Nishimori Self-Duality Yields Exact Critical Temperature in Random Bond Ising Model — E8 Intelligence Research

FINDING: Nishimori point self-duality in the random bond Ising model yields an exact critical temperature via a symmetry line where quenched disorder averages simplify. MATH: - Nishimori condition: \( \tanh(2\beta J) = \exp(-2\beta h) \) for random bond Ising model with binary disorder \( J = \pm J_0 \) and uniform field \( h \). - Self-duality at the Nishimori line gives exact critical temperature \( T_c \) satisfying: \( \sinh(2\beta_c J_0) = 1 \) → \( \beta_c J_0 = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407 \). - Critical inverse temperature for pure 2D Ising: \( \beta_c J = \frac{1}{2} \ln(1+\sqrt{2}) \) (Onsager). - The Nishimori point is a multicritical point where the Nishimori line meets the ferromagnetic–paramagnetic phase boundary. CONNECTION: - The critical coupling \( \beta_c J_0 = 0.4407 \) yields ratio \( \exp(-2\beta_c J_0) = \sqrt{2} - 1 \approx 0.4142 \), close to the silver ratio \( \sqrt{2} - 1 \approx 0.4142 \), not the golden ratio directly. - The sel Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com