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

TL;DR

The paper addresses robust identification of scaling exponents from mutual information in 1D quantum spin chains by introducing the geometric scaling relation (GSR), which links a local scale factor $G = {\partial_r d_E}$ to $I$ via $G \propto I^{\kappa}$ with $\kappa = 1/X - 1/2$. For power-law correlations $I(r) \sim r^{-X}$, the framework shows that $G$ is position-independent only at the benchmark $X=2$, indicating a unique uniform geometric scale. Analytical derivation and numerical validation on the XXZ and XX models demonstrate that a coordinate-agnostic slope test, given by the linear relation $\log G = \kappa \log I + \log A$, yields $\kappa=0$ at $X=2$ and nonzero slopes otherwise. The approach provides a practical, amplitude- and coordinate-independent diagnostic that can be applied as a post-processing tool to numerical or experimental mutual-information data to identify scaling regimes and reduce ambiguities from nonuniversal amplitudes.

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.

Figures (2)

• Figure 1: Informational distance and geometric conversion factor in the XXZ chain. (DMRG, $L=96$, OBC, $D=128$). (a) Informational distance $d_E(r)$ computed from center-pair mutual information. (b) Geometric conversion factor $G(r) = \partial_r d_E$. In the critical phase ($\Delta=1$), $G(r)$ is flat in the bulk window ($r\ge 10$, shaded), indicating a uniform scale ($\kappa \approx 0$). The dashed line shows the bulk mean $\bar{G} \approx 1.564$. In the gapped phase ($\Delta=2$), $G(r)$ grows rapidly and varies strongly with position, then turns over at the largest separations where $I(r)$ is at the numerical noise floor ($I(r) \lesssim 10^{-8}$); in this regime the derivative is not quantitatively reliable, but the strong non-uniformity of $G(r)$ relative to the critical case is already evident at intermediate $r$.
• Figure 2: Coordinate-agnostic slope test at the $X=2$ benchmark. Exact results for the XX chain ground state (thermodynamic limit). The plot shows $\log G(r) - \langle \log G \rangle$ versus $\log I(r)$ for bulk data (odd $r \ge 15$). The slope $\kappa$ is extracted using least-squares (LS, orange line with 95% CI band) and robust Theil-Sen (TS, green dashed) estimators. Both yield $\kappa \approx 0$ (inset), confirming the GSR prediction $\kappa = 0$ at $X=2$. The extremely small CoV($G$) quantifies the uniformity of the geometric scale.