Non-quantized Zak phases, PT/APT symmetry transitions, and doubly degenerate exceptional points in a non-Hermitian spin–orbit coupled SSH model

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 (PT) symmetric phase to a parity-time and anti-parity-time (PT&APT) symmetry-breaking phase, and finally to an anti-parity-time (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.