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Progress of Theoretical Physics

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

Hiroshi Katsumori, Shingo Ohya

Published Jun 1, 2000• 0 citations
Publisher

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/kJ 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.

DOI
10.1143/ptp.103.1137

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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/kJ 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.
A new graphical method is developed to calculate the critical temperature of 2- and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221(2), is in a good agreement with the accurate value of 0.22165 reported by others.
The value of the internal energy per spin is independent of the strip width for a certain class of spin systems on two-dimensional infinite strips. It is verified that the Ising model on the kagome lattice belongs to this class through an exact transfer-matrix calculation of the internal energy for the two smallest widths. More generally, one can suggest an upper bound for the critical coupling strength K(c)(q) for the q-state Potts model from exact calculations of the internal energy for the two smallest strip widths. Combining this with the corresponding calculation for the dual lattice and using an exact duality relation enables us to conjecture the critical coupling strengths for the three- and four-state Potts models on the kagome lattice. The values are K(c)(q=3)=1.0565094269290 and K(c)(q=4)=1.1493605872292, and the values can, in principle, be obtained to an arbitrary precision. We discuss the fact that these values are in the middle of earlier approximate results and furthermore differ from earlier conjectures for the exact values.
The internal energy density of the Potts model on a semi-infinite strip with a width L has been conjectured to have no finite-size corrections at the critical point K = K c . By factorizing the transfer matrix for the kagome lattice with larger widths, we have found that this conjecture is not correct in that the internal energy density varies slightly with L at the critical point. From this size dependence of the internal energy density, we obtain an upper bound as K c < 1.0565615, which is close to a recent estimate K JS = 1.0565600(7) by Jacobsen and Scullard [J. Phys. A 45, 494003 (2012)]. We also obtain a lower bound as K c > 1.0560 by calculating the correlation length along the strips.
Abstract We study the phase diagram of the triangular-lattice Q -state Potts model in the real ( Q , v ) -plane, where v = e J − 1 is the temperature variable. Our first goal is to provide an obviously missing feature of this diagram: the position of the antiferromagnetic critical curve. This curve turns out to possess a bifurcation point with two branches emerging from it, entailing important consequences for the global phase diagram. We have obtained accurate numerical estimates for the position of this curve by combining the transfer-matrix approach for strip graphs with toroidal boundary conditions and the recent method of critical polynomials. The second goal of this work is to study the corresponding A p − 1 RSOS model on the torus, for integer p = 4 , 5 , … , 8 . We clarify its relation to the corresponding Potts model, in particular concerning the role of boundary conditions. For certain values of p , we identify several new critical points and regimes for the RSOS model and we initiate the study of the flows between the corresponding field theories.