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

Abstract

We consider the q-state Potts model on families of self-dual strip graphs G D 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 ∑ j=1 N F, L y , λ c F, L y , j ( λ F, L y , j ) L x , with F= Z, P. For arbitrary L y , we determine (i) the general coefficient c F, L y , 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, L y , λ . We point out interesting connections between the n Z ( L y , d) and Temperley–Lieb algebras, and between the N F, L y , λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽ L y ⩽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 which B 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 .