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

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.