Data for “Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes”

Zenodo repository containing data and analysis code for the paper ‘Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes’

Abstract:
Motivated by the observation of a learning-induced tricritical point, at which three phases with strong, weak,
and broken Z2 symmetry meet, we revisit the phase diagram of a deformed toric code wavefunction subjected
to weak measurements. This setting is exactly dual to the classical Bayesian inference problem of the 2D
classical Ising model under bond-energy measurements. 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 / Bayes 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 (EA)
correlation function is exactly equal to that of the spin correlation function at the unmeasured Ising critical
point – an observation that is readily supported in our numerical simulations. In addition, we obtain a number of
exact bounds on the power-law exponents of higher measurement-averaged moments of the spin-spin correlation
function. We also show that the scaling dimension for the second moment of the dual spin correlation function
at the higher Nishimori critical point vanishes, and that the scaling dimensions of all moments higher than
second are non-positive. An analogous higher Nishimori critical point exists also for the Bayesian inference
problem of the general D-dimensional classical Ising model when D > 1, which again allows us to obtain
exact results for the EA correlator, and to use its existence to determine the phase diagram of the classical Ising
critical point (or, equivalently, the corresponding conformal quantum critical ‘Rokhsar-Kivelson’ wavefunction)
subjected to bond-energy measurements in general dimension D. Finally, coming back to the two-dimensional
case, we establish – using the tools employed in the proof of a recent c-effective theorem [arXiv:2507.07959]
– that the Casimir effective central charge ceff, a characteristic of the universality class, decreases under the
renormalization group (RG) flow from the higher Nishimori critical point to the unmeasured 2DIsing critical
point, and is thus greater than 1/2. This is corroborated by extensive numerical simulations finding a Casimir
effective central charge ceff = 0.522(1), and a sharp decrease towards ceff = 1/2 as one moves towards the
Ising critical point. The analytical result also explains, with a certain ph