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Journal of Physics A Mathematical and General

Possible interpretation of the Sykes-Gaunt exponent for the mean cluster size near Tc

Dietrich Stauffer

Published Apr 1, 1977• 6 citations
Publisher

Abstract

Sykes and Gaunt (see ibid., vol.9, p.2131 (1976)) showed the mean cluster size in the two-dimensional Ising model to diverge as (T c -T) - theta with theta approximately=1.91 larger than the Ising susceptibility exponent gamma =1.75. This difference theta - gamma is explained here at least partly by Binder's cluster model (see Ann. Phys. NY. vol.98, p.390 (1976)) for critical points where theta >or= gamma + beta =1.875.

DOI
10.1088/0305-4470/10/4/005

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Rational Fraction Critical Exponents and √2 in the 2D Ising Model Exact Solution — E8 Intelligence Research

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FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Jul 19, 2026•0 citations
Sykes and Gaunt (see ibid., vol.9, p.2131 (1976)) showed the mean cluster size in the two-dimensional Ising model to diverge as (T c -T) - theta with theta approximately=1.91 larger than the Ising susceptibility exponent gamma =1.75. This difference theta - gamma is explained here at least partly by Binder's cluster model (see Ann. Phys. NY. vol.98, p.390 (1976)) for critical points where theta >or= gamma + beta =1.875.
The derivation of series expansions for the mean size of finite clusters in the Ising model is described briefly. From the analysis of low temperature series it is concluded that for a two-dimensional lattice in zero magnetic field the mean size probably diverges at the Ising critical temperature, T c , as (T c -T) - theta , with theta =1.91+or-0.01. It appears therefore that theta < gamma '=1.75 the corresponding Ising susceptibility exponent. For a three-dimensional lattice it is tentatively concluded that the mean size diverges at some temperature T*<T c .
The authors re-examine, using a confluent singularity analysis, existing low-temperature series for the mean size of finite clusters for the pure spin- 1 / 2 nearest-neighbour Ising model in two dimensions. Although the triangular lattice is problematical, results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta =1.87+or-0.04 (consistent with the theoretical prediction theta = gamma + beta ), together with competing analytic and non-analytic ( Delta 1 =1.35+or-0.25) correction-to-scaling terms.
We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field $\ensuremath{\Delta}$. In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters $\ensuremath{\xi}$ is finite for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremath{\approx}1.65$ and divergent for $\ensuremath{\Delta}\ensuremath{\leqslant}{\ensuremath{\Delta}}_{c}$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, $\ensuremath{\xi}(\ensuremath{\Delta})\ensuremath{\sim}{(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c})}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\ensuremath{\approx}2$, is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.
A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.
FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com