Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective
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arXiv
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| Format: | Preprint |
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2025
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| author | Guan, Yuntao Bradlyn, Barry |
| author_facet | Guan, Yuntao Bradlyn, Barry |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_23028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective Guan, Yuntao Bradlyn, Barry Mesoscale and Nanoscale Physics Statistical Mechanics Quantum Physics 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. |
| title | Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective |
| topic | Mesoscale and Nanoscale Physics Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2507.23028 |