Journal of Applied Physics

Tricritical scaling in the spherical model limit

Stéphane Sarbach, Michael Ellis Fisher

Published Mar 1, 1978• 13 citations

Abstract

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.

DOI: 10.1063/1.324986

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

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Journal of Applied Physics

Tricritical scaling in the spherical model limit

Stéphane Sarbach, Michael Ellis Fisher

Published Mar 1, 1978• 13 citations

Publisher

Abstract

DOI: 10.1063/1.324986

Related papers

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

Stéphane Sarbach, Michael Ellis Fisher

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.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.

Physical review. B, Condensed matterSep 1, 197849 citations

Nonuniversality of Tricritical Behavior

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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 $^{3}\mathrm{He}$ - $^{4}\mathrm{He}$ mixtures. Novel qualitative predictions are made for the threephase monohedron below ${T}_{t}$ .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 $^{3}\mathrm{He}$-$^{4}\mathrm{He}$ mixtures. Novel qualitative predictions are made for the threephase monohedron below ${T}_{t}$.

Physical Review LettersOct 16, 197864 citations

Tricritical scaling and renormalisation of ϕ6operators in scalar systems near four dimensions

Ian D. Lawrie

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.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.

Journal of Physics A Mathematical and GeneralJun 1, 197916 citations

Critical Behavior in Tricritical Phase Transition. I: Critical Exponents ( , , , , )

Y. Yamazaki, Masuo Suzuki

Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.

Progress of Theoretical PhysicsAug 1, 19773 citations

Tricritical behavior in two dimensions. II. Universal quantities from the ε expansion

A. L. Lewis, F. W. Adams

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 $n$ -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.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 $n$-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.

Physical review. B, Condensed matterNov 1, 197837 citations

Critical Behavior in Tricritical Phase Transition. II: Equation of State, Critical Exponents ( , ) and Logarithmic Corrections

Y. Yamazaki, Masuo Suzuki

Progress of Theoretical PhysicsNov 1, 19774 citations

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.

^{3}\mathrm{He}

^{4}\mathrm{He}

mixtures. Novel qualitative predictions are made for the threephase monohedron below

{T}_{t}

n

-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.