  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.
