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Physical review. B, Solid state

Equivalence of two exactly soluble models for tricritical points

St. Sarbach, T. Schneider

Published Jan 1, 1976• 9 citations
Publisher

Abstract

An exactly soluble $n$-component continuous spin model with tricritical points is shown to be equivalent, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, to a one-component model with anharmonic long-range interactions.

DOI
10.1103/physrevb.13.464

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Tricritical scaling in the spherical model limit

Stéphane Sarbach, Michael Ellis Fisher

The tricritical scaling properties of the n-component continuous-spin model are investigated in the sphercal model limit, n→∞, with special emphasis on the odd fields h and h3. A full scaling description of the tricritical region for 3<d<4 necessitates the recognition of a dangerous irrelevant variable, p, related to the inverse range of the pair interaction. As the lambda line in zero field (h=h3=0) is approached, p becomes important and determines the correct spherical model exponents. Setting p=0 gives only classical theory; away from the lambda line p yields corrections to scaling of Gaussian character. The wing critical exponents arising for h≠0, h3=0 and T<Tt are classical. Furthermore, for h3≠0 classical behavior is found even on the lambda line above Tt which, in fact, continues into one of the wing critical lines. Some consequences for real tricritical behavior are reviewed.

Journal of Applied Physics•Mar 1, 1978•13 citations
An exactly soluble nnn-component continuous spin model with tricritical points is shown to be equivalent, in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞, to a one-component model with anharmonic long-range interactions.
An nnn-component continuous-spin model with a tricritical point is solved exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. Its properties are derived and are compared with the results of experiments and approximate calculations. The model leads to an analytical description of the features of tricritical points and suggests the use of an approximation procedure for discrete spin systems which is a substantial qualitative and quantitative improvement on mean-field theory.
The asymptotic tricritical equation of state, including the three-phase coexistence monohedron, is analyzed in detail for the exactly soluble multicomponent or spherical limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$, of the continuous-spin model with terms of order s3{s}^{3}s3, s4{s}^{4}s4, and s6{s}^{6}s6, in d=3d=3d=3 spatial dimensions. Various nonuniversal scaling functions and amplitude ratios, depending on the range parameter $z\ensuremath{\propto}\frac{1}{{R}_{0}^{3}}$, are evaluated explicitly and reveal the nature and magnitude of the deviations from the classical, phenomenological theory of tricriticality (which is developed systematically in an Appendix). The relationship to results for finite nnn is discussed briefly.
We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the nnn-component continuous-spin model with a cubic field term ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}}_{3}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}|}^{2}$. Full "four-field" tricritical behavior is exhibited for dimensionalities $d\ensuremath{\ge}3$. However, orthodox tricritical scaling is shown to be impossible for 3<d<43<d<43<d<4: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ line (H=H3=0,T>TtH={H}_{3}=0, T>{T}_{t}H=H3​=0,T>Tt​) it is essential to allow for a dangerous irrelevant scaling variable $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where R0{R}_{0}R0​ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ so that ppp is marginal for d=3d=3d=3: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane (H=H3=0H={H}_{3}=0H=H3​=0) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For T>TtT>{T}_{t}T>Tt​ bicritical crossover from spherical to classical critical behavior occurs when H3{H}_{3}H3​ varies but scaling is fully obeyed. Some inferences for systems with finite nnn are drawn.
Abstract Two infinite range Ising models are solved exactly using a microcanonical ensemble in the longrange order parameter. Classical tricritical points exist for both models. The first model involving two‐spin and four‐spin terms is relevant to the compressible Ising model. The other model is the infinite range version of the Blume‐Capel spin‐one Ising model. In both the cases the exact free‐energy functions calculated are examples of exact classical Landau free‐energy functions with tricritical point behaviour.
The tricritical scaling properties of the n-component continuous-spin model are investigated in the sphercal model limit, n→∞, with special emphasis on the odd fields h and h3. A full scaling description of the tricritical region for 3<d<4 necessitates the recognition of a dangerous irrelevant variable, p, related to the inverse range of the pair interaction. As the lambda line in zero field (h=h3=0) is approached, p becomes important and determines the correct spherical model exponents. Setting p=0 gives only classical theory; away from the lambda line p yields corrections to scaling of Gaussian character. The wing critical exponents arising for h≠0, h3=0 and T<Tt are classical. Furthermore, for h3≠0 classical behavior is found even on the lambda line above Tt which, in fact, continues into one of the wing critical lines. Some consequences for real tricritical behavior are reviewed.