2025/11/29 by Leighton-Trudel, Beau
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2512.00649
We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance dE = K0/√(I), where I is the mutual information. We define a geometric conversion factor, G ≡ ∂r dE, which quantifies the local scale. We show that G relates directly to I via G ∝ Iκ. For systems with power-law correlations I(r) ∼ r-X, the metric scaling exponent is κ= 1/X - 1/2. A key consequence is that the geometric scale G is uniform (position-independent) if and only if κ= 0, which occurs precisely at X = 2. This identifies X = 2 as the unique condition for a uniform and metric informational distance. We validate this relation using DMRG simulations of the 1D XXZ chain and exact results for the XX model. We demonstrate two falsifiable diagnostics: (i) G(r) is flat in the bulk at criticality (X ≈ 2) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of log G versus log I at the XX benchmark (X = 2) yields κ≃ 0. This approach provides a coordinate-independent method for identifying scaling regimes that helps to reduce ambiguity from non-universal amplitudes and from the fitting choices in standard power-law analyses, and defines a simple post-processing pipeline that can be applied directly to numerical or experimental mutual-information data.