An Energy Curve Family of a Semi-Infinite Lattice and the Transition Point in the Potts Model

Abstract

An energy curve family for a semi-infinite square lattice (l ×∞ ) of the q-state Potts model with periodic boundary conditions has an l-independent common fixed point, which gives the exact critical values of the coupling parameter K = J/kT and the internal energy E (K), as a natural consequence of the self-dual property of the model. An energy curve intersection procedure (ECIP) is proposed to find the critical properties of the Potts model through numerical investigation of the transfer matrix. The ECIP shows that the lattices (l ×∞ ) for two different small values of l can reproduce the transition temperature and the critical internal energy of the bulk lattice ( ∞×∞ ) without use of the duality argument. As an applicable example of the ECIP, a semi-infinite simple cubic lattice (l × m ×∞ ) of the Potts model is preliminarily studied to obtain approximate critical quantities of the layer lattice (l ×∞×∞ ). A brief comment on the specific heat exponent µ estimated using the ECIP is added. Previous numerical studies 1) - 3) of the largest eigenvalue of the transfer matrix for a semi-infinite lattice of the Potts model in two dimensions (l ×∞ ) and in three dimensions (l × m ×∞ ) involved a specific heat anomaly which created sizedependent pseudocritical temperatures that seemed to approach the critical temperatures as l →∞ or l, m →∞ . In the present paper, we first point out that an energy curve family for a semiinfinite square lattice (l ×∞ ) of the q-state Potts model with periodic boundary conditions has an l-independent common fixed point, as a natural consequence of the self-dual property of the model. We propose an energy curve intersection procedure (ECIP) for semi-infinite lattices for the purpose of determining numerically the critical quantities of the bulk lattice ( ∞×∞ ) of the Potts model in two dimensions. With the ECIP, we find that lattices (l ×∞ ) for two different small values of l are sufficient to reproduce the transition temperature and critical energy of the bulk lattice, without appealing to the duality argument. This problem is discussed in §2. The ECIP may be used to predict an unknown transition point of a certain statistical system which has a not-yet-known duality property. In §3 the ECIP is applied to a semi-infinite simple cubic Potts lattice (l × m ×∞ ) in order to obtain approximate values for some critical quantities of a layer lattice (l ×∞×∞ ). In ∗) A part of this work in the early stages was reported at the International Conference on