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Physical review. B, Condensed matter

Renormalized perturbation theory for anisotropic spin systems: Crossover behavior of the susceptibility

W. K. Theumann

Published Aug 1, 1981• 2 citations
Publisher

Abstract

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 $g$, 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.

DOI
10.1103/physrevb.24.1504

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Physical review. B, Condensed matter•Sep 1, 1984•5 citations
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 \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d. In terms of the true temperature variable t=T\ensuremath−Tc(g)t=T\ensuremath{-}{T}_{c}(g)t=T\ensuremath−Tc​(g), for anisotropy ggg, and crossover variable x=gt\ensuremathφx=\frac{g}{{t}^{\ensuremath{\varphi}}}x=t\ensuremathφg​, the effective susceptibility exponent \ensuremathγeff(t,x){\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)\ensuremathγeff​(t,x) is calculated for two \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4-field models, one with a sharp momentum cutoff \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ and the other with a smooth term \ensuremathΛ\ensuremath−2(\ensuremath∇2\ensuremathφ)2{\ensuremath{\Lambda}}^{\ensuremath{-}2}{({\ensuremath{\nabla}}^{2}\ensuremath{\varphi})}^{2}\ensuremathΛ\ensuremath−2(\ensuremath∇2\ensuremathφ)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 \ensuremathγeff(t,x){\ensuremath{\gamma}}_{\mathrm{eff}}(t,x)\ensuremathγeff​(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.
A cutoff-dependent formulation of renormalized perturbation theory is used to study the Gaussian-to-Heisenberg crossover, describing the changeover from tricritical to critical-line behavior, for the specific heat, to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, in extension of earlier work by Bruce and Wallace for the susceptibility. The role of the nonhomogeneous part of the renormalization-group equation for the bare vertex function ${\ensuremath{\Gamma}}^{(2,0)}$ - proportional to the specific heat - with two ${\ensuremath{\varphi}}^{2}$ insertions is discussed in detail in an isotropic ${\ensuremath{\varphi}}^{4}$ field model. An explicit crossover scaling function for the specific heat is calculated and the effective exponent ${\ensuremath{\alpha}}_{\mathrm{eff}}$ is found to differ considerably from an earlier first-order calculation.
Abstract We derive a theory of crossover scaling based on a scaling variable gξg, where g is the anisotropy parameter inducing the crossover and ξg is the correlation length in the presence of g. Our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom. ξg is a nonlinear scaling field for this renormalization group and interpolates between (T – Tc(g))–v0 and (T – Tc(g))–v∞ (v0 and v∞ being the isotropic and anisotropic exponents respectively). By expanding about a ‘floating’ fixed point we can make corrections to scaling small throughout the crossover. In this formulation effective scaling exponents obey standard scaling laws, e. g. γeff = veff(2 – ɳeff). We discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility, which is found to conform to the general form derived from the g dependent renormalization group.
A continuous ${\ensuremath{\varphi}}^{3}$ field theory with quadratic symmetry breaking and isotropic trilinear interaction ${g}_{0} \ensuremath{\Sigma}{d}_{\mathrm{ijk}}{\ensuremath{\varphi}}_{i}{\ensuremath{\varphi}}_{j}{\ensuremath{\varphi}}_{k}$ is studied, to one-loop order in dimension $d=6\ensuremath{-}\ensuremath{\epsilon}$, for an anisotropic (n+1n+1n+1)-state Potts model. Effective critical exponents for the crossover induced by quadratic symmetry breaking are calculated, and the results, in the limit n=0n=0n=0, are applicable to the thermally driven crossover in random bond-diluted ferromagnets near the percolation threshold. The limitation of a single renormalized ggg is explicitly pointed out.
The crossover behavior of the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ of spin systems with anisotropic exchange coupling is discussed on the basis of an "extended" scaling hypothesis which gives ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)\ensuremath{\approx}A{t}^{\ensuremath{-}\ensuremath{\gamma}}{X}_{\ensuremath{\alpha}}(\frac{Bg}{{t}^{\ensuremath{\varphi}}})$, where ggg is the anisotropy parameter, $t=\frac{(T\ensuremath{-}{T}_{c0})}{{T}_{c0}}$, with Tc0{T}_{c0}Tc0​ and $\ensuremath{\gamma}$ being the iso tropic critical temperature, and susceptibility exponent, while AAA and BBB are model-dependent amplitudes. Analysis of high-temperature series expansions for ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(T,g)$ as polynomials in ggg for rhombic and axial anisotropy in the fcc, bcc, and simple cubic classical Heisenberg (n=3n=3n=3) and classical XYXYXY and planar-spin models (N=2N=2N=2), verifies the scaling with crossover exponents $\ensuremath{\varphi}=1.25\ifmmode\pm\else\textpm\fi{}0.015$ (n=3n=3n=3) and $\ensuremath{\varphi}=1.175\ifmmode\pm\else\textpm\fi{}0.015$ (n=2n=2n=2). The universality of the scaling functions ${X}_{\ensuremath{\alpha}}(x)$ is demonstrated both for small xxx and in the anisotropic limit $T\ensuremath{\rightarrow}{T}_{c}(g)$, where $x\ensuremath{\rightarrow}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}$; for n=3n=3n=3, accurate representations are constructed in the form ${X}_{\ensuremath{\alpha}}(x)\ensuremath{\simeq}{P}_{\ensuremath{\alpha}}(\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}}){(1\ensuremath{-}\frac{x}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{x}})}^{\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ is the anisotropic susceptibility exponent, while ${P}_{\ensuremath{\alpha}}(z)$ varies smoothly from ${P}_{\ensuremath{\alpha}}(0)=1$ to ${P}_{\ensuremath{\alpha}}(1)\ensuremath{\simeq}1.10$.
The thermodynamic properties of an anisotropic multicomponent spin system are investigated near the bicritical point in the disordered phase using a renormalization group approach. Explicit forms for the cross-over scaling functions for the specific heat and the susceptibilities are found to lowest orders in epsilon =4-d in terms of z=gt - phi where g is the non-ordering field and t=(T-T b )/T b with T b the bicritical temperature. The scaling functions have the expected properties in the limits z to O and z to zt/dt, the value on the critical lines, and interpolate smoothly between these limits.
The effect of quadratic symmetry breaking (QSB) on two representations for the Potts vectors in a continuum-field model are studied to two-loop order in renormalized perturbation theory in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, in extension of an earlier group-theoretical analysis by Wallace and Young. The explicit dependence of the crossover exponent $\ensuremath{\varphi}$ that corresponds to QSB that destroys the equivalence between pairs of Potts vectors is obtained as a function of ddd and nnn for the ppp-state model with p=n+1p=n+1p=n+1. It is shown that this exponent follows from the calculation of vertex functions in a representation due to Wallace and Young, whereas a second crossover exponent $\overline{\ensuremath{\varphi}}$, that can be identified with the critical exponent $\ensuremath{\beta}$, and which corresponds to QSB that favors a single Potts vector against the others, follows from a calculation using the representation of the Potts vectors due to Priest and Lubensky.