Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

We present the exact solution of the 1D classical short-range Potts model with invisible states. Besides the q states of the ordinary Potts model, this possesses r additional states which contribute to the entropy, but not to the interaction energy. We determine the partition function, using the transfer-matrix method, in the general case of two ordering fields: h₁ acting on a visible state and h₂ on an invisible state. We analyse its zeros in the complex-temperature plane in the case that h₁=0. When Imh₂=0 and r≥0, these zeros accumulate along a line that intersects the real temperature axis at the origin. This corresponds to the usual “phase transition” in a 1D system. However, for Imh₂≠0 or r<0, the line of zeros intersects the positive part of the real temperature axis, which signals the existence of a phase transition at non-zero temperature.