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.Comment: 36 pages, latex, three postscript figure