Ground state entropy of Potts antiferromagnets and the approach to the two-dimensional thermodynamic limit

Abstract

We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the $q$-state Potts antiferromagnet on infinitely long strips with width ${L}_{y}$ of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the $y$ direction, denoted ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y}.$ We show that the approach of $W$ to its 2D thermodynamic limit as ${L}_{y}$ increases is quite rapid; for moderate values of $q$ and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for ${\mathrm{FBC}}_{y}$ and ${\mathrm{PBC}}_{y},$ respectively. The approach of $W$ to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for ${\mathrm{FBC}}_{y}$ $({\mathrm{PBC}}_{y}).$ It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.