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Physical review. B, Condensed matter

Gaussian-to-Heisenberg crossover behavior of the specific heat in renormalized perturbation theory

W. K. Theumann

Published Mar 1, 1980• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.21.1930

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Journal of Physics A Mathematical and General•Sep 1, 1977•9 citations
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 \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=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 \ensuremathΓ(2,0){\ensuremath{\Gamma}}^{(2,0)}\ensuremathΓ(2,0) - proportional to the specific heat - with two \ensuremathφ2{\ensuremath{\varphi}}^{2}\ensuremathφ2 insertions is discussed in detail in an isotropic \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 field model. An explicit crossover scaling function for the specific heat is calculated and the effective exponent \ensuremathαeff{\ensuremath{\alpha}}_{\mathrm{eff}}\ensuremathαeff​ is found to differ considerably from an earlier first-order calculation.
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.
By expressing the free energy as a line integral along a renormalization-group trajectory, a technique for calculating the crossover scaling functions which describe tricritical systems, anisotropic spin systems, etc., is developed. This formalism is applied to the model recursion relations of Riedel and Wegner, which simulate crossover behavior. A simple mechanism for the breakdown of dimensionality dependent hyperscaling relationships emerges from the analysis. The specific-heat crossover scaling function describing crossovers from Gaussian to Heisenberg critical behavior is constructed to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$.
The expectations of phenomenological crossover scaling theory are worked out for two-point correlation functions in zero field above the primary critical point. Renormalization-group recursion relations are then used to construct the two-point correlation function for the changeover from tricritical to critical-line behavior, above the tricritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, for $3\ensuremath{\le}d<4$. An explicit expression is obtained for the normalized double-scaling function ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}({q}^{2}{\ensuremath{\xi}}^{2}, \frac{g}{{t}^{\ensuremath{\phi}}})$, for the general scaling variable ${q}^{2}{\ensuremath{\xi}}^{2}$, and the crossover variable $w=\frac{g}{{t}^{\ensuremath{\phi}}}$, $q\ensuremath{\ll}{a}^{\ensuremath{-}1}$ being the wave vector; $\ensuremath{\xi}\ensuremath{\gg}a$ is a second-moment correlation length and aaa the lattice spacing. In the tricritical regime, $w\ensuremath{\simeq}0$, ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}$ (x2,w{x}^{2}, wx2,w) has the Ornstein-Zernicke form and the changeover to the critical regime $w\ensuremath{\simeq}\ensuremath{\infty}$ is explicitly discussed. The limitation of the results, within our calculations to first order in $\ensuremath{\epsilon}$, are pointed out.
By considering the relationship of the matching techniques of Bruce and Wallace to the differential renormalization-group generators, we find that a restatement of the former gives improved results with the same number of perturbative terms. In particular, the vertex functions and specific heat of a nnn-component spin system are given exactly in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ even at first order in perturbation theory (T>TcT>{T}_{c}T>Tc​). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general nnn-component disordered phase functions are rederived to $O({\ensuremath{\epsilon}}^{2})$, where $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. The cross-over equations for the n=1n=1n=1 Ising-like case are derived for the Helmholtz potential A(M)A(M)A(M), the magnetic field hM\frac{h}{M}Mh​, the inverse susceptibility ${\ensuremath{\Gamma}}_{2}$, and the correlation length $\ensuremath{\xi}$ to $O({\ensuremath{\epsilon}}^{2})$.
The crossover behaviour between Gaussian and Heisenberg critical behaviour is studied in an isotropic n-vector model, in which the effect of a lattice cutoff, a= Lambda -1 , is imitated by means of (nabla 2 phi ) 2 and (nabla nabla 2 phi ) 2 terms in the effective Hamiltonian. A field theoretic method is used to construct, to first order in in =4-d, a crossover scaling function for the susceptibility containing the corrections to scaling which involve Lambda directly. The leading corrections vanish, near the Heisenberg fixed point, as ((T-T c ) v / Lambda ) 2 where v is the corresponding correlation length exponent, and near the Gaussian fixed point as ((T-T c ) 1 2// Lambda ) 2 .