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 subjectedto weak measurements. This setting is exactly dual to the classical Bayesian inference problem of the 2Dclassical Ising model under bond-energy measurements. Here we demonstrate that this tricritical point lieson a distinct higher Nishimori line, which has an emergent gauge-invariant formulation, just like the ordinaryNishimori line but with a higher replica symmetry as a replica stat-mech model in the replica number R -> 2limit, where disorder is averaged according to the Born / Bayes rule. As such, the learning tricritical pointis in fact a higher Nishimori critical point. Using this identification, we obtain a number of exact results atthis 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 criticalpoint – an observation that is readily supported in our numerical simulations. In addition, we obtain a number ofexact bounds on the power-law exponents of higher measurement-averaged moments of the spin-spin correlationfunction. We also show that the scaling dimension for the second moment of the dual spin correlation functionat the higher Nishimori critical point vanishes, and that the scaling dimensions of all moments higher thansecond are non-positive. An analogous higher Nishimori critical point exists also for the Bayesian inferenceproblem of the general D-dimensional classical Ising model when D > 1, which again allows us to obtainexact results for the EA correlator, and to use its existence to determine the phase diagram of the classical Isingcritical 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-dimensionalcase, 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 therenormalization group (RG) flow from the higher Nishimori critical point to the unmeasured 2DIsing criticalpoint, and is thus greater than 1/2. This is corroborated by extensive numerical simulations finding a Casimireffective central charge ceff = 0.522(1), and a sharp decrease towards ceff = 1/2 as one moves towards theIsing critical point. The analytical result also explains, with a certain physically motivated assumption, thenumerically observed increase of the Casimir effective central charge under the RG flow from the ordinaryNishimori critical point to the clean Ising critical point in the 2Drandom-bond Ising model. ##### Repository structure ##### ### Code:In order to reproduce the same results, make sure to run Julia version 1.12.5 and instantiate the julia projectsaved in the Project.toml and Manifest.toml. You can do this by navigating to the `code` folder in your terminaland starting Julia with the following command: julia --projectThen switch to the package manager mode by pressing `]` and run the command: instantiateThis should install the same package versions that were used to generate and analyse the data in this repository. The jupyter notebooks `code/fig*.ipynb` contain the code to generate the figures in the paper,including the analysis of the data. ### Data:For all the data we inlcude the averaged data. The raw data of each sample is way too largefor this Zenodo repository. There are 3 kinds of data in this repository: 1. correlation data -- The spin-spin correlation <sigma_i sigma_j>, as well as the EA correlation <sigma_i sigma_j>^2 for an open bc square lattice in the following files: - "data/correlation/betac.jld2" -- along the beta = betac line (up to system size L = 512) - "data/correlation/different_betas.jld2" -- for more values of beta (up to system size L = 256) 2. freeenergy data -- The free energy density of a cylinder with circumference L and length 1000 * L in the following files: - "data/freeenergy/ising_to_tricritical.jld2" -- along the beta = betac line between the Ising point and the higher Nishimori point (up to system size L = 24) - "data/freeenergy/ising_nishimori_tricritical.jld2" -- for the three critical points in the phase diagram (Ising, Nishimori and higher Nishimori point) (up to system size L = 32) 3. arXiv:2504.12385 -- Data from the paper arXiv:2504.12385 in order to draw the phase diagram from this paper. The original data can also be found here (https://zenodo.org/records/15227834).