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Physical Review Letters

One—Scale-Factor Universality for Critical Phenomena

Donald D. Betts, Douglas S. Ritchie

Published Mar 31, 1975• 3 citations
Publisher

Abstract

An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of ${V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}$, where ${q}_{X}$ is the coordination number, ${{K}_{C}}^{X}$ is the dimensionless inverse critical temperature, and ${n}_{X}$, ${g}_{X}$, and ${l}_{X}$ are the universality scale factors. ${V}_{X}$ is an exact invariant for the two-dimensional $S=\frac{1}{2}$ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, $\frac{{V}_{X}}{{l}_{X}}$ is an exact invariant.

DOI
10.1103/physrevlett.34.788

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An extended universality hypothesis is proposed for critical phenomena in simple systems. The hypothesis implies the invariance of VX=nXqXKCXgXlX{V}_{X}=\frac{{n}_{X}{q}_{X}{{K}_{C}}^{X}}{{g}_{X}{l}_{X}}VX​=gX​lX​nX​qX​KC​X​, where qX{q}_{X}qX​ is the coordination number, KCX{{K}_{C}}^{X}KC​X is the dimensionless inverse critical temperature, and nX{n}_{X}nX​, gX{g}_{X}gX​, and lX{l}_{X}lX​ are the universality scale factors. VX{V}_{X}VX​ is an exact invariant for the two-dimensional S=12S=\frac{1}{2}S=21​ Ising model and invariant to high precision for several three-dimensional models. For the spherical model, however, VXlX\frac{{V}_{X}}{{l}_{X}}lX​VX​​ is an exact invariant.
An extension of the critical-phenomena universality hypothesis assumed previously to be valid for less singular terms is shown to be invalid. A reformulated extension of the universality hypothesis is found to be valid exactly for the spin-1/2 Ising model on the triangular, square, and honeycomb lattices, and, further support is supplied by high- and low-temperature expansions for the susceptibility. The extension fails however for the Ising model on the kagome\r{} lattice and for the spherical model. At present there are insufficient data to test the extended principle for other models or for experimental systems.
This paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on a broad class of complex phenomena. We shall see that while scale invariance has been tested for many years, universality is relatively more rarely discussed. In particular, we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena, and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems.
New universality classes in two-dimensional critical phenomena have recently been discovered using methods of quantum field theory and conformal invariance. These can equally well describe classical two-dimensional systems at their critical temperature or one-dimensional gapless quantum systems at (or near) zero temperature. (The two types of systems are related by the transfer matrix or Feynman path integral.) It is argued that isotropic one-dimensional antiferromagnets are realizations of these new universality classes. Exact predictions for the low temperature specific heat, susceptibility, correlation length, and other critical properties are made. These agree exactly with exactly solveable models and well with experimental results on CuCl2⋅2(NC5H5).
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 CO2{\mathrm{O}}_{2}O2​, Xe, and $\ensuremath{\beta}$-brass and the "Heisenberg-like" systems RbMnF3{\mathrm{F}}_{3}F3​ 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 Tc{T}_{c}Tc​, 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 4He^{4}\mathrm{He}4He, which is $X\ensuremath{-}Y$-like, and for the Heisenberg antiferromagnet RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​. Comparisons are made between the values of the amplitude ratios coming from $\ensuremath{\epsilon}$ expansions, series, and experiments.