Scaling of a Mutual-Information Distance in One-dimensional Quantum Spin Chains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911294806818816 |
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| author | Leighton-Trudel, Beau |
| author_facet | Leighton-Trudel, Beau |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling of a Mutual-Information Distance in One-dimensional Quantum Spin Chains Leighton-Trudel, Beau Statistical Mechanics 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. |
| title | Scaling of a Mutual-Information Distance in One-dimensional Quantum Spin Chains |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.00649 |