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Journal of Physics C Solid State Physics

First-order transition for the (1+1)-dimensional q⩾4 Potts model from finite lattice extrapolation

Ferenc Iglói, Jenő Sólyom

Published May 30, 1983• 28 citations
Publisher

Abstract

The (1+1)-dimensional Hamiltonian Potts model is studied for q>or=4 using finite lattice extrapolation techniques. The ground-state energy and its first derivative give information about the free energy and the latent heat of the classical two-dimensional Potts model, while the gap in the excitation energy corresponds to the inverse of the correlation length. It is shown that the finite latent heat for q>4, when the transition is of first order, comes from the crossing of levels; nevertheless there are no excitations for which the gap would vanish in the thermodynamic limit i.e. the correlation length is also finite.

DOI
10.1088/0022-3719/16/15/005

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The (1+1)-dimensional Hamiltonian Potts model is studied for q>or=4 using finite lattice extrapolation techniques. The ground-state energy and its first derivative give information about the free energy and the latent heat of the classical two-dimensional Potts model, while the gap in the excitation energy corresponds to the inverse of the correlation length. It is shown that the finite latent heat for q>4, when the transition is of first order, comes from the crossing of levels; nevertheless there are no excitations for which the gap would vanish in the thermodynamic limit i.e. the correlation length is also finite.
Finite-lattice sequence extrapolation methods are applied to the Q-state Potts models in (1+1)-D Hamiltonian field theory. Thermal exponents nu =0.71+or-0.02, alpha =0.53+or-0.02 are obtained for the four-state model. Methods are discussed for treating the first-order transitions for Q>4, and a sequence extrapolation is used to estimate the latent heat. Exact analytic formulae are derived for the ground-state energy, the mass gap, and the latent heat at the transition point; these are compared with the numerical results.
Both 'high-' and 'low-temperature' perturbation series are used to locate and characterize the first-order transition in the three-state Potts model in (3+1) dimensions on the simple-cubic, body-centred cubic and face-centred cubic lattices. Estimates are presented for the vacuum energy, 'latent heat', magnetization, susceptibility and mass gap at the transition. The results show a remarkable degree of universality between the different lattices, and a 'law of corresponding states' is followed very closely.
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.
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.