Potts model partition functions for self-dual families of strip graphs

We consider the q-state Potts model on families of self-dual strip graphs GD of the square lattice of width L-y and arbitrarily great length L-x, with periodic longitudinal boundary conditions. The general partition function Z and the T = 0 antiferromagnetic special case P (chromatic polynomial) have the respective forms Sigma (NF,Ly,lambda)(j=1) C-F,C-Ly,C-j(lambda (F,Ly,j))(Lx), with F = Z, P. For arbitrary L-y, we determine (i) the general coefficient C-F,C-Ly,C-j in terms of Chebyshev polynomials, (ii) the number n(F)(L-y,d) of terms with each type of coefficient, and (iii) the total number of terms N-F,N-Ly,N-lambda. We point out interesting connections between the n(Z)(L-y, d) and Temperley-Lieb algebras, and between the N-F,N-Ly,N-lambda and enumerations of directed lattice animals. Exact calculations of P are presented for 2 less than or equal to L-y less than or equal to 4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W(q). Generalizing q from Z(+) to C, we determine the continuous locus B in the complex q plane where W(q) is singular. We find the interesting result that for all L-y values considered, the maximal point at wh ich a crosses the real q-axis, denoted q(c), is the same, and is equal to the value for the infinite square lattice, q(c) = 3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of q(c). (C) 2001 Elsevier Science B.V. All rights reserved.