Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality

Abstract: Motivated by a previous Ising study, we identify a ${\it higher}$ Nishimori line in the learning phase diagram of the $2D$ $q$-state Potts model $(2 < q\leq 4)$ under bond-energy measurements. This ${\it higher}$ Nishimori line meets the critical temperature line of the Potts model, in a ${\it higher}$ 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 ${\it decrease}$ 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 ${\it Elitzur's \; theorem}$ that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed $\mathbb{Z}_q$ toric code where the tricritical ${\it higher}$ Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.

• The paper extends higher Nishimori criticality to q-state Potts models with 2 < q ≤ 4, identifying the learning tricritical point as an R → 2 replica-symmetric critical point with exact universal constraints.
• Numerical simulations of the 3-state Potts model locate the ordinary and higher Nishimori points at γ = 0.764(1) and 0.581(1), while measuring an EA exponent of 0.261(1), close to the predicted 4/15.
• The results map directly to measurement thresholds in deformed Z3 toric codes, show stable ordinary Nishimori universality through gauge-symmetry arguments, and identify open questions involving q > 4 and clock-model extensions.

Overview

The paper "Learning Potts Models and $\mathbb{Z}_3$ Toric Codes: Higher and Ordinary Nishimori Criticality" (2608.20268) extends the recently developed framework of higher Nishimori criticality from the two-dimensional Ising model to the $q$-state Potts model with $2 < q \leq 4$. The central objects of study are Bayesian inference (learning) phase diagrams of classical statistical-mechanical models probed by bond-energy measurements, and their exact duals: measurement phase diagrams of deformed $\mathbb{Z}_q$ toric-code wavefunctions under Born-rule measurements of single qubits/qutrits. The paper establishes that the learning tricritical point of the Potts model—where paramagnetic, ferromagnetic, and 'spin-glass' phases meet—is a higher Nishimori critical point governed by an enlarged replica symmetry in the $R \to 2$ limit, and derives a set of exact universal results there.

Learning setup and quantum dual

The classical problem considers a $2D$ Potts model at inverse temperature $\beta$, whose configuration is inferred from noisy bond-energy measurements on every link. For the discrete $q$-state protocol, each measurement outcome $m_{ij} \in \mathbb{Z}_q$ estimates the bond variable $\omega_i \omega_j^{-1}$ correctly with probability $q$0, with $q$1 interpolating between uninformative and projective measurements. The posterior distribution under Bayes' rule defines conditioned expectation values, and measurement records are averaged over to obtain moments such as the Edwards–Anderson (EA) correlator $q$2.

The quantum counterpart is obtained via the Rokhsar–Kivelson state of the Potts Boltzmann weight, which after gauging maps to a deformed $q$3 toric code state with domain-wall line tension $q$4. The discrete protocol corresponds to weak diagonal Kraus measurements on links; tracing out the measurement record yields a generalized dephasing channel. Under this dictionary, the ordinary Nishimori point $q$5 at $q$6 bounds the error-correction threshold of the undeformed code against dephasing noise, while the tricritical point $q$7 governs readout of the critically deformed code.

Replica theory and the higher Nishimori line

Measurement-averaged moments are expressed through a replica field theory taken in the physical limit $q$8, distinct from the quenched-disorder limit $q$9 appropriate to the random-bond Potts model (RBPM). At second order in measurement strength, the replicated Hamiltonian contains a temperature term and an inter-replica coupling $2 < q \leq 4$0. On the line

$2 < q \leq 4$1

a gauging transformation introducing an auxiliary $2 < q \leq 4$2-th replica renders the Hamiltonian invariant under local $2 < q \leq 4$3 gauge transformations acting on all replicas simultaneously, with permutation symmetry over all $2 < q \leq 4$4 copies. In the limit $2 < q \leq 4$5 this is the spin-glass problem; $2 < q \leq 4$6 gives the gauge-invariant ordinary Nishimori line of the RBPM; and $2 < q \leq 4$7 gives the gauge-invariant formulation of the higher Nishimori line in the learning problem ($2 < q \leq 4$8). This structure mirrors the Ising case but is new for Potts.

For a specialized Gaussian measurement protocol with continuous complex-valued outcomes, the truncated replica Hamiltonian holds exactly (no $2 < q \leq 4$9 corrections), so the higher Nishimori line exists microscopically. The associated Nishimori identity implies that along $\mathbb{Z}_q$0,

$\mathbb{Z}_q$1

where the last quantity is the unmeasured correlation function. Consequently, tuning to the critical temperature $\mathbb{Z}_q$2, the intersection point

$\mathbb{Z}_q$3

must lie at the meeting point of the ferromagnet–paramagnet and paramagnet–'spin-glass' boundaries, i.e., it is a genuine tricritical higher Nishimori point. The same equality forces the EA correlator at $\mathbb{Z}_q$4 to decay with the exponent of the unmeasured Potts spin correlator—for $\mathbb{Z}_q$5, $\mathbb{Z}_q$6.

Numerical results for the discrete protocol

Using a hybrid Monte-Carlo/tensor-network scheme on lattices up to $\mathbb{Z}_q$7, averaging over roughly $\mathbb{Z}_q$8 measurement records per point, the authors map the learning phase diagram of the $\mathbb{Z}_q$9-state model under the natural discrete protocol, for which no exact higher Nishimori line exists at the microscopic level. The key findings are:

Quantity	Value
Ordinary Nishimori point $R \to 2$0	$R \to 2$1
Higher (tricritical) Nishimori point $R \to 2$2	$R \to 2$3
EA exponent $R \to 2$4 at $R \to 2$5	$R \to 2$6
Prediction (higher Nishimori condition)	$R \to 2$7

The measured EA exponent at the tricritical point agrees with the analytic prediction within the finite-size systematic scale calibrated at the clean critical point (where the same-size deviation is $R \to 2$8). This constitutes the central numerical evidence that the enlarged $R \to 2$9 replica symmetry, explicitly broken by the discrete protocol's higher-order terms, emerges in the infrared—an emergent higher Nishimori line is also tracked inside the paramagnetic phase by matching the EA correlation length to the unmeasured spin-spin correlation length.

Two further numerical observations concern crossover physics. Along the $2D$0 line, the EA exponent drifts continuously toward the three-loop $2D$1-expansion value $2D$2 expected at the attractive fixed point $2D$3, but no plateau resolves because the crossover length $2D$4 reaches only $2D$5 at accessible sizes—the flow remains preasymptotic. Along the paramagnet–'spin-glass' boundary connecting $2D$6 to $2D$7, the extracted correlation-length exponent rises from the ordinary Nishimori value $2D$8 to $2D$9 near the tricritical point, consistent with an RG flow out of the unstable higher Nishimori fixed point into the stable ordinary one. The authors note honestly that the numerical estimate of $\beta$0 sits slightly below the truncated-replica-theory estimate ($\beta$1), and attribute part of the emergent-line shift to overestimation of inverse correlation lengths.

Exact results and stability arguments

Beyond the EA exponent, the higher Nishimori identity fixes the modulus-squared average of any correlation function at $\beta$2 to equal its unmeasured counterpart, whose scaling is known exactly from rational CFT. Rigorous bounds follow for higher moments: $\beta$3 satisfies $\beta$4. It is notable that these exact statements apply at a strongly disordered, frustrated multicritical point.

The paper also supplies a general stability argument for the ordinary Nishimori universality class in monitored systems. On the infinite-temperature line, the $\beta$5 replica theory possesses a local ('gauge') symmetry under simultaneous flips of all replicas at a site. A thermal perturbation violates this local symmetry but respects only global $\beta$6; by Elitzur's theorem, most perturbation-induced terms vanish identically, and the remainder merely shifts the nonuniversal transition location. The thermal direction is therefore trivially RG irrelevant—a variation of the Wegner/Fradkin–Shenker argument for pure-gauge transitions—which explains why ordinary Nishimori transitions form stable universality classes across monitored problems, including wavefunction- and Hamiltonian-deformed toric codes.

Casimir effective central charges

Applying the $\beta$7-effective theorem and its extensions non-perturbatively, the Casimir effective central charge monotonically decreases along measurement-induced RG flows: from the clean Potts critical point ($\beta$8 for $\beta$9) down to the attractive fixed point $q$0, and from $q$1 down to $q$2. The three-loop $q$3-expansion gives $q$4, yielding $q$5 for $q$6. Combined with a physically motivated assumption about the replica central charge, the argument implies the inequalities reverse in the $q$7 random-bond problem, where $q$8 increases along flows to $q$9—consistent with existing numerical results. The resulting organization of non-unitary daughter theories around unitary parents ($m_{ij} \in \mathbb{Z}_q$0 Ising, $m_{ij} \in \mathbb{Z}_q$1 Potts) exhibits a reproducible ordering: ordinary Nishimori, percolation, clean, higher Nishimori. The placement of $m_{ij} \in \mathbb{Z}_q$2 above the clean Potts point is explicitly flagged as speculative rather than theorem-guaranteed.

Limitations and open questions

Several caveats are stated directly. First, for the physically natural discrete protocol, the higher Nishimori line is emergent only, its existence resting on the RG irrelevance of $m_{ij} \in \mathbb{Z}_q$3 terms at the tricritical point—an assumption supported numerically but not proven analytically. Second, the attractive fixed point $m_{ij} \in \mathbb{Z}_q$4 is not resolved as a plateau within accessible system sizes owing to long crossovers, leaving its universal data unverified beyond the $m_{ij} \in \mathbb{Z}_q$5-expansion. Third, the case $m_{ij} \in \mathbb{Z}_q$6 remains open: the clean first-order transition persists in the learning problem (measurement-averaged first moments inherit the ordered unmeasured behavior), in sharp contrast with the Aizenman–Wehr rounding in the RBPM, and the corresponding large-$m_{ij} \in \mathbb{Z}_q$7 learning phase diagram has not been mapped. Finally, the extension to $m_{ij} \in \mathbb{Z}_q$8 clock models with $m_{ij} \in \mathbb{Z}_q$9, where the higher Nishimori line should terminate in two multicritical points bracketing a critical segment with continuously varying exponents, and the identification of the far-left analogue of the weak self-dual toric-code criticality in the Potts fine structure, are left to future work.

Conclusion

This work demonstrates that higher Nishimori criticality, previously established for the Ising learning problem, extends to the $\omega_i \omega_j^{-1}$0-state Potts model with $\omega_i \omega_j^{-1}$1: the learning tricritical point is a higher Nishimori point characterized by an emergent $\omega_i \omega_j^{-1}$2 replica symmetry, with exactly computable universal exponents confirmed numerically under a generic discrete measurement protocol. Together with the Elitzur-theorem stability argument for ordinary Nishimori transitions and the monotonicity properties of Casimir effective central charges, the results organize the disorder-induced daughter theories of both learning and random-bond Potts models into a coherent fine-structured hierarchy, and translate directly into thresholds for reading out deformed $\omega_i \omega_j^{-1}$3 toric-code memories.