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Physical review. B, Solid state

Tricritical behavior of the Ising antiferromagnet with next-nearest-neighbor ferromagnetic interactions: Mean-field-like tricritical exponents?

Michael Wortis, Fredric Harbus, Eugene Stanley

Published Apr 1, 1975• 20 citations
Publisher

Abstract

The tricritical behavior of a model studied by Harbus and Stanley is reanalyzed. The Harbus-Stanley analysis of the direct susceptibility suggested anomalous (i.e., non-mean-field-like) tricritical exponents. The tricritical point cannot be unambiguously located by present series data. We point out that, if the true tricritical temperature were actually somewhat lower than the Harbus-Stanley value, then standard ratio analysis would give tricritical exponents consistent with the Gaussian-tricritical-fixed-point analysis of Riedel and Wegner. However, the present situation remains inconclusive.

DOI
10.1103/physrevb.11.2689

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The tricritical behavior of a model studied by Harbus and Stanley is reanalyzed. The Harbus-Stanley analysis of the direct susceptibility suggested anomalous (i.e., non-mean-field-like) tricritical exponents. The tricritical point cannot be unambiguously located by present series data. We point out that, if the true tricritical temperature were actually somewhat lower than the Harbus-Stanley value, then standard ratio analysis would give tricritical exponents consistent with the Gaussian-tricritical-fixed-point analysis of Riedel and Wegner. However, the present situation remains inconclusive.
Monte Carlo data for a simple cubic Ising antiferromagnet with nearest- and next-nearest-neighbor interactions reveal asymptotic tricritical behavior of the order parameter which is mean-field-like modified by logarithmic corrections and high-temperature susceptibilities which are mean-field-like without corrections, in agreement with renormalization-group calculations. Crossover between tricritical and critical behavior is observed in the temperature variation of the order parameter and high-temperature susceptibility.
The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (J1<0{J}_{1}<0J1​<0) but also next-nearest-neighbor ferromagnetic exchange (J2>0{J}_{2}>0J2​>0). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power a‾2{\overline{a}}_{2}a2​ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.
Using a Monte Carlo renormalization-group (MCRG) method we study tricritical behavior in two very different two-dimensional spin models: the Blume-Capel ferromagnet and the next-nearest-neighbor Ising antiferromagnet. The MCRG method is used to locate accurately tricritical points independent of the convergence of eigenvalue estimates. Despite the different symmetries and the different renormalization transformations used for the two models, in both cases we find four relevant tricritical eigenvalues which are essentially identical for both models. For the antiferromagnet, there is no indication of any decomposition of the tricritical point as predicted by mean-field theory, for ratios of intrasublattice to intersublattice coupling as small as (1/8).