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 qui te different, we point out an interesting connection between their com plex-temperature phase diagrams. We carry out an exact determination o f the complex-temperature phase diagram for the ID Ising model for arb itrary spin s and show that in the u(s) = e(-K/s2) plane (i) it consis ts of N-c,N-1D = 4s(2) infinite regions separated by an equal number o f boundary curves where the free energy is nonanalytic; (ii) these cur ves extend from the origin to complex infinity, and in both limits are oriented along the angles theta(n) = (1 + 2n)pi/4s(2), for n = 0,..., 4s(2) - 1; (iii) of these curves, there are N-c,N-NE,N-1D = N-c,N-NW,N -1D = [s(2)] in the first and second (NE and NW) quadrants; and (iv) t here is a boundary curve (line) along the negative real u(s) axis if a nd only if s is half-integral. We note a close relation between these results and the number of area of zeros protruding into the FM phase i n our recent calculation of partition function zeros for the 2D spin a Ising model.