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Physical review. B, Solid state

Crossover scaling functions and renormalization-group trajectory integrals

David R. Nelson

Published May 1, 1975• 91 citations
Publisher

Abstract

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 $\ensuremath{\epsilon}=4\ensuremath{-}d$.

DOI
10.1103/physrevb.11.3504

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Physical review. B, Solid state•Mar 1, 1976•49 citations
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 \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d.
We express the free energy of a system in its critical region as a line integral along a renormalization group trajectory. The kernal associated with the trajectory integral is just the spin independent part of the Hamiltonian generated by each renormalization group iteration. If the trajectory integral is parametrized in terms of nonlinear scaling fields, a closed expression for the crossover scaling function is obtained. Crossover scaling functions associated with free energy are calculated explicitly for recursion relation models treated by Riedel and Wegner, and the relevance to ε‐expansions near four dimensions is discussed. A breakdown of hyperscaling above the borderline dimensions (d=4 for critical points, and d=3 for tricritical points) arises naturally in the formalism.
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 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.
Recent theoretical work has led to closed form expressions for the thermodynamic functions which characterize complicated multricritcal phenomeana. Renormalization group recursion relations are used to map multicritical systems out of the critical regime, at which point Landau theory with fluctuation corrections can be employed. We review here applications of this approach to tricritical points, spin flop transitions, dipolar systems, and Heisenberg ferromagnets below Tc. Although the calculations have presently been carried out only to first order ε=4−d, they nevertheless give a good qualtiative understanding of multicritical behavior. In particular, they provide a detailed picture of the crossover between competing kinds of critical behavior, and allow treatment of ordered phases with spontaneously broken symmetries.
Renormalization-group recursion relations are used to calculate to order $\ensuremath{\epsilon}=4\ensuremath{-}d$ the free-energy, ordering-susceptibility, and magnetization-crossover scaling functions for tricritical points in isotropic nnn-vector models. The equations of state describe critical and tricritical behavior in both ordered and disordered phases.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.