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Journal of Physics A Mathematical Nuclear and General

Tricritical behaviour in an Ising system and the Potts model

R.V. Ditzian, J. Oitmaa

Published Mar 21, 1974• 43 citations
Publisher

Abstract

The susceptibility of the Ising S=1 model with biquadratic interactions is expanded in a high temperature series on the FCC lattice for a range of the biquadratic interaction parameter that includes the Potts model (1952). A tricritical point and a point above which the magnetization M remains zero are approximately located. The Potts model seems to fall in the region of the first-order phase transition in M.

DOI
10.1088/0305-4470/7/5/002

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The susceptibility of the Ising S=1 model with biquadratic interactions is expanded in a high temperature series on the FCC lattice for a range of the biquadratic interaction parameter that includes the Potts model (1952). A tricritical point and a point above which the magnetization M remains zero are approximately located. The Potts model seems to fall in the region of the first-order phase transition in M.
Reports studies of the critical behaviour of the Ising model with first and second neighbour interactions J 1 , J 2 on the triangular lattice. High-temperature series have been used to determine the locations of the ferromagnetic critical line and the critical line in the region (J 1 <0, J 2 >0). On the latter line the ordering susceptibility appears to diverge algebraically with the exponent gamma approximately 2.4. The existence of a sinusoidal phase and an associated Lifshitz point are conjectured.
Thirteen terms are presented of the low-temperature series for the free energy of the three-state Potts model in arbitrary external fields. Extrapolation of the series for specific heat, order parameter, and susceptibilities indicates that the transition in zero field is continuous (in contrast to the Landau prediction) with exponents alpha = alpha '=0.05+or-0.10, beta =0.10+or-0.01, gamma ' approximately=1.5+or-0.2 and gamma ' perpendicular to =1.1+or-0.1. These conclusions suggest the presence of an 'anomalous' tricritical point in the model; the associated phase diagram is discussed and contrasted with phenomenological predictions.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (JJJ) and biquadratic (KKK) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in JJJ, KKK, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (HHH) and another odd interaction (LLL) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.
For paper II, see ibid, vol. 27, p.1503 (1994). The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.