Exact results for a $Z_3$ clock-type model and some close relatives

In this paper, we generalized the Peschel-Emery line of the interacting transverse field Ising model to a model based on three-state clock variables. Along this line, the model has exactly degenerate ground states, which can be written as product states. In addition, we present operators that transform these ground states into each other. Such operators are also presented for the Peschel-Emery case. We numerically show that the generalized model is gapped. Furthermore, we study the spin-S generalization of interacting Ising model and show that along a Peschel-Emery line they also have degenerate ground states. We discuss some examples of excited states that can be obtained exactly for all these models.