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Physical review. B, Condensed matter

Tricriticality and the failure of scaling in the many-component limit

Stéphane Sarbach, Michael Ellis Fisher

Published Sep 1, 1978• 49 citations
Publisher

Abstract

We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the $n$-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 $3<d<4$: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ line ($H={H}_{3}=0, T>{T}_{t}$) it is essential to allow for a dangerous irrelevant scaling variable $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where ${R}_{0}$ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ so that $p$ is marginal for $d=3$: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane ($H={H}_{3}=0$) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For $T>{T}_{t}$ bicritical crossover from spherical to classical critical behavior occurs when ${H}_{3}$ varies but scaling is fully obeyed. Some inferences for systems with finite $n$ are drawn.

DOI
10.1103/physrevb.18.2350

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Physical review. B, Condensed matter•Nov 1, 1978•37 citations
We analyze exactly in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞, the nnn-component continuous-spin model with a cubic field term H\ensuremath→3\ifmmode⋅\else\textperiodcentered\fis\ensuremath→∣s\ensuremath→∣2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}}_{3}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}|}^{2}H\ensuremath→​3​\ifmmode⋅\else\textperiodcentered\fis\ensuremath→∣s\ensuremath→∣2. Full "four-field" tricritical behavior is exhibited for dimensionalities d\ensuremath≥3d\ensuremath{\ge}3d\ensuremath≥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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ 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∝1R0dp\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}p\ensuremath∝R0d​1​, where R0{R}_{0}R0​ is the range of the pair interactions. The appropriate crossover exponent is \ensuremathφp=3\ensuremath−d{\ensuremath{\varphi}}_{p}=3\ensuremath{-}d\ensuremathφp​=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.
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.
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.
Deviations from the classical phenomenological theory of tricriticality are analyzed on the basis of the exact solution (in four field variables) of the spherical-model limit ($n\ensuremath{\rightarrow}\ensuremath{\infty}$) in three dimensions. The scaling functions are nonuniversal but are parametrized by a single variable, which vanishes for infinite-range interactions. Special relations between critical amplitude ratios correlate well with observations on antiferromagnets and 3He^{3}\mathrm{He}3He-4He^{4}\mathrm{He}4He mixtures. Novel qualitative predictions are made for the threephase monohedron below Tt{T}_{t}Tt​.
A cubic field coupling to s mod s mod 2 , in n-component spin models induces a bicritical crossover from n-isotropic to Ising-like (m=1) critical behaviour for 1<n< infinity , but to classical behaviour in the limit n to infinity . The mechanism for this is explained and related to the van der Waals or infinite-range crossover. By following the analysis of Nelson and Domany (1976), the bicritical scaling function for the parallel susceptibility in d dimensions is obtained correct to order epsilon =4-d and for general (m,n) in a form correctly reproducing all limits; the effective exponent, gamma eff , is examined numerically in various regimes.
Renormalisation of a field theory model for tricritical behaviour in which the operator phi 6 is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi p =- 1 / 2 (1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi ) 2 in the field-theoretic Hamiltonian.
The tricirical exponents ${\ensuremath{\eta}}_{t}$ and ${\ensuremath{\varphi}}_{t}$ are calculated through order ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{2}$, respectively, in the isotropic nnn-component model, where $\ensuremath{\epsilon}=3\ensuremath{-}d$. The estimate for ${\ensuremath{\eta}}_{t}$, in two dimensions for the Ising case, is 0.027; the series for ${\ensuremath{\varphi}}_{t}$ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in $\ensuremath{\epsilon}$, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through ${\ensuremath{\epsilon}}^{2}$; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where $\ensuremath{\theta}$ coexisting phases become critical.