Bulk--Boundary Correspondence and Boundary Zero Modes in a Non-Hermitian Kitaev Chain Model

We study a non-Hermitian Kitaev chain model that contains three sources of non-Hermiticity: a constant imaginary potential, asymmetry between hopping amplitudes $t_{\rm R}$ and $t_{\rm L}$ in the right and left directions, and imbalance in pair potentials $\Delta_{\rm c}$ and $\Delta_{\rm a}$ for pair creation and annihilation, respectively. We show that bulk--boundary correspondence holds in this system; two topological invariants defined in bulk geometry under a modified periodic boundary condition correctly describe the presence or absence of a pair of boundary zero modes in boundary geometry under an open boundary condition. One topological invariant characterizes a topologically nontrivial phase with a line gap and the other characterizes that with a point gap. The latter appears only in the asymmetric hopping case of $t_{\rm R} \neq t_{\rm L}$. These two nontrivial phases are essentially equivalent except for their gap structures. Indeed, the boundary zero modes do not disappear across the boundary between them. We also show that the boundary zero modes do not satisfy the Majorana condition if $\Delta_{\rm c} \neq \Delta_{\rm a}$ and/or $t_{\rm R} \neq t_{\rm L}$.