关键词: spin-orbit coupling, parity-time symmetry, Zak phase

Abstract: We theoretically investigate a one-dimensional Su-Schrieffer-Heeger (SSH) model with spin-orbit coupling (SOC) and sublattice-dependent gain and loss. As the gain and loss increase, the system transitions from a parity-time (${\rm PT}$) symmetric phase to a parity-time and anti-parity-time (${\rm PT\&APT}$) symmetry-breaking phase, and finally to an anti-parity-time (${\rm APT}$) symmetric phase. Notably, when the intracell and intercell hopping, intracell and intercell SOC parameters are all equal to half the gain-loss parameter, the model exhibits a doubly degenerate exceptional point (EP). When the SOC is equal for intracell and intercell interactions, a stronger hopping mechanism within cells compared to that between cells results in an increase in SOC that transitions the Zak phase from zero to a non-quantized value, eventually arriving at one. In contrast, a reduction in the strength of intracell hopping leads the Zak phase to transition from two to a non-quantized value, eventually arriving at one. If the intracell and intercell SOC are not aligned, altering these couplings leads to a shift in the Zak phase from two to a non-quantized level, then to one, re-entering the non-quantized region, and eventually arriving at zero. We suggest a practical experimental setup for our model that can be implemented using electrical circuits.

Key words: spin-orbit coupling, parity-time symmetry, Zak phase