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arXiv:hep-lat/9807005 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Large- q expansion of the specific heat for the two-dimensionalq-state Potts model

H. Arisue, Kanehito Tabata

Published Jan 1, 1999• 8 citations
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Abstract

We have calculated the large-$q$ expansion for the specific heat at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in $1/\sqrt{q}$ using the finite lattice method. The series obtained allows us to give highly convergent estimates of the specific heat for $q>4$ on the first-order transition point. The result confirms the correctness of the conjecture by Bhattacharya et al. [Nucl. Phys. B 435, 526 (1995); J. Phys. I 7, 1155 (1997)] on the asymptotic behavior of the specific heat for $\stackrel{\ensuremath{\rightarrow}}{q}4.$

DOI
10.1103/physreve.59.186

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We have calculated the large-qqq expansion for the specific heat at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in 1/q1/\sqrt{q}1/q​ using the finite lattice method. The series obtained allows us to give highly convergent estimates of the specific heat for q>4q>4q>4 on the first-order transition point. The result confirms the correctness of the conjecture by Bhattacharya et al. [Nucl. Phys. B 435, 526 (1995); J. Phys. I 7, 1155 (1997)] on the asymptotic behavior of the specific heat for q\ensuremath→4.\stackrel{\ensuremath{\rightarrow}}{q}4.q\ensuremath→​4.
We have calculated the large-q expansion for the energy cumulants at the phase transition point in the two-dimensional q-state Potts model to the 23rd order in 1/q1/\sqrt{q}1/q​ using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for q>4q>4q>4 on the first order transition point. The result confirm us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior of the cumulants for q→4+q \to 4_+q→4+​.
Large-qqq expansion of the energy and magnetization cumulants for the two-dimensional qqq-state Potts model
We have calculated the large-qqq expansion for the energy cumulants and the magnetization cumulants at the phase transition point in the two-dimensional qqq-state Potts model to the 21st or 23rd order in 1/sqrtq1/\\sqrt{q}1/sqrtq using the finite lattice method. The obtained series allow us to give very precise estimates of the cumulants for q>4q>4q>4 on the first order transition point. The result confirms us the correctness of the conjecture by Bhattacharya et al. on the asymptotic behavior not only of the energy cumulants but also of the magnetization cumulants for qto4+q \\to 4_+qto4+​.
We present a recursive method to calculate a large q expansion of the 2d q-states Potts model free energies based on the Fortuin-Kasteleyn representation of the model. With this procedure, we compute directly the ordered phase partition function up to order 10 in 1/sqrt{q}. The energy cumulants at the transition can be obtained with suitable resummation and come out large for q less or around 15. As a consequence, expansions of the free energies around the transition temperature are useless for not large enough values of q. In particular the pure phase specific heats are predicted to be much larger, at q < 15, than the values extracted from current finite size scaling analysis of extrema, whereas they agree very well with recent values extracted at the transition point.
We present a recursive method to calculate a large q expansion of the 2d q-states Potts model free energies based on the Fortuin-Kasteleyn representation of the model. With this procedure, we compute directly the ordered phase partition function up to order 10 in 1/sqrt{q}. The energy cumulants at the transition can be obtained with suitable resummation and come out large for q less or around 15. As a consequence, expansions of the free energies around the transition temperature are useless for not large enough values of q. In particular the pure phase specific heats are predicted to be much larger, at q < 15, than the values extracted from current finite size scaling analysis of extrema, whereas they agree very well with recent values extracted at the transition point.
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.
Phase transition in the two-dimensional q-state Potts model with random ferromagnetic couplings is studied in the large-q limit by a combinatorial optimization algorithm and by approximate mappings. We conjecture that the critical behavior of the model is controlled by the isotropic version of the infinite randomness fixed point of the random transverse-field Ising spin chain and the critical exponents are exactly given by beta=(3-sqrt[5])/4, beta(s)=1/2, and nu=1. The specific heat has a logarithmic singularity, but at the transition point there are very strong sample-to-sample fluctuations. Discretized randomness results in discontinuities in the internal energy.