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Local nearest-neighbor correlation as a transferable critical indicator: From Ising universality to Binder-blind regimes

Qian Zhao

Published Aug 15, 2026• 1 citation
Publisher

Abstract

Critical phenomena are traditionally diagnosed via global order parameters and their cumulants. These methods fail when the order parameter is unknown, undefined, or trivial—as in the Kosterlitz-Thouless transition, magnetization-conserved dynamics, or geometric percolation. We introduce information purity P, a local observable constructed from the absolute nearest-neighbor correlation Δ=⟨|Xi|⟩. The quantity requires only O(Nz) operations per measurement, needs no global magnetization accumulation, and remains well defined in systems with strictly zero global order.Finite-size scaling of equilibrium Monte Carlo data yields dimension-dependent critical values Pc(2D)=0.517±0.007 (2D Ising) and Pc(3D)≈0.36–0.38 (3D Ising), confirming that Pc is not super-universal but serves as a universality-class fingerprint. In the 2D XY model, where the Binder cumulant remains locked at 2/3 across the critical window, P(T) exhibits a measurable slope enhancement near the Kosterlitz-Thouless temperature. For the q-state Potts model, Pc(q) tracks the known critical correlation uc(q). An operational protocol is provided for percolation, a system without a temperature parameter, demonstrating that the local-correlation logic underlying P transfers to non-thermal phase transitions.All calculations employ standard statistical mechanics methods: vectorized Metropolis and Wolff dynamics, binning error analysis, integrated autocorrelation time estimation, and finite-size scaling extrapolation. The results establish P as a computationally inexpensive, locally defined complement to global cumulants, extending the standard critical diagnostics toolkit into regimes where traditional order-parameter-based measures are uninformative or undefined.

DOI
10.5281/zenodo.21943503

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Critical phenomena are traditionally diagnosed via global order parameters and their cumulants. These methods fail when the order parameter is unknown, undefined, or trivial—as in the Kosterlitz-Thouless transition, magnetization-conserved dynamics, or geometric percolation. We introduce information purity P, a local observable constructed from the absolute nearest-neighbor correlation Δ=⟨|Xi|⟩. The quantity requires only O(Nz) operations per measurement, needs no global magnetization accumulation, and remains well defined in systems with strictly zero global order.Finite-size scaling of equilibrium Monte Carlo data yields dimension-dependent critical values Pc(2D)=0.517±0.007 (2D Ising) and Pc(3D)≈0.36–0.38 (3D Ising), confirming that Pc is not super-universal but serves as a universality-class fingerprint. In the 2D XY model, where the Binder cumulant remains locked at 2/3 across the critical window, P(T) exhibits a measurable slope enhancement near the Kosterlitz-Thouless temperature. For the q-state Potts model, Pc(q) tracks the known critical correlation uc(q). An operational protocol is provided for percolation, a system without a temperature parameter, demonstrating that the local-correlation logic underlying P transfers to non-thermal phase transitions.All calculations employ standard statistical mechanics methods: vectorized Metropolis and Wolff dynamics, binning error analysis, integrated autocorrelation time estimation, and finite-size scaling extrapolation. The results establish P as a computationally inexpensive, locally defined complement to global cumulants, extending the standard critical diagnostics toolkit into regimes where traditional order-parameter-based measures are uninformative or undefined.
Critical phenomena are traditionally characterized by global order parameters and their fluctuations, yet local correlations carry information about universality classes that global averages suppress. We introduce a local-order purity P defined via the absolute nearest-neighbor correlation ⟨|σiσj|⟩, distinct from the standard correlation u=⟨σiσj⟩. The quantity P captures non-Gaussian fluctuation effects through the rigorous bound Δ≥|u|. For the 2D and 3D Ising universality classes, P takes universal critical values Pc(2D)≈0.517 and Pc(3D)≈0.36–0.38, serving as class-specific fingerprints. In the 2D XY model, P(T) shows a slope enhancement near the Kosterlitz-Thouless temperature where the Binder cumulant is featureless, providing a low-cost complementary diagnostic. Under Kawasaki dynamics with conserved magnetization, P(t) exhibits depth-dependent four-stage coarsening. The locking time follows τlock∝|Tc−Tq|−α with α≈1.73 across four quench depths. The cumulative information dissipation εin(t) decomposes into linear drift, exponential relaxation, and long-range correlated noise (R2>0.999). The quantity εin correlates strongly with domain-wall activity (r>0.79) but only weakly with thermodynamic entropy production, constituting an operationally defined measure of order-parameter-space roughness. These results establish P as a computationally efficient, locally defined critical indicator applicable to both equilibrium and nonequilibrium statistical mechanics.
Fluctuations are an intrinsic feature of many-body systems, and their full statistical distributions reveal a wealth of information about the underlying physics. Of particular interest are non-Gaussian, extreme-value statistics that arise when nontrivial correlations and criticality dominate over the central limit theorem. Strikingly, in two-dimensional (2D) quantum fluids, such effects have been predicted to manifest in the order parameter distribution in the Berezinskii-Kosterlitz-Thouless (BKT) superfluid phase, which approaches a universal extreme-value form in the low-temperature limit. Here, we measure the order parameter statistics of 2D Bose gases across the BKT critical point using matter-wave interferometry. This allows us to confirm the predicted convergence of the observed statistics to a universal Gumbel distribution at low temperatures, to the 0.1% level of the probability density. Furthermore, the intrinsic precision of the atom interferometer allows the robust extraction of higher-moment observables such as skewness and kurtosis; in particular, we report direct measurements of the Binder cumulant which allows us to precisely identify the onset of the phase transition. Extending this approach to the investigation of non-equilibrium systems, we probe vortex unbinding dynamics following a quench across the BKT critical point and identify parameter-independent scaling behaviour of higher moments.
The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics.Here, we study how the minimum number of variables needed to accurately describe the important features of a data set-the intrinsic dimension (I d )-behaves in the vicinity of phase transitions.We employ state-of-the-art nearest-neighbors-based I d estimators to compute the I d of raw Monte Carlo thermal configurations across different phase transitions: first-order, second-order, and Berezinskii-Kosterlitz-Thouless.For all the considered cases, we find that the I d uniquely characterizes the transition regime.The finite-size analysis of the I d allows us to not only identify critical points with an accuracy comparable to methods that rely on a priori identification of order parameters but also to determine the corresponding (critical) exponent ν in the case of continuous transitions.For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods.Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme and suggest direct parallelism between conventional order parameters in real space and the intrinsic dimension in the data space.
We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.
We study the behaviour of the Binder cumulant on finite square lattices at the Kosterlitz–Thouless phase transition. We determine the fixed-point value of the Binder cumulant and the coefficient of the leading logarithmic correction. These calculations are supplemented with Monte Carlo simulations of the classical XY (plane rotator) model, the Villain model and the dual of the absolute value solid-on-solid model. Using the single-cluster algorithm, we simulate lattices up to L = 4096. For the lattice sizes reached, subleading corrections are needed to fit the data for the Binder cumulant. We demonstrate that the combined analysis of the Binder cumulant and the second moment correlation length over the lattice size allows for an accurate determination of the Kosterlitz–Thouless transition temperature on relatively small lattices. We test the new method on the example of the two-component ϕ 4 model on the lattice.
Quantum magic---the resource that separates universal quantum computation from efficiently simulable Clifford circuits---has emerged as a diagnostic of many-body quantum states, yet the stabilizer Rényi entropy (SRE) that quantifies it remains largely inaccessible in interacting fermion systems. Estimating the global SRE requires a sum over exponentially many Majorana strings, which in determinant quantum Monte Carlo can be sampled without a sign problem only for the interacting density matrix obtained after averaging the auxiliary fields: sampling strings and fields simultaneously incurs a sign problem even for sign-problem-free models, so that the known sign-free alternative is a nested Monte Carlo that does not scale. We propose instead the two-point SRE---a practical stabilizer-Rényi correlator built from rank-2 SREs of one- and two-site reduced density matrices, which are fixed exactly by single-particle Green functions and density correlations already measured in standard simulations---and show that it obeys universal finite-size scaling at fermionic quantum critical points. In the one-dimensional half-filled spinless ttt-VVV chain, the correlator distinguishes algebraic and exponential decay regimes and tracks the inverse-logarithmic finite-size drift characteristic of the Berezinskii--Kosterlitz--Thouless transition. On the honeycomb lattice, sign-problem-free quantum Monte Carlo yields Gross--Neveu--Ising scaling with finite anomalous dimension at zero temperature and two-dimensional Ising collapse at the thermal transition. Our results suggest stabilizer-Rényi microscopy as a spatially resolved, quantitatively universal probe of critical correlations in interacting fermionic matter, on par with conventional order-parameter correlators and accessible to quantum-simulator measurements.