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arXiv:cond-mat/9711214 · Journal of the Physical Society of Japan

Numerical Latent Heat Observation of theq=5Potts Model

Tomotoshi Nishino, Kouichi Okunishi

Published Apr 15, 1998• 15 citations
arXivHTMLPDF

Abstract

Site energy of the five-state ferromagnetic Potts model is numerically calculated at the first-order transition temperature using corner transfer matrix renormalization group (CTMRG) method. The calculated energy of the disordered phase $U^{+}$ is clearly different from that of the ordered phase $U^{-}$. The obtained latent heat $L = U^{-} - U^{+}$ is 0.027, which quantitatively agrees with the exact solution.

DOI
10.1143/jpsj.67.1492

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Site energy of the five-state ferromagnetic Potts model is numerically calculated at the first-order transition temperature using corner transfer matrix renormalization group (CTMRG) method. The calculated energy of the disordered phase U+U^{+}U+ is clearly different from that of the ordered phase U−U^{-}U−. The obtained latent heat L=U−−U+L = U^{-} - U^{+}L=U−−U+ is 0.027, which quantitatively agrees with the exact solution.
Three-dimensional (3D) q-state Potts models (q=3, 4, and 5) are studied by the tensor product variational approach, which is a recently developed variational method for 3D classical lattice models. The variational state is given by a 2D product of local factors, and is improved by way of self-consistent calculations assisted by the corner transfer matrix renormalization group. It should be noted that no a priori condition is imposed for the local factor. Transition temperatures and latent heats are calculated from the observations of thermodynamic functions in both ordered and disordered phases.
The surface and bulk properties of the two-dimensional Q>4Q>4Q>4 state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of Q>4.Q>4.Q>4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.
The changeover from first-order to second-order phase transitions in q-state Potts models is obtained at q_{c}=2 in spatial dimension d=3 and essentially at q_{c}=4 in d=2, using a physically intuited simple adaptation of the Migdal-Kadanoff renormalization-group transformation. This simple procedure yields the latent heats at the first-order phase transitions. In both d=2 and 3, the calculated phase transition temperatures, respectively compared with the exact self-duality and Monte Carlo results, are dramatically improved. The method, when applied to a slab of finite thickness, yields dimensional crossover.
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.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.
We provide a convincing numerical demonstration of the first-order nature of the transition in the three-state ferromagnetic Potts model in three dimensions by combining Monte Carlo simulations of the bulk free-energy barrier between the ordered and disordered states with finite-size scaling. From the histogram of internal energy and the number of clusters we extrapolate in q, the number of states, and find qc{\mathit{q}}_{\mathit{c}}qc​=2.45\ifmmode\pm\else\textpm\fi{}0.1. We also find that the free-energy barrier \ensuremath{\Delta}F(q,L)\ensuremath{\sim}(q-qc{\mathit{q}}_{\mathit{c}}qc​)2{)}^{2})2L2.3{\mathit{L}}^{2.3}L2.3 and conclude that there is no essential singularity in the correlation length or latent heat in contrast to two dimensions.