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Journal of Physics A Mathematical and General

Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Ann Bruce, D. J. Wallace

Published Jul 1, 1976• 74 citations
Publisher

Abstract

Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2 ) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.

DOI
10.1088/0305-4470/9/7/015

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Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2 ) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.
Feynman graph methods are used to analyse crossover behaviour in an n-component Heisenberg system, with a spin anisotropy that results asymptotically in m-component critical behaviour. An expression for the universal n- and m-dependent amplitude locating the singularity of the crossover scaling function is derived to second order in epsilon =4-d and found to give fair agreement with the results of recently derived series expansions. The same techniques are used to obtain, to 0( epsilon 2 ), a universal amplitude ratio characterizing the shape of the phase boundary in the vicinity of spin-flop bicritical point.
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 .
Renormalisation group recursion relations and matching techniques are used to evaluate to O( epsilon ) the crossover scaling function describing the competition between isotropic short-range and dipolar critical behaviours in the two-point correlation function of a magnetic system above T c in zero field. The scaling function, obtained in closed form, interpolates between the small-q (q xi <<1) and large-q (q xi >>1) limits where asymptotic expressions are written. The results for the effective critical exponents are compared with neutron scattering measurements.
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 Firstly the renormalization group technique is applied to an isotropic Heisenberg spin system with long‐range interaction and a set of critical exponents are calculated. These exponents are expressed in terms of a potential parameter s̀. All critical exponents except β, the magnetization exponent, depend on s̀. Secondly, the anisotropy is considered and the related crossover exponent φ is evaluated. This later parameter turns out to be equal to γ, the susceptibility exponent, as predicted by some authors.
The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.