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Exploring many-body quantum geometry beyond the quantum metric with correlation functions: A time-dependent perspective

Yuntao Guan and Barry Bradlyn*

  • *Contact author: bbradlyn@illinois.edu
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Phys. Rev. Research 8, 013291 – Published 17 March, 2026

DOI: https://doi.org/10.1103/3xjs-c7v7

Abstract

The quantum geometric tensor and quantum Fisher information have recently been shown to provide a unified geometric description of the linear response of many-body systems. However, a similar geometric description of higher-order perturbative phenomena including nonlinear response in generic quantum systems is lacking. In this work, we develop a general framework for the time-dependent quantum geometry of many-body systems by treating external perturbing fields as coordinates on the space of density matrices. We use the Bures distance between the initial and time-evolved density matrix to define geometric quantities through a perturbative expansion. To lowest order, we derive a time-dependent generalization of the Bures metric related to the spectral density of linear response functions, unifying previous results for the quantum metric in various limits and providing a geometric interpretation of Fermi’s golden rule. At next order in the expansion, we define a time-dependent Bures-Levi-Civita connection for general many-body systems. We show that the connection is the sum of one contribution that is related to a second-order nonlinear response function, and a second contribution that captures the higher geometric structure of first-order perturbation theory. We show that in the quasistatic, zero-temperature limit for noninteracting fermions, this Bures connection reduces to the known expression for band-theoretic Christoffel symbols. Our work provides a systematic framework to explore many-body quantum geometry beyond the quantum metric and highlights how higher-order correlation functions can probe this geometry.

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Phys. Rev. Research 8, 013291– Published 17 March, 2026
  • Received 11 August 2025
  • Accepted 26 February 2026
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