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Canadian Journal of Physics

Tricritical phenomena in an Ising model with nearest- and next-nearest-neighbor interactions

Michael Plischke, D. Zobin

Published Jul 1, 1977• 2 citations
Publisher

Abstract

We report on the analysis of low and high temperature series for the Ising model with nearest-neighbor antiferromagnetic and next-nearest-neighbor ferromagnetic interactions on the bcc lattice. The high temperature series are complete to β7, the low temperature series to u37. We determine the phase diagram, locate the tricritical point, and estimate the tricritical exponents. The tricritical exponents are only in fair agreement with the predictions of tricritical mean field theory.

DOI
10.1139/p77-150

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We report on the analysis of low and high temperature series for the Ising model with nearest-neighbor antiferromagnetic and next-nearest-neighbor ferromagnetic interactions on the bcc lattice. The high temperature series are complete to β7, the low temperature series to u37. We determine the phase diagram, locate the tricritical point, and estimate the tricritical exponents. The tricritical exponents are only in fair agreement with the predictions of tricritical mean field theory.
We have generalized the code method of Sykes et al. and applied it to the Ising model with nearest and next nearest neighbor interactions. On the bcc lattice, we have obtained the first seven low temperature polynomials for arbitrary sign of the interactions. Special cases of this model are the Ising ferromagnet and the Ising antiferromagnet with next nearest neighbor ferromagnetic interactions. The latter system exhibits a tricritical point which we plan to study using our low temperature data and high temperature series to be obtained in the future.
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.
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.
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 authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.