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arXiv:cond-mat/9903380 · Europhysics Letters (EPL)

Integrability of the critical point of the Kagomé three-state Potts model

J C Anglès d'Auriac

Published Oct 1, 1999• 1 citation
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Abstract

The three-state Potts model on a Kagomé lattice with three equal coupling constants is studied in the vicinity of the critical point and away from it by means of Random Matrix Theory. Circumstantial evidence indicating that at the critical point it is integrable is given.
DOI
10.1209/epl/i1999-00107-2

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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.
A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.
Abstract The Potts model on a Kagome lattice is considered. The Monte Carlo method is used to obtain the temperature dependences of the thermodynamic parameters: heat capacity C, order parameter m, and susceptibility χ. The calculations were performed for systems with periodic boundary conditions. Systems with linear dimensions L × L = N, L = 20–90, were considered. Based on the fourth-order Binder cumulant method, the critical temperature (Tc) was calculated for the three-vertex Potts model on a Kagome lattice. It is shown that the obtained value of Tc, within the statistical error, is in good agreement with the results obtained by the transfer matrix and polynomial approximation methods.
We use Monte Carlo methods to study the three-state Potts model on a two-dimensional lattice with fivefold coordination. The lattice consists of triangular and hexagonal units and has properties intermediate between the triangular and kagom\'e lattices. The critical behavior of both ferromagnetic and antiferromagnetic cases is considered. As expected, the critical behavior of the ferromagnetic model is independent of the lattice details and lies in the universality class of the three-state ferromagnetic Potts model. A critical end point is found at the coexistence between the ferromagnetic and antiferromagnetic phases, and the transition to the antiferromagnetic state is always first order.
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 determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.