The +/-J Ising spin glass [probabilities p and (1-p) associated with ferromagnetic and antiferromagnetic couplings, respectively] is studied by applying a real-space renormalization-group technique on a hierarchical lattice that approaches the square lattice. Within such a procedure, there is no spin-glass phase and only two finite-temperature phases are found, namely, the paramagnetic and ferromagnetic ones. In spite of a reasonably small computational effort, an accurate paramagnetic-ferromagnetic boundary is presented: the estimate for the slope at p=1 is in very good agreement with the well-known exact result, whereas the coordinates of the Nishimori point are determined within a high precision. Below the Nishimori point, such a boundary is not strictly vertical-contrary to the usual belief-in such a way that a small reentrance is found at low temperatures.
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.
Phase transitions in a modified Nishimori model, including the model considered by Kitatani, on a two-dimensional square lattice are investigated using a tensor-network-based sampling scheme. In this model, generating bond configurations is computationally demanding because of the correlated random interactions. The employed sampling method enables hierarchical and independent sampling of both bonds and spins. This approach allows high-precision calculations for system sizes up to
L=256. The results provide clear numerical evidence that the spin-glass and ferromagnetic transitions are separated on the Nishimori line, supporting the existence of an intermediate Mattis-like spin-glass phase. This finding is consistent with the reentrant transition numerically observed in the two-dimensional Edwards-Anderson (EA) model. Furthermore, critical exponents estimated via finite-size-scaling analysis indicate that the universality class of the transitions differs from that of the standard independent and identically distributed EA model.
The Ising spin glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture that relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Toward this end we precisely determine the global phase diagrams for these six hierarchical spin glasses, using up to $2.5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities
d and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with
d=2 or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.
Motivated by a previous Ising study, we identify a
higher Nishimori line in the learning phase diagram of the
2D q-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.
We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [$tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1$ for the $\ifmmode\pm\else\textpm\fi{}J$ distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori's manifold is expected to control multicriticality for a wider class of models.