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arXiv:hep-th/9303075 · Europhysics Letters (EPL)

The Free Energy of the Potts Model: From the Continuous to the First-Order Transition Region

Tanmoy Bhattacharya A, R. Lacaze, A Morel

Published Sep 10, 1993• 15 citations
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Abstract

We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.

DOI
10.1209/0295-5075/23/8/002

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We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.
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First-order phase transition in 1d1d1d Potts model with long-range interactions
The first-order phase transition in the one-dimensional qqq-state Potts model with long-range interactions decaying with distance as 1/r1+σ1/r^{1+σ}1/r1+σ has been studied by Monte Carlo numerical simulations for 0<σ≤10 < σ\le 10<σ≤1 and integer values of q>2q > 2q>2. On the basis of finite-size scaling analysis of interface free energy ΔFLΔF_LΔFL​, specific heat and Binder's fourth order cumulant, we obtain the first-order transition which occurs for σσσ below a threshold value σc(q)σ_c(q)σc​(q).
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We performed Monte Carlo simulations of the two-dimensional q-state Potts model with q=10, 15, and 20 to study the energy and magnetization cumulants in the ordered and disordered phase at the first-order transition point βtβ_tβt​. By using very large systems of size 300 x 300, 120 x 120, and 80 x 80 for q=10, 15, and 20, respectively, our numerical estimates provide practically (up to unavoidable, but very small statistical errors) exact results which can serve as a useful test of recent resummed large-q expansions for the energy cumulants by Bhattacharya `et al.' [J. Phys. I (France) 7 (1997) 81]. Up to the third order cumulant and down to q=10 we obtain very good agreement, and also the higher-order estimates are found to be compatible.