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arXiv:1405.1179 · Chinese Physics Letters

Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice

Shun Wang, Zhi-Yuan Xie, Jing Chen, Bruce Normand, Tao Xiang

Published Jul 1, 2014• 44 citations
arXivHTMLPDF

Abstract

We investigate the 2- and 3-state ferromagnetic Potts models on the simple cubic lattice using the tensor renormalization group method with higher-order singular value decomposition (HOTRG). HOTRG works in the thermodynamic limit, where we use the Z q symmetry of the model, combined with a new measure for detecting the transition, to improve the accuracy of the critical point for the 2-state model by two orders of magnitude, obtaining T c = 4.51152469(1). The 3-state model is far more complex, and we improve the overall understanding of this case by calculating its thermodynamic quantities with high accuracy. Our results verify that the first-order nature of the phase transition and the HOTRG transition temperature benchmarks the most recent Monte Carlo result.

DOI
10.1088/0256-307x/31/7/070503

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Abstract It has been proven that the phase transition of the q-state ferromagnetic Potts model on the square lattice is of the second order for q=4q=4q=4. However, some observables of the q=4q=4q=4 model show features of a first-order phase transition. In particular, a recent large-scale Monte Carlo investigation of the 4-state Potts model observed that the two-peak structure of the energy density distribution becomes more noticeable when L increases. This finding suggests that the signal of coexistence of phases grows stronger with L. Motivated by these unusual critical behaviors, we study the energy density E and the specific heat CvC_vCv​ of the 4-state Potts model on the square lattice using the technique of bond-weighted tensor renormalization group (BTRG). As a comparison, we also investigate the q=2q=2q=2 and q=5q=5q=5 ferromagnetic Potts models on the square lattice using the same method. Remarkably, our results imply that there may be a small energy gap for the q=4q=4q=4 model. Although the appearance of the small energy gap can be explained and would disappear with a more sophisticated investigation, our finding suggests that further investigation is required to determine whether a message of a first-order phase transition is genuine or an artificial effect.
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We employ the second renormalization group method of tensor-network states to investigate thermodynamic properties of the ferromagnetic and antiferromagnetic Potts model on triangular lattices. From the temperature dependence of the internal energy and the specific heat, both the critical temperatures and critical exponents are evaluated. For the q = 3 antiferromagnetic Potts model, the critical temperature is found to be T c = 0.627163±0.000003, which is at least one order of magnitude more accurate than that obtained by other methods.
We study the q-state Potts model on the simple cubic lattice with ferromagnetic interactions along one lattice direction, and antiferromagnetic interactions along the two other directions. As the temperature T decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O(n) model with n=q-1. This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller T we find, for q=4 and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at T=0, and the system effectively reduces to a disordered two-dimensional Potts antiferromagnet. These results are obtained by means of Monte Carlo simulations and finite-size scaling.