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Physical review. B, Condensed matter

Tricritical coexistence in three dimensions: The multicomponent limit

Stéphane Sarbach, Michael Ellis Fisher

Published Oct 1, 1979• 41 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.20.2797

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Journal of Physics A Mathematical and General•Jun 1, 1979•16 citations
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→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞, 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∝1R03z\ensuremath{\propto}\frac{1}{{R}_{0}^{3}}z\ensuremath∝R03​1​, 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​.
We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the nnn-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<43<d<43<d<4: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ 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 $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where R0{R}_{0}R0​ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ 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.
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
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.
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!
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.