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AIP conference proceedings

Universality of the Critical Correlations

Martin V. Ferer, Dietrich Stauffer, Michael Wortis, Hugh C. Wolfe, C. D. Graham, James Jennings Rhyne

Published Jan 1, 1973• 2 citations

Abstract

We write the singular parts of the free energy and correlation functions in terms of universal quantities, identical for all members of the same symmetry class, and in terms of two system‐dependent scale‐factors. Besides the universality and homogeneity of the functions, we also assume that the free energy in a volume defined by the correlation length is a universal constant. We can then write the critical singularities of properties derived from these functions in terms of universal quantities and the two scale factors. This two scale‐factor universality has important experimental consequences because once any two properties have been measured for a system the two scale‐factors may be determined via comparison with some reference system in the same symmetry class. Then the remaining properties of the system are predicted by our formalism using the two scale‐factors and the known properties of the reference system. Both our extra assumption and the predictions of two‐scale factor universality are exact for the d = 2 Ising model and agree satisfactorily with results for d = 3 systems.

DOI: 10.1063/1.2947031

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We write the singular parts of the free energy and correlation functions in terms of universal quantities, identical for all members of the same symmetry class, and in terms of two system‐dependent scale‐factors. Besides the universality and homogeneity of the functions, we also assume that the free energy in a volume defined by the correlation length is a universal constant. We can then write the critical singularities of properties derived from these functions in terms of universal quantities and the two scale factors. This two scale‐factor universality has important experimental consequences because once any two properties have been measured for a system the two scale‐factors may be determined via comparison with some reference system in the same symmetry class. Then the remaining properties of the system are predicted by our formalism using the two scale‐factors and the known properties of the reference system. Both our extra assumption and the predictions of two‐scale factor universality are exact for the d = 2 Ising model and agree satisfactorily with results for d = 3 systems.

A universality hypothesis relates the amplitude of the singular parts of the coherence length and the specific heat (or surface tension). For the spin-\textonehalf{} Ising model it is exact in two dimensions and numerically accurate to within 1% in three. It is consistent with measurements on the "Ising-like" systems C

{\mathrm{O}}_{2}

, Xe, and $\ensuremath{\beta}$-brass and the "Heisenberg-like" systems RbMn

{\mathrm{F}}_{3}

and EuS to within experimental uncertainties (\ensuremath{\sim} 20%). It provides a sensitive and experimentally convenient indicator of symmetry ("universality") class.

The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close to their critical point. Values of the universal ratios involving correlation length and specific-heat amplitudes are obtained from the $\ensuremath{\epsilon}$ expansion, for Ising, $X\ensuremath{-}Y$, and Heisenberg models. In the latter two cases the correlation function has a power-law behavior at large distances below

{T}_{c}

, and the (transverse) correlation length is defined in terms of the stiffness constant ${\ensuremath{\rho}}_{s}$. Experimental values of the correlation lengths and amplitude ratios are determined for superfluid

^{4}\mathrm{He}

, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet

{\mathrm{RbMnF}}_{3}

. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.

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.

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.

The structure of finite-size scaling within the framework of the field-theoretic renormalization-group approach is examined. New size-dependent ultraviolet divergences are introduced into the theory. It is shown, nonetheless, that ``two-scale-factor'' universality reflects itself in the finite-size free energy and produces a scaling form such as that discussed by Privman and Fisher. A consequence is that the singular part of the finite-system free energy at the bulk critical point is universal. Extensions to surface and interfacial free energies, first proposed from the results of Ising simulations, are supported qualitatively.