Physical Review Letters

Nonuniversality of Tricritical Behavior

Michael Ellis Fisher, Stéphane Sarbach

Published Oct 16, 1978• 64 citations

Abstract

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 ${T}_{t}$.

DOI: 10.1103/physrevlett.41.1127

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Universality and tricritical points in three dimensions

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Journal of Physics C Solid State PhysicsFeb 10, 198024 citations

Physical Review Letters

Nonuniversality of Tricritical Behavior

Michael Ellis Fisher, Stéphane Sarbach

Published Oct 16, 1978• 64 citations

Publisher

Abstract

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}

DOI: 10.1103/physrevlett.41.1127

Related papers

Tricritical coexistence in three dimensions: The multicomponent limit

Stéphane Sarbach, Michael Ellis Fisher

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 ${s}^{3}$ , ${s}^{4}$ , and ${s}^{6}$ , in $d=3$ 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 $n$ is discussed briefly.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 ${s}^{3}$, ${s}^{4}$, and ${s}^{6}$, in $d=3$ 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 $n$ is discussed briefly.

Physical review. B, Condensed matterOct 1, 197941 citations

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

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.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 PhysicsMar 1, 197813 citations

Nonuniversality of tricritical behavior of Fermi-Bose mixtures

K. K. Singh

Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}( ${b}^{3}$ /${\ensuremath{\lambda}}_{B}^{4}$ ${K}_{F}$ ${)}^{1/2}$ where ${K}_{F}$ , ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(${b}^{3}$/${\ensuremath{\lambda}}_{B}^{4}$${K}_{F}$${)}^{1/2}$ where ${K}_{F}$, ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.

Physical review. B, Condensed matterJan 1, 19873 citations

Phase transitions and criticality: Finite-size effects, conformal covariance and other current problems

Michael Ellis Fisher

This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!

Journal of Magnetism and Magnetic MaterialsFeb 1, 19863 citations

Universality and tricritical points in three dimensions

M. J. Stephen

Journal of Physics C Solid State PhysicsFeb 10, 198024 citations

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}

^{3}\mathrm{He}

^{4}\mathrm{He}

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

{T}_{t}

{s}^{3}

{s}^{4}

, and

{s}^{6}

, in

d=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

n

is discussed briefly.

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.

Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(

{b}^{3}

/${\ensuremath{\lambda}}_{B}^{4}$

{K}_{F}

{)}^{1/2}

where

{K}_{F}

, ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.