arXiv subjects

Beau Leighton-Trudel

Publications and source records attributed to Beau Leighton-Trudel.

3 recordsLinked to original sources

arXiv2025

Geometric Characterization of Anisotropic Correlations via Mutual Information Tomography

Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale $C_{\mathrm{vac}}$: $ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)$. We prove that this map yields a smooth Riemannian structure $g_{ij}$ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent $X_{\text{loc}}=2$). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness $g_{ij} \propto δ_{ij}$, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split $g_{ij} = V^{2/D}γ_{ij}$, which rigorously separates local density fluctuations (volume $V$) from directional anisotropy (shape/shear $γ_{ij}$). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor $γ_{ij}$ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining $X_{\text{loc}}=2$ branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims.

cond-mat.stat-mech↗

arXiv2025

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 $d_E = K_0/\sqrt{I}$, 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.

cond-mat.stat-mech↗

arXiv2025

Emergent Distance and Metricity of Mutual Information in 1D Quantum Chains

We develop and formalize a phase diagnostic based on the information-distance $d_E = K_0/\sqrt{I}$ (mutual information $I$) for 1D quantum chains. Calibrating with the Euclidean benchmark $I(r)\propto r^{-2}\mapsto d_E(r)\propto r$ makes the triangle-inequality test parameter-free and scale-invariant. Under site-averaged, monotone scaling conditions on the 1D line we establish a criterion linking the decay of $I(r)$ to metric behavior of $d_E(r)$: power laws $I(r)\sim r^{-X}$ with $0<X\le 2$ yield subadditivity (metric scaling), while exponential clustering leads to superadditivity. As an analytic check complementing our earlier numerical study, we verify these predictions in the 1D transverse-field Ising chain using an exact Jordan-Wigner/Bogoliubov-de Gennes solution: at criticality $I(r)$ follows a power law close to the $X=2$ benchmark and the equal-legs triangle defect $Δ(r,r)=d_E(2r)-2d_E(r)$ is asymptotically non-positive; in gapped regimes $I(r)$ decays exponentially and $Δ(r,r)\gg 0$. The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.

cond-mat.stat-mech↗