Density Matrix Geometry and Sum Rules

TL;DR

This paper introduces a time-dependent quantum geometric tensor for thermal density matrices, linking geometric properties to physical responses and sum rules at finite temperature, and suggests experimental probes of quantum geometry.

Contribution

It develops a unified geometric framework for finite-temperature sum rules using a new time-dependent quantum geometric tensor.

Findings

Unified interpretation of sum rules via quantum geometry

Connection between geometry and fluctuation-dissipation theorem

Proposals for experimental measurement of quantum geometry

Abstract

Geometry plays a fundamental role in a wide range of physical responses, from anomalous transport coefficients to their related sum rules. Notable examples include the quantization of the Hall conductivity and the Souza-Wilkens-Martin (SWM) sum rule -- both valid at zero temperature, independent of interactions and disorder. The finite-temperature generalization of the SWM sum rule has been explored in the literature, revealing deep connections to the geometry of density matrices. Building on recent advances in time-dependent geometric frameworks, we propose a time-dependent quantum geometric tensor for thermal density matrices. This formalism provides a unified interpretation of known sum rules within the framework of the fluctuation-dissipation theorem, further elucidating their fundamental geometric origin. In addition, it provides experimentally accessible methods to probe quantum geometry beyond the zero-temperature regime.