In-plane noncollinear exchange coupling mediated by helical edge states in quantum spin Hall systems

We comprehensively investigate gap solitons and Bloch waves in one-dimensional nonlinear periodic systems. Our results show that there exists a composition relation between them: Bloch waves at either the center or edge of the Brillouin zone are infinite chains composed of fundamental gap solitons (FGSs). We argue that such a relation is related to the exact relation between nonlinear Bloch waves and nonlinear Wannier functions. With this composition relation, many conclusions can be drawn for gap solitons without any computation. For example, for the defocusing nonlinearity, there are n families of FGS in the nth linear Bloch band gap; for the focusing case, there are infinite number of families of FGSs in the semi-infinite gap and other gaps. In addition, the stability of gap solitons is analyzed. In literature, there are numerical results showing that some FGSs have cutoffs on propagation constant (or chemical potential), i.e., these FGSs do not exist for all values of propagation constant (or chemical potential) in the linear band gap. We develop an intuitive picture to describe this cutoff.