Multi-State Geometry of Density Matrices and Rectification Sum Rules

This paper addresses the need for a unified geometric theory of nonlinear response in quantum materials that extends beyond clean, noninteracting systems at zero temperature. The author develops a theory of multi-state geometry for density matrices, introducing a complex, quantum generalization of the Amari-Chentsov (cQAC) tensor from classical information theory. By applying this framework to second-order response theory, the study derives sum rules for DC rectification that remain valid regardless of the strength of disorder or interactions.

The central result is a zero-temperature sum rule that expresses the frequency-integrated DC rectification response of an insulator as the difference between a ground-state third cumulant and a complex distortion tensor. This complex distortion tensor is a multi-state geometric quantity derived from the cQAC tensor, which captures the geometry of the perturbed density matrix. The author demonstrates that this framework generalizes existing single-particle sum rules for shift and nonlinear Hall currents to many-body systems. Numerical verification in a generalized Kane-Mele model confirms that the geometric contribution can dominate the integrated response. Furthermore, the paper shows that while the clean separation of these terms is temperature-dependent, low-temperature measurements can still effectively probe the underlying multi-state geometry due to corrections that are exponentially small in the energy gap.

This work provides a powerful, model-independent lens for understanding nonlinear optical and transport phenomena. By moving beyond the single-particle Bloch band picture, the author provides a rigorous way to account for interactions and disorder in quantum materials. This geometric approach not only unifies various nonlinear effects but also establishes a clear path for using rectification measurements as a diagnostic tool to probe the multi-state geometry of complex quantum states.