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Physical review. B, Condensed matter

Crossover from nonclassical to Ornstein-Zernike behavior for the order-parameter correlation function

Z. Y. Chen

Published Sep 1, 1989• 6 citations
Publisher

Abstract

A theoretical procedure is presented for the temperature- and wave-number-dependent order-parameter correlation function which incorporates singular scaling behavior near the critical point and the analytic Ornstein-Zernike function away from the critical point. The theory is based on an analysis of the renormalization-group equation. Near the critical point, the crossover formalism produces a Wegner correction-to-scaling term for the correlation function.

DOI
10.1103/physrevb.40.4656

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A theoretical procedure is presented for the temperature- and wave-number-dependent order-parameter correlation function which incorporates singular scaling behavior near the critical point and the analytic Ornstein-Zernike function away from the critical point. The theory is based on an analysis of the renormalization-group equation. Near the critical point, the crossover formalism produces a Wegner correction-to-scaling term for the correlation function.
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A previously proposed generalization of the usual Ornstein-Zernike equation for near-critical correlations is examined. It is shown that near the critical point, to leading order, this equation is compatible with realistic critical behavior in both spatial and thermal variables. The scaling predictions of this equation are compared to known results for d=2d=2d=2 and d=3d=3d=3 Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.
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 $\ensuremath{\epsilon}=4\ensuremath{-}d$.
Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general nnn-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate kkk. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.
We discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two-parameter scaling form G(k)=kc(-2)g(kxi,k/kc), where k is the wave vector and xi is the correlation length, and the interaction-dependent nonuniversal momentum scale kc remains finite at the critical fixed point. The correlation function describes the entire critical regime and captures the classical to critical crossover. One-parameter scaling is recovered only in the limit k/kc-->0. We present an approximate calculation of g(x,y) for the Ising universality class using the functional renormalization group.
The singularity of the static correlation function at the critical temperature in a ferromagnetic system is analysed by means of a set of coupled nonlinear integral equations which are obtained from the density field formalism of Sherrington (1966) and its extension to general order by Bhagavan and Lambert (1969). With the assumption of the onset of a phase transition, these equations yield a singular correlation function. Its singularity, which is determined in a selfconsistent way, shows a logarithmic deviation from the Ornstein-Zernike results. The singularity obtained is therefore fundamentally different from that given by the Orstein-Zernike theory.