Phenomenological renormalization-group calculations for 12- and 16-vertex models on a square lattice

Abstract

We performed phenomenological renormalization-group calculations for ferroelectric 12- and 16-vertex models on a square lattice with periodic and helical boundary conditions. We considered strips of infinite length and finite widths ($n=1\ensuremath{-}7,8, or 9$). The extrapolated values for the transition temperature of the 12-vertex model, which has been used for assessing the transition in squaric acid, are lower than the predictions of the Bethe approximation. The estimates for the critical exponent $\ensuremath{\nu}$ do not allow a definite conclusion about its asymptotic behavior, although the Ising value $\ensuremath{\nu}=1$ seems more plausible. The estimates for the 16-vertex model considered in this paper, which is equivalent to an anisotropic nearest-neighbor Ising model, show an excellent convergence to the exact values. Also, we analyze the finite-size scaling behavior of the critical free energy of both models.