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Journal of Physics A Mathematical and General

Correlation functions near the critical point

Franz Wegner

Published May 1, 1975• 24 citations
Publisher

Abstract

Using renormalization group arguments the author expands n-point correlation functions (for non-exceptional wavevectors) in expectation values of translational invariant short-range operators O i . He uses the fact that the Fourier components of our operators become negligible for wavevectors q large in comparison to the momentum cut-off. The correlation functions show the same non-analyticities at the critical point as the expectation values (O i ). The expansion coefficients are regular in the thermodynamic variables for q not=0. They can be expressed in terms of (a) functions which become singular at q=0 and yield the scaling behaviour, and (b) functions which are regular at q=0. The expansion coefficients of the two-point correlation function are sums of both types of functions.

DOI
10.1088/0305-4470/8/5/007

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Using renormalization group arguments the author expands n-point correlation functions (for non-exceptional wavevectors) in expectation values of translational invariant short-range operators O i . He uses the fact that the Fourier components of our operators become negligible for wavevectors q large in comparison to the momentum cut-off. The correlation functions show the same non-analyticities at the critical point as the expectation values (O i ). The expansion coefficients are regular in the thermodynamic variables for q not=0. They can be expressed in terms of (a) functions which become singular at q=0 and yield the scaling behaviour, and (b) functions which are regular at q=0. The expansion coefficients of the two-point correlation function are sums of both types of functions.
A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below Tc{T}_{c}Tc​ than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.
Abstract In Chapter 15, the scaling behaviour of correlation functions at criticality, T = Tc, has been derived. This chapter is devoted to the critical domain, where the correlation length is large with respect to the microscopic scale, but finite. In dimensions d < 4, above Tc, the property of strong scaling is derived: in the critical domain above Tc, all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. However, because the correlation length is singular at Tc, in this formalism, the critical temperature cannot be crossed. Alternatively, one can expand correlation functions in formal power series of the deviation (T −Tc) from the critical temperature, in presence of a magnetic field. The sum of the expansion satisfies renormalization group (RG) equations also valid for T < Tc and in a magnetic field, from which follow scaling properties in the whole critical domain. The universal two-point function can be expanded when T approaches Tc, using the short-distance expansion (SDE). A few terms of the ϵ expansion (ϵ is the deviation from dimension 4) of a few universal quantities are reported. Calculations at fixed dimension and summation of perturbative expansions are described. The conformal bootstrap based on the SDE and conformal invariance at the infrared (IR) fixed point provides an alternative method to determine critical exponents.
Starting from Wilson's differential renormalisation group equation with smooth momentum cut-off the author calculates the spin-spin correlation (S q S -q ) near the critical point for the one-component vector model in lowest non-trivial order in epsilon . It is shown that it differs only by a short-range term and a trivial factor from the self-correlation of the wavevector-dependent eigen operator O h (q,g). This self-correlation is calculated following the procedure given by Wegner (1975) and expanded in expectation value of translationally invariant eigenoperators. Comparisons with the results of various authors are made.
Chapter 25 established the scaling behaviour of correlation functions at criticality, T = Tc . This chapter studies the critical domain which is defined by the property that the correlation length is large with respect to the microscopic scale, but finite. Using results proven in Chapter 10, it first demonstrates strong scaling above Tc : in the critical domain above Tc , all correlation functions, after rescaling, can be expressed in terms of universal correlation functions, in which the scale of distance is provided by the correlation length. The first part of the chapter is restricted to Ising-like systems. It then generalizes the results to N-component order parameters in Section 26.6. In Section 26.7, shows how to expand the universal two-point function when T approaches Tc , using the short distance expansion. Finally, the appendix discusses the energy correlation function when the specific heat exponent vanishes.
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.