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

TL;DR

This paper introduces a scale-invariant, parameter-free method using mutual information to diagnose phases in 1D quantum chains, validated analytically on the transverse-field Ising model.

Contribution

It formalizes a new phase diagnostic based on information-distance and establishes its connection to metric behavior, supported by exact solutions.

Findings

Power-law decay of mutual information indicates metric behavior at criticality.

Exponential decay of mutual information signals gapped phases.

Triangle defect analysis distinguishes between critical and gapped regimes.

Abstract

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 \(\Delta(r,r)=d_E(2r)-2d_E(r)\) is asymptotically non-positive; in gapped regimes \(I(r)\) decays exponentially and \(\Delta(r,r)\gg 0\). The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.