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Journal of Physics A Mathematical and General

Q-state Potts models in Hamiltonian field theory for Q⩾4 in (1+1) dimensions

Chris J. Hamer

Published Nov 1, 1981• 48 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/14/11/020

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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.
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.
A method is presented to accelerate the convergence of finite-lattice sequences to their bulk limit. The calculation of highly accurate estimates of the critical parameters of the bulk system is then possible. Applied to the Hamiltonian version of the Z3 model (three-state Potts model) in (1+1) dimensions, these techniques yield estimates for the exponents gamma =1.444+or-0.0001, nu =0.8333+or-0.0003 and alpha =0.33+or-0.01. For the Z5 model, the presence of a Kosterlitz-Thouless transition is confirmed.
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.
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.