cooperation of the dimerization and the nonlinear term.