Complex-Temperature Phase Diagrams of 1D Spin Models with Next-Nearest-Neighbor Couplings

We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the 1D Ising model with nearest-neighbor (NN) and next-to-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of $J_{nnn}/J_{nn}$ and compare these with the case of pure nearest-neighbor (NN) couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.