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.
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.
We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.
We revisit a learning-induced tricritical point, at which three phases with strong, weak, and broken Z2 symmetry meet, in the phase diagram of a deformed toric code wavefunction subjected to weak measurements. This setting is exactly dual to a classical Bayesian inference phase diagram of the 2D classical Ising model. Here we demonstrate that this tricritical point lies on a distinct higher Nishimori line, which has an emergent gauge-invariant formulation, just like the ordinary Nishimori line but with a higher replica symmetry as a replica stat-mech model in the replica number R→2 limit, where disorder is averaged according to the Born rule. As such, the learning tricritical point is in fact a higher Nishimori critical point. Using this identification, we obtain a number of exact results at this higher Nishimori critical point; e.g., we show that the power-law exponent of the Edwards-Anderson correlation function is exactly equal to that of the spin correlation function at the unmeasured Ising critical point and verify this in numerical simulations. Using the tools of the proof of a c-effective theorem [arXiv:2507.07959], we show that the Casimir effective central charge ceffdecreases under renormalization group (RG) flow from the higher Nishimori critical point to the unmeasured 2D Ising critical point, and is thus greater than 1/2. This is corroborated by extensive numerical simulations finding ceff=0.522(1). The analytical result also explains, with a physically motivated assumption, the numerically observed increase of the Casimir effective central charge under the RG flow from the ordinary Nishimori critical point to the clean Ising critical point in the random-bond Ising model. We also discuss higher Nishimori criticality in general dimensions D>1.
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(−2βJ)=p/(1−p) where p is bond probability (ferromagnetic), J coupling strength, β=1/kBT. Self-duality at Nishimori point gives exact critical temperature Tc satisfying sinh(2βcJ)=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(2βcJ)=1 yields βcJ=21ln(1+2)≈0.4407. The critical temperature ratio kBTc/J=2/ln(1+2)≈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
We numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape of the effective central charge obtained by the transfer matrix method based on the Zamolodchikov's C-theorem. We find a line of random fixed points (FPs) for nc<n<1, with nc∼0.5, for which the central charge and critical exponents agree well with the results of the 1−n perturbative expansion. Furthermore, for n>nc, we find a line of multicritical FPs at strong randomness. The FP at n=1 has c=0.4612(4), which suggests that it belongs to the universality class of the Nishimori point in the ±J random-bond Ising model. For n>2, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP.
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.
Learning Potts Models and Z3 Toric Codes: Higher and Ordinary Nishimori Criticality
Motivated by a previous Ising study, we identify a higher Nishimori line in the learning phase diagram of the 2Dq-state Potts model (2<q≤4) under bond-energy measurements. This higher Nishimori line meets the critical temperature line of the Potts model, in a higher Nishimori critical point -- a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete q-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic decrease along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on Elitzur′stheorem that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed Zq toric code where the tricritical higher Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.