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Journal of Physics A Mathematical and General

Cutoff corrections to Gaussian-Heisenberg crossover behaviour

Ian D. Lawrie

Published Sep 1, 1977• 9 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/10/9/014

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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 .
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.
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.
The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.
Confluent corrections to scaling are explicitly incorporated in the analysis of high-temperature series for the S= 1 / 2 (quantum-mechanical) and S= infinity (classical) isotropic Heisenberg models on the FCC lattice. For S= infinity , strong confluent corrections are found in the susceptibility, the second moment of the correlation function, as well as the anisotropy crossover function. No evidence for confluent, non-analytic corrections to scaling is found in the analysis of the S= 1 / 2 susceptibility. The best value for the S= infinity susceptibility exponent is gamma ( infinity )=1.42 -0.01 +0.02 , which-taken with the best previous estimate gamma ( 1 / 2 )=1.43-is consistent with universality. However, for S= 1 / 2 , it is felt that (because of apparent non-confluent singularities) gamma is known no better than gamma ( 1 / 2 ) approximately=1.41-1.51. The S= infinity correlation-length exponent is estimated to be nu =0.725+or-0.015, and the crossover exponent is estimated to be phi =1.30+or-0.03. Finally, the S= infinity correction-to-scaling exponent is found to be Delta 1 =0.54+or-0.10.
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.