Solutions of PT symmetric tight-binding chain and its equivalent Hermitian counterpart

We study the Non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an $N$-site tight-binding chain with two conjugated imaginary potentials $\pm i\gamma$ at two end sites, which Hamiltonian has parity-time symmetry ($\mathcal{PT}$ symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases, an unbroken symmetry phase with a purely real energy spectrum in the region $\gamma \prec \gamma_{c}$ and a spontaneously-broken symmetry phase with $N-2$ real and 2 imaginary eigenvalues in the region $\gamma \succ \gamma_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of $\gamma_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.