A connection between complex-temperature properties of the 1D and 2D spin s Ising model

Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s = e ^{-K s2} plane (i) it consists of N_c,1D = 4s² infinite regions separated by an equal number of boundary curves where the free energy is nonanalytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles θ_n = (1 + 2n)π 4s², for n = 0,...,4s² - 1; (iii) of these curves, there are N_c,NE,1D = N_c,NW,1D = [s²] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.