Fig. 2: Phase diagram and dynamics of the nonlinear QWZ model.
From: Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number
a, Schematic of the nonlinear QWZ model. The model has two sublattices (black circles) at each lattice point encircled by the blue ellipse. The green lines represent the linear couplings. We use the notation Ψi(x, y) to represent the state variable at each sublattice, where (x, y) is the location of the representative point of each lattice point denoted by the red cross. b, Analytical demonstration of the phase diagram of the nonlinear QWZ model. The horizontal axis represents the parameter of the mass term and the vertical axis corresponds to the strength of nonlinearity. The colour of each separated region represents the difference in the nonlinear Chern number. c, Numerical demonstration of the absence of edge modes in the topologically trivial parameter region. We simulate the dynamics of the prototypical model of a nonlinear Chern insulator starting from an initial state localized at the left edge. We impose the open boundary condition in the x direction and the periodic boundary condition in the y direction. The figure shows the snapshot at t = 1. The colour shows the absolute value of the components of the state vector at each site. The parameters used are u = 3 and κ = 0.1, and the average amplitude is w = 0.1, which corresponds to the red square in b. d, Numerical demonstration of the existence of the long-lived localized state in the weakly nonlinear topological insulator. The figure shows the snapshot of the simulation at t = 1. The sites at the left edge show large amplitudes, which indicates the existence of the edge-localized state. The parameters used are u = −1 and κ = 0.1, and the average amplitude is w = 0.1, which corresponds to the blue circle in b.
