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arXiv:2512.00649

Scaling of a Mutual-Information Distance in One-dimensional Quantum Spin Chains

Beau Leighton-Trudel

Published Nov 29, 2025• 0 citations
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Abstract

We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance $d_E = K_0/\sqrt{I}$, where $I$ is the mutual information. We define a geometric conversion factor, $G \equiv \partial_r d_E$, which quantifies the local scale. We show that $G$ relates directly to $I$ via $G \propto I^κ$. For systems with power-law correlations $I(r) \sim r^{-X}$, 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 ($X \approx 2$) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of $\log G$ versus $\log I$ at the XX benchmark ($X = 2$) yields $κ\simeq 0$. 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.

DOI
10.48550/arxiv.2512.00649

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We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance dE=K0/Id_E = K_0/\sqrt{I}dE​=K0​/I​, where III is the mutual information. We define a geometric conversion factor, G≡∂rdEG \equiv \partial_r d_EG≡∂r​dE​, which quantifies the local scale. We show that GGG relates directly to III via G∝IκG \propto I^κG∝Iκ. For systems with power-law correlations I(r)∼r−XI(r) \sim r^{-X}I(r)∼r−X, the metric scaling exponent is κ=1/X−1/2κ= 1/X - 1/2κ=1/X−1/2. A key consequence is that the geometric scale GGG is uniform (position-independent) if and only if κ=0κ= 0κ=0, which occurs precisely at X=2X = 2X=2. This identifies X=2X = 2X=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)G(r)G(r) is flat in the bulk at criticality (X≈2X \approx 2X≈2) but varies strongly when gapped; (ii) a coordinate-agnostic slope test of log⁡G\log GlogG versus log⁡I\log IlogI at the XX benchmark (X=2X = 2X=2) yields κ≃0κ\simeq 0κ≃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.
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