Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals

Abstract

We prove a general rigorous lower bound for $W(Λ,q)=\exp(S_0(Λ,q)/k_B)$, the exponent of the ground state entropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedean lattice $Λ$. We calculate large-$q$ series expansions for the exact $W_r(Λ,q)=q^{-1}W(Λ,q)$ and compare these with our lower bounds on this function on the various Archimedean lattices. It is shown that the lower bounds coincide with a number of terms in the large-$q$ expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions $W_r(Λ,q)$ for large $q$ on the various lattices $Λ$. Plots of $W_r(Λ,q)$ are given, and the general dependence on lattice coordination number is noted. Lower bounds and series are also presented for the duals of Archimedean lattices. As part of the study, the chromatic number is determined for all Archimedean lattices and their duals. Finally, we report calculations of chromatic zeros for several lattices; these provide further support for our earlier conjecture that a sufficient condition for $W_r(Λ,q)$ to be analytic at $1/q=0$ is that $Λ$ is a regular lattice.