  Since the realization of quantum systems described by non-Hermitian
Hamiltonians with parity-time (PT) symmetry, interest in non-Hermitian, quantum
many-body models has steadily grown. Most studies to-date map to traditional
quantum spin models with a non-Hermiticity that arises from making the model
parameters complex or purely imaginary. Here, we present a new set of models
with non-Hermiticity generated by splitting a Hermitian term into two
Jordan-normal form parts, and the perturbations are confined to one or two
sites. We present exact diagonalization results for the PT-threshold in such
models, and provide an analytical approach for understanding the numerical
results. Surprisingly, with non-Hermitian potentials confined to two or even a
single site, we find a robust PT threshold that seems insensitive to the size
of the quantum spin chain. Our results provide a pathway to experimentally
feasible non-Hermitian quantum spin chains where the confluence of many-body
effects and non-Hermiticity effects can be observed.
