Journal of Physics C Solid State Physics

Universality and tricritical points in three dimensions

M. J. Stephen

Published Feb 10, 1980• 24 citations

Abstract

Amplitude ratios near tricritical points in three dimensions are discussed and shown to be universal and in agreement with the classical values when logarithmic corrections are included. The spherical model of a tricritical point is shown to be a special case exhibiting non-universality because of the absence of logarithmic corrections.

DOI: 10.1088/0022-3719/13/4/005

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Journal of Physics C Solid State Physics

Universality and tricritical points in three dimensions

M. J. Stephen

Published Feb 10, 1980• 24 citations

Publisher

Abstract

DOI: 10.1088/0022-3719/13/4/005

Related papers

Nonuniversality of Tricritical Behavior

Michael Ellis Fisher, Stéphane Sarbach

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 ${T}_{t}$.

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Universal amplitude ratios of the renormalization group: Two-dimensional tricritical Ising model

Davide Fioravanti A, Giuseppe Mussardo, Pascal Simon

The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.

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We study d-dimensional ferromagnetic spherical models with power-law, J(R)\ensuremath{\sim}1/${\mathit{R}}^{\mathit{d}+\mathrm{\ensuremath{\sigma}}}$ (\ensuremath{\sigma}2), or short-range (\ensuremath{\sigma}\ensuremath{\simeq}2) interactions and on-site terms of the form ${\mathit{Ds}}^{2}$ -\ensuremath{\Vert}U\ensuremath{\Vert} ${\mathit{s}}^{4}$ + ${\mathit{Vs}}^{6}$ , and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.We study d-dimensional ferromagnetic spherical models with power-law, J(R)\ensuremath{\sim}1/${\mathit{R}}^{\mathit{d}+\mathrm{\ensuremath{\sigma}}}$ (\ensuremath{\sigma}2), or short-range (\ensuremath{\sigma}\ensuremath{\simeq}2) interactions and on-site terms of the form ${\mathit{Ds}}^{2}$-\ensuremath{\Vert}U\ensuremath{\Vert}${\mathit{s}}^{4}$+${\mathit{Vs}}^{6}$, and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.

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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

^{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.

{\mathit{Ds}}^{2}

-\ensuremath{\Vert}U\ensuremath{\Vert}

{\mathit{s}}^{4}

{\mathit{Vs}}^{6}

, and establish that the phase diagrams for all d and \ensuremath{\sigma} with \ensuremath{\sigma}d2\ensuremath{\sigma}, may exhibit critical end points terminating nonclassical critical lines; tricritical points may arise only for 3/2\ensuremath{\sigma}\ensuremath{\le}d. The phase boundary ${\mathit{D}}_{\mathrm{\ensuremath{\sigma}}}$(T,h), where h is the magnetic field, between the noncritical, spectator phase and the ordering and critical phase is shown to exhibit singularities at the end point with amplitudes obeying the universal ratio laws advanced on phenomenological grounds in Part I.

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.