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 Ly and arbitrarily great length Lx, 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=1NF,Ly,λcF,Ly,j(λF,Ly,j)Lx, with F=Z,P. For arbitrary Ly, we determine (i) the general coefficient cF,Ly,j in terms of Chebyshev polynomials, (ii) the number nF(Ly,d) of terms with each type of coefficient, and (iii) the total number of terms NF,Ly,λ. We point out interesting connections between the nZ(Ly,d) and Temperley–Lieb algebras, and between the NF,Ly,λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽Ly⩽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 Ly values considered, the maximal point at which B crosses the real q-axis, denoted qc, is the same, and is equal to the value for the infinite square lattice, qc=3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of qc.