Journal of Physics C Solid State Physics

Cubic fields, bicritical crossover, the spherical and van der Waals limits

Pere Seglar, Michael Ellis Fisher

Published Dec 30, 1980• 27 citations

Abstract

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.

DOI: 10.1088/0022-3719/13/36/015

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Journal of Physics C Solid State Physics

Cubic fields, bicritical crossover, the spherical and van der Waals limits

Pere Seglar, Michael Ellis Fisher

Published Dec 30, 1980• 27 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/13/36/015

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.

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Physical review. B, Condensed matterNov 1, 197837 citations

Dimensional crossover in quantal and finite-size Ising models

Ian D. Lawrie

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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.

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Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Ann Bruce, D. J. Wallace

Journal of Physics A Mathematical and GeneralJul 1, 197674 citations

Universality and crossover in an Ising-like model

M A Moore, D. M. Saul, Michael Wortis

The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.The critical behaviour of an anisotropic n-vector model with purely Ising-like coupling is studied on the FCC lattice by means of high-temperature series. The model includes the s= 1 / 2 (n=1) and s= infinity (n=3) Ising models as special cases and exhibits rigorously mean-field behaviour at n= infinity . The crossover region 1/n to 0, t identical to (1-v/v c ) to 0, is examined in detail. The susceptibility exponent gamma remains Ising-like for all finite n. The critical temperature and critical susceptibility amplitude vary as 1/n.

Journal of Physics C Solid State PhysicsJan 7, 197417 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.

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.