A preprint reports evidence for a second Nishimori critical point in a three-state Potts model. At this point, the higher Nishimori line meets the Potts critical line, bringing paramagnetic, ferromagnetic and spin-glass phases together.
In the paper’s terminology, a Nishimori line is a special relationship between the model’s temperature setting and the strength of its measurements. For a specially designed Gaussian protocol, that relationship is exact: the higher line is β = Δ, while the ordinary line is at β = 0.
From exact equations to a numerical test
The exact result is specific to a specialized Gaussian bond-energy protocol. Under that protocol, a replica calculation says that the first and second ways of averaging Potts-spin correlations over measurement records have the same critical scaling.
The researchers then tested a discrete three-state protocol numerically. Finite-size scaling of coherent information placed the higher point at γ = 0.581(1) and the ordinary point at γ = 0.764(1). The phase-diagram analysis used linear sizes from 8 to 128; direct EA-correlator measurements used 256 × 256 systems and averaged about 10^5 measurement realizations.
What the numbers suggest
At the learning tricritical point, the fitted Edwards–Anderson (EA) exponent, which tracks the scaling of spin-glass correlations, was 0.261(1). That was consistent with the predicted 4/15 after the reported finite-size calibration and provides the paper’s central numerical evidence for an emergent higher Nishimori line in the discrete protocol.
Along the boundary between the paramagnetic and spin-glass phases, the effective exponent was near 1.45 away from the tricritical region and rose toward about 2.5 near the higher point. The authors interpret that change as a crossover from higher to ordinary Nishimori criticality.
The proposed picture includes an intermediate L(1) fixed point. A three-loop epsilon expansion predicts an EA exponent of about 0.315 there, but the available simulations did not resolve a plateau at that value.
The paper also tracks an effective central charge, a quantity used to compare critical flows. It decreases along measurement-induced R → 1 flows from the clean Potts point to L(1), and from the higher point to L(1); the corresponding R → 0 random-bond flow is argued to increase.
Beyond the phase diagram
The authors’ Elitzur-theorem argument says that moving slightly away from β = 0 leaves the ordinary transition in the same universality class—the same large-scale pattern of critical behavior—as the ordinary point at β = 0.
The paper also translates the result into a toric-code threshold comparison, a theoretical setting relevant to quantum error correction. The measurement threshold γcmeas = 0.764(1) corresponds to a measurement-error threshold pmeas_c = 0.2709(8). The paper says this measurement threshold upper-bounds, rather than necessarily equals, the dephasing-noise threshold.
Why the finding remains provisional
The exact higher line belongs to the specially designed Gaussian protocol; for the discrete protocol, the nearby line is an emergent numerical result rather than an exact microscopic condition. The detailed numerical confirmation is concentrated on q = 3, and no L(1) exponent plateau was resolved at the available system sizes.
Paper data and sources
Original title: Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality
Authors: Rushikesh A. Patil, Malte Pütz, Rohit Mukherjee et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text