  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.
