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Journal of Physics A Mathematical and General

Spin-spin correlation function near the critical point

K Stutzer

Published Dec 1, 1978• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0305-4470/11/12/010

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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.
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.
The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.
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 $n\ensuremath{\rightarrow}\ensuremath{\infty}$.
In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.
The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u 0c , the Feynman graph coupling constant, is discussed.