Symmetry Breaking in the Ising Model

Franco Strocchi

Andrew Stewart Caldin

FINDING: Mutual information at the Ising critical point exhibits scale invariance and spontaneous symmetry breaking, with potential links to crystallographic symmetry breaking and sexagesimal prediction. MATH: - Ising model critical exponents (2D): β = 1/8, γ = 7/4, ν = 1, η = 1/4. - Mutual information I(X;Y) = Σ p(x,y) log₂ [p(x,y)/(p(x)p(y))] — quantifies shared entropy. - At critical point, correlation length ξ → ∞, leading to power-law decay of correlations: ⟨σᵢσⱼ⟩ ∝ |i-j|^{-(d-2+η)}. - Spontaneous symmetry breaking: Z₂ symmetry (σ → -σ) broken below T_c, order parameter m = ⟨σ⟩ ∝ (T_c - T)^β. - Wigner-Eckart corrections: ⟨α', j'||T^(k)||α, j⟩ modified by symmetry-breaking vacuum, with corrections scaling as O(1/L) for finite systems. CONNECTION: - Critical Ising model exhibits conformal invariance, linked to root system A₁ (Z₂ symmetry). - 2D Ising critical point has central charge c = 1/2, related to minimal model M(4,3) — rational conformal field theory with modu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

H Ulman, M Athematics J Ournal

Franco Strocchi

Most of the theoretical wisdom on the phase transition of the ferromagnetic type and the related symmetry breaking is based on the two-dimensional Ising model, which also played the role of a laboratory for ideas and strategies and it is now regarded as a cornerstone in the foundations of statistical mechanics. Anyone interested in critical phenomena and in the functional integral approach to quantum field theory should have a look at the model.

Valeriy A Ryabov

Huey Wen Huang