Density Matrix Geometry and Response Functions

yzj5306@psu.edu (Yiyang Jiang) — Mon, 28 Sep 2026 16:30:00 -0400


Presenter	Yuntao Guan (Prof. Barry Bradlyn’s group, UIUC)
Date	September 28, 2026 · 4:30–6:00 PM
Location	Davey 339
Topic	Density matrix geometry and response functions
Papers	Y. Guan and B. Bradlyn, Phys. Rev. Research 8, 013291 (2026); B. Bradlyn, arXiv:2608.06326 (2026)

The quantum geometric tensor packs the Berry curvature and the quantum metric into one object, and it has become a standard tool for Hall responses, Wannier localization, flat-band superconductivity and optical sum rules. But the usual version has four limitations. It is a linear-order object. It lives at zero temperature. And it is tied either to Bloch bands of noninteracting electrons or to twist angles of an interacting ground state. The talk replaces the pure state by the density matrix, and the quantum distance by the Bures distance between the thermal state and the state driven by an arbitrary time-dependent field. Expanding that distance in the field strength defines a Bures metric at second order and a Bures connection at third order, and both come out as explicit spectral functions. Specific limits of the metric recover earlier results: a finite-temperature Souza–Wilkens–Martin (SWM) sum rule, the quantum Fisher information, and Fermi’s golden rule recast as geometry. At zero temperature, the connection reduces to the band-theory Christoffel symbol. The second half follows the connection into nonlinear optics. The sum rule for the shift current contains a genuinely multi-state term, built from transitions between two excited states. That term is the non-metricity of the mixture/exponential connections of quantum information geometry.

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Density Matrix Geometry and Response Functions