Computable information density (CID), the ratio of the length of a losslessly compressed data file to that of the uncompressed file, is a measure of order and correlation in both equilibrium and nonequilibrium systems. Here we show that correlation lengths can be obtained by decimation, thinning a configuration by sampling data at increasing intervals and recalculating the CID. When the sampling interval is larger than the system's correlation length, the data becomes incompressible. The correlation length and its critical exponents are thus accessible with no a priori knowledge of an order parameter or even the nature of the ordering. The correlation length measured in this way agrees well with that computed from the decay of two-point correlation functions g_{2}(r) when they exist. But the CID reveals the correlation length and its scaling even when g_{2}(r) has no structure, as we demonstrate by "cloaking" the data with a Rudin-Shapiro sequence.
Computable information density (CID), the ratio of the length of a losslessly compressed data file to that of the uncompressed file, is a measure of order and correlation in both equilibrium and nonequilibrium systems. Here we show that correlation lengths can be obtained by decimation, thinning a configuration by sampling data at increasing intervals and recalculating the CID. When the sampling interval is larger than the system's correlation length, the data becomes incompressible. The correlation length and its critical exponents are thus accessible with no a priori knowledge of an order parameter or even the nature of the ordering. The correlation length measured in this way agrees well with that computed from the decay of two-point correlation functions g_{2}(r) when they exist. But the CID reveals the correlation length and its scaling even when g_{2}(r) has no structure, as we demonstrate by "cloaking" the data with a Rudin-Shapiro sequence.
Wu et al. (arXiv:2610.35790) introduced entropy imaging for quantum matter: a microscopy image is serialized along a Hilbert curve, compressed with a lossless algorithm, and the computable information density (CID) is used as a measure of spatial complexity. Applied to microwave impedance microscopy of graphene, CID peaks near quantum Hall transitions, tracks fractional transitions, and resolves fractional puddles in a local map, while a trivial band-edge crossover does not show the same enhancement. This preprint offers an information-dynamics reading of that result. In this reading, a microscopy image is a projection, and CID is treated as an indicator of the arrangement information in the projection. The reading changes no number in Wu et al. No new data are reported. All numbers are Wu et al.'s as reported. Three falsifiable predictions are stated, testing the CID diagnostic.
Phase transitions can generate intricate spatial organization invisible to spatially averaged probes, yet quantitative measures of this organization remain limited. Here we develop "entropy imaging," an information-theoretic technique that harnesses lossless compression to quantify spatial complexity, predictability, and order in quantum materials. Using this approach, emergent spatial textures can be imaged and converted into an experimentally-accessible observable: the computable information density (CID), a compression-based measure of information complexity that approaches the Shannon entropy rate in the large-system limit. Applied to millikelvin microwave impedance microscopy of graphene, the method reveals a pronounced maximum in the CID near quantum Hall transitions, a quantitative measure of the information-rich spatial organization of the critical landscape. A disordered Hofstadter model reproduces this behavior and shows that the CID can identify the quantum critical point from a sequence of images, without ensemble averaging over disorder realizations. The same framework tracks fractional quantum Hall transitions and resolves isolated fractional puddles that coexist with a disordered background. By contrast, the trivial band-edge crossover in a MoSe2/WS2 moiré heterobilayer exhibits qualitatively different behavior, suggesting that the pronounced CID enhancement observed at the quantum Hall critical point is not simply a generic feature of a conductive-insulating crossover. We further show that the geometric information encoded in CID is not reducible to a correlation length, and it can even distinguish spatial textures with identical multifractal statistics. Together, these results establish lossless compression as a framework for quantifying emergent spatial organization in quantum materials, with portability across imaging modalities and physical systems.
The finite-size-scaling method in the equilibrium Monte Carlo (MC) simulations and the finite-time-scaling method in the nonequilibrium relaxation simulations are compromised. MC time data of various physical quantities are scaled by the MC time data of the dynamic correlation length. It corresponds to changing the system size in the finite-size-scaling method. This scaling method is tested in the three-dimensional ferromagnetic Ising model and in the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Ising spin-glass model. The transition temperature and the critical exponents are obtained. We also comment on the definition of the dynamic correlation length in the nonequilibrium relaxation process. The Ornstein-Zernike formula is not always appropriate.
Considering an example of the long-range Kitaev model, we are looking for a correlation length in a model with long range interactions whose correlation functions away from a critical point have power-law tails instead of the usual exponential decay. It turns out that quasiparticle spectrum depends on a distance from the critical point in a way that allows to identify the standard correlation length exponent, ν. The exponent implicitly defines a correlation length ξ that diverges when the critical point is approached. We show that the correlation length manifests itself also in the correlation function but not in its exponential tail because there is none. Instead ξ is a distance that marks a crossover between two different algebraic decays with different exponents. At distances shorter than ξ the correlator decays with the same power law as at the critical point while at distances longer than ξ it decays faster, with a steeper power law. For this correlator it is possible to formulate the usual scaling hypothesis with ξ playing the role of the scaling distance. The correlation length also leaves its mark on the subleading anomalous fermionic correlator but, interestingly, there is a regime of long range interactions where its short distance critical power-law decay is steeper than its long distance power law tail.
The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.
In this work, we study the neural scaling laws of RydbergGPT, an autoregressive transformer model trained on qubit projective measurement data gathered from interacting Rydberg atom arrays. The quantum system is known to exhibit a finite-size remnant of a critical point as the laser detuning parameter is varied. We find that near the critical point the transformer loss as a function of training dataset size is well described by a power-law with a loss floor correction. However, away from criticality the quality of the power-law description is substantially reduced. We then compare the statistical structure of both Rydberg measurements and natural-language corpora using an entropy-normalised, finite sample corrected mutual information "two-point" function. We find that near-critical statistics of the two point functions are closest to those observed in natural-language, whilst other qubit configurations far from the critical point have two-point functions that decay more rapidly. This supports the hypothesis that multi-scale dependence contributes to stable neural scaling, and that scaling behaviour should be viewed as a property of the model-data pair.