Non-Hermitian spin chains with inhomogeneous coupling

An open $U_q(sl_2)$-invariant spin chain of spin $S$ and length $N$ with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter $\gamma$ are determined for which the spectrum of the model is real. For a certain range of $\gamma$, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous $U_q(sl_2)$-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.