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

Andrew Stewart Caldin

Published Jul 24, 2026• 0 citations
Publisher

Abstract

FINDING: Nishimori point self-duality in random-bond Ising model yields exact critical temperature via gauge symmetry, not numerical approximation. | MATH: The Nishimori line condition: \( \exp(-2\beta J) = p/(1-p) \) where \( p \) is bond probability (ferromagnetic), \( J \) coupling strength, \( \beta = 1/k_B T \). Self-duality at Nishimori point gives exact critical temperature \( T_c \) satisfying \( \sinh(2\beta_c J) = 1 \) for square lattice, identical to pure 2D Ising criticality. For correlated disorder, mapping to pure Ising model preserves critical exponents. | CONNECTION: The self-duality condition \( \sinh(2\beta_c J) = 1 \) yields \( \beta_c J = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407 \). The critical temperature ratio \( k_B T_c / J = 2/\ln(1+\sqrt{2}) \approx 2.269 \). No direct golden ratio (0.618, 1.618) appears, but the self-dual point sits at a fixed point of the Kramers-Wannier duality transformation, which maps high- and low-temperature expansions via \( \tanh(\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.21524604

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FINDING: Nishimori point self-duality in random-bond Ising model yields exact critical temperature via gauge symmetry, not numerical approximation. | MATH: The Nishimori line condition: ( \exp(-2\beta J) = p/(1-p) ) where ( p ) is bond probability (ferromagnetic), ( J ) coupling strength, ( \beta = 1/k_B T ). Self-duality at Nishimori point gives exact critical temperature ( T_c ) satisfying ( \sinh(2\beta_c J) = 1 ) for square lattice, identical to pure 2D Ising criticality. For correlated disorder, mapping to pure Ising model preserves critical exponents. | CONNECTION: The self-duality condition ( \sinh(2\beta_c J) = 1 ) yields ( \beta_c J = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407 ). The critical temperature ratio ( k_B T_c / J = 2/\ln(1+\sqrt{2}) \approx 2.269 ). No direct golden ratio (0.618, 1.618) appears, but the self-dual point sits at a fixed point of the Kramers-Wannier duality transformation, which maps high- and low-temperature expansions via ( \tanh(\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.
We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T *=0.9527(1), p *=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T *, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T *. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν−η=d−2=0.
We investigate the critical behavior of the random-bond $\ifmmode\pm\else\textpm\fi{}J$ Ising model on a square lattice at the multicritical Nishimori point in the $T\text{\ensuremath{-}}p$ phase diagram, where TTT is the temperature and ppp is the disorder parameter (p=1p=1p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by $2p\ensuremath{-}1=\text{tanh}(1/T)$, along which the multicritical point lies. The multicritical Nishimori point is located at ${p}^{\ensuremath{\ast}}=0.890\text{ }81(7)$, ${T}^{\ensuremath{\ast}}=0.9528(4)$, and the renormalization-group dimensions of the operators that control the multicritical behavior are y1=0.655(15){y}_{1}=0.655(15)y1​=0.655(15) and y2=0.250(2){y}_{2}=0.250(2)y2​=0.250(2); they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.
The two-dimensional q-state Potts model is subjected to a Z(q) symmetric disorder that allows for the existence of a Nishimori line. At q=2, this model coincides with the +/- J random-bond Ising model. For q>2, apart from the usual pure- and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q=3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.
We reexamine the disorder-dominated multicritical point of the two-dimensional $\ifmmode\pm\else\textpm\fi{}J$-Ising model, known as the Nishimori point (NP). At the NP we investigate numerically and analytically the behavior of the disorder correlator, familiar from the self-dual description of the pure critical point of the two-dimensional Ising model. We consider the logarithmic average and the qthq\mathrm{th}qth moments of this correlator in the ensemble average over randomness, for continuous q in the range 0<q<2.5,0<q<2.5,0<q<2.5, and demonstrate their conformal invariance. At the NP we find, in contrast to the self-dual pure critical point, that the disorder correlators exhibit multiscaling in q which is different from that of spin-spin correlators and that their scaling dimension becomes negative for q>1q>1q>1 and q<0.q<0.q<0. Using properties on the Nishimori line we show that the first moment (q=1)(q=1)(q=1) of the disorder correlator is exactly one for all separations. The spectrum of scaling dimensions at the NP is not parabolic in q.
Abstract The Nishimori point of the random bond Ising model is a prototype of renormalization group fixed points with strong disorder. We show that the exact correlation length and crossover critical exponents at this point can be identified in two and three spatial dimensions starting from properties of the Nishimori line. These are the first exact exponents for frustrated random magnets, a circumstance to be also contrasted with the fact that the exact exponents of the Ising model without disorder are not known in three dimensions. Our considerations extend to higher dimensions and models other than Ising.