Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes

Abstract

We revisit a learning-induced tricritical point, at which three phases with strong, weak, and broken $Z_2$ 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 $\textit{higher Nishimori line}$, 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 $R\rightarrow2$ limit, where disorder is averaged according to the Born rule. As such, the learning tricritical point is in fact a $\textit{higher Nishimori critical point}$. Using this identification, we obtain a number of $\textit{exact results}$ at this $\textit{higher}$ 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 $c_{\text{eff}}$ $\textit{decreases}$ under renormalization group (RG) flow from the $\textit{higher}$ 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 $c_{\text{eff}} = 0.522(1)$. 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 $\textit{higher}$ Nishimori criticality in general dimensions $D>1$.