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Physical review. B, Solid state

Equations of state for bicritical points. III. Cubic anisotropy and tetracriticality

Eytan Domany, Michael Ellis Fisher

Published Apr 1, 1977• 36 citations
Publisher

Abstract

Crossover scaling functions for a two-component spin system with quadratic anisotropy $g$ and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point ($T={T}_{m},g=0$). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.

DOI
10.1103/physrevb.15.3510

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Journal of Physics C Solid State Physics•Dec 30, 1980•27 citations
Crossover scaling functions for a two-component spin system with quadratic anisotropy ggg and cubic anisotropy v\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}v\ifmmode~\else˜\fi are calculated to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d and in v\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}v\ifmmode~\else˜\fi, near the transitions within the ordered phases below the multicritical point (T=Tm,g=0T={T}_{m},g=0T=Tm​,g=0). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable v\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}v\ifmmode~\else˜\fi. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.
A scaling theory is introduced for bicritical points, such as antiferromagnetic spinflop points (with analogies to the upper $\ensuremath{\lambda}$ point in 4He^{4}\mathrm{He}4He), where two distinct critical lines meet. Experimentally testable predictions follow from renormalization-group calculations which indicate that the bicritical exponents should be Heisenberg like for systems with $n\ensuremath{\le}3$ components; the crossover exponent $\ensuremath{\varphi}(\ensuremath{\simeq}1.25 \mathrm{for} n=3)$ is directly observable. For n>3n>3n>3 an intermediate ("supersolid") low-temperature phase may appear, the bicritical point then becoming tetracritical.
The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.
A cutoff-dependent formulation of renormalized perturbation theory is extended to study the crossover of the parallel susceptibility in systems with quadratic symmetry breaking near a bicritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. In terms of the true temperature variable $t=T\ensuremath{-}{T}_{c}(g)$, for anisotropy ggg, and crossover variable $x=\frac{g}{{t}^{\ensuremath{\varphi}}}$, the effective susceptibility exponent ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is calculated for two ${\ensuremath{\varphi}}^{4}$-field models, one with a sharp momentum cutoff $\ensuremath{\Lambda}$ and the other with a smooth term ${\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}$. Corrections to the first model due to insertions of composite operators of canonical dimension six, inside the scaling regime, are explicitly calculated and shown to yield small but not negligible contributions to crossover behavior. The role of new, finite renormalizations is discussed and it is shown that either a fairly monotonic ${\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)$ is obtained or a displaced maximum may appear depending on the outcome of a higher-order renormalization-group study, which is not done here.
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.
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.