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

We introduce a geometric scaling relation that characterizes the local scale behavior of correlations using the informational distance $d_E=K_0\mathrm{/}\sqrt{I}$, where $I$ is the mutual information. We define a geometric conversion factor, $G\equiv {\mathrm{\partial }}_r{d}_E$, which quantifies the local scale. We show that $G$ relates directly to $I$ via $G\propto I^{\kappa }$. For systems with power-law correlations $I(r)\sim r^{-X}$, the metric scaling exponent is $\kappa =1\mathrm{/}X-1\mathrm{/}2$. A key consequence is that the geometric scale $G$ is uniform (position-independent) if and only if $\kappa =0$, 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 $\kappa \simeq 0$. 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.