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Journal of Physics A Mathematical and General

Universality in the two-dimensional continuous spin model

Ann Bruce

Published Oct 1, 1985• 76 citations
Publisher

Abstract

Monte Carlo methods have been used to analyse the critical behaviour of the d=2 'border' phi 4 model which, according to a recent series expansion study, violates expectations based upon universality. The results suggest that the model does, in fact, fall into the Ising universality class, at least as regards those parameters which characterise the critical point configurations of the ordering variable.

DOI
10.1088/0305-4470/18/14/009

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Monte Carlo methods have been used to analyse the critical behaviour of the d=2 'border' phi 4 model which, according to a recent series expansion study, violates expectations based upon universality. The results suggest that the model does, in fact, fall into the Ising universality class, at least as regards those parameters which characterise the critical point configurations of the ordering variable.
Monte Carlo methods are used to explore the universal configurational structure of two-dimensional spin- 1/2, spin-1 and border- phi 4 models. Comparison of spin- 1/2 and spin-1 data provides evidence that the magnetisation distribution (effectively the Helmholtz free-energy function) and its coupling derivative (effectively the internal-energy function) constitute readily accessible signatures of a universality class. It is shown that, when allowance is made for relatively large corrections-to-scaling effects, the behaviour of the border- phi 4 model may be satisfactorily matched to that of the other two models, substantiating the view that the border model does indeed belong to the Ising universality class.
Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.
Although the two-dimensional spin- 1 / 2 Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin- 1 / 2 model, but appears to exhibit non-analytic corrections to scaling as expected for a phi 4 model and unlike the spin- 1 / 2 model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi 4 universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta 1 <1.3 is found. The validity of this result is enhanced by a comparison with the S= 1 / 2 case.
The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (J2{J}_{2}J2​) and four - (J4{J}_{4}J4​) spin interactions. In the limit J2=0{J}_{2}=0J2​=0 the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.
In this work, we evaluate the probability distribution function of the order parameter for the two-dimensional site-diluted Ising model. Extensive Monte Carlo simulations have been performed for different spin concentrations p (0.70<or=p<or=1) on square lattices with linear sizes L (20<or=L<or=100). The results indicate that the universality class of the diluted Ising model seems to be independent of the amount of dilution. Logarithmic corrections of the finite-size critical temperature behavior of the model can also be inferred even for such small lattices.