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.
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.
The geometry of quantum states has emerged as a key ingredient in understanding the linear and nonlinear responses of quantum materials. To date, however, the connection between geometry and nonlinear response is best understood for clean, noninteracting systems at zero temperature. In this work, we develop a theory of multi-state geometry for density matrices and use it to derive sum rules for second-order rectification, making no assumptions about the strength of disorder or interactions. We first show that perturbation theory for thermal density matrices gives rise to two dual information-theoretic connections and an almost complex structure. We introduce a complex, quantum generalization of the Amari-Chentsov tensor of classical information theory, the cQAC tensor, which captures the multi-state geometry of the perturbed density matrix. We derive a zero-temperature sum rule for the frequency-integrated DC rectification response of an insulator as a difference between a ground state third cumulant and the complex distortion tensor, a multi-state geometric quantity built from the cQAC tensor. This generalizes known single-particle sum rules for the shift and nonlinear Hall currents to many-body systems and general perturbations. Specializing to the shift current, we resolve the geometric contribution for multiband insulators into particle-like and hole-like terms. We verify the sum rule numerically in a generalized Kane-Mele model, finding that the geometric contribution can dominate the integrated response. Finally, we show that although the splitting of the sum rule into cumulant and geometric contributions does not survive at nonzero temperature, the measured sum rule for insulators differs from its zero-temperature form by corrections exponentially small in the gap, allowing low-temperature rectification measurements to probe the multi-state geometry of insulators.
Geometry of quantum states is a fundamental research area with applications ranging from topological materials in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are often studied, while mixed states are generally needed, e.g., to have nonzero temperature. Mixed-state geometry, however, is much more challenging and has so far lacked any structure relating distance and curvature, like the Kähler structure does for pure states. This work introduces that structure. We show that it has the physical meaning of fluctuation-dissipation---by identifying distance with fluctuations and curvature with linear response---and that it becomes the known structure for pure states, but is non-Kähler otherwise. This paints a simple picture that unifies fluctuation-dissipation and geometry of quantum states. It also applies on parameter spaces, as encountered in condensed matter or estimation theory. Our picture implies that the geometry contains all linear responses and fluctuations, proves the fluctuation-dissipation relation, explains that quantum geometry appears in any sum rule, and allows one to geometrically characterize generic transport behaviors. Closed and open systems are treated with equal ease.
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.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.
The quantum geometric tensor (QGT) reveals local geometric properties and associated topological information of quantum states. Here a generalization of the QGT to mixed quantum states at finite temperatures based on the Sj\"oqvist distance is developed. The resulting Sj\"oqvist QGT is invariant under gauge transformations of individual spectrum levels of the density matrix. A Pythagorean-like relation connects the distances and gauge transformations, which clarifies the role of the parallel-transport condition. The real part of the QGT naturally decomposes into a sum of the Fisher-Rao metric and Fubini-Study metric, allowing a distinction between different contributions to the quantum distance. The imaginary part of the QGT is proportional to a weighted summation of the Berry curvatures, which leads to a geometric phase for mixed states under certain conditions. We present three examples of different dimensions to illustrate the temperature dependence of the QGT and a discussion on possible implications.
Quantum geometry has emerged as a unifying concept across condensed matter physics, underlying phenomena from nonlinear topological response to flat-band superconductivity. While usually formulated within band theory, quantum geometry remains meaningful in disordered interacting systems~\cite{resta1999electron}. Here we show that the first negative moment of the optical conductivity -- proportional to the zero temperature quantum Fisher information as a bound on the multipartite entanglement -- provides an experimental probe of quantum geometry across the three-dimensional metal-insulator quantum phase transition in phosphorus-doped silicon. We extract a quantum geometric length ℓ that characterizes the local wavefunctions. Far from the transition, this length is almost coincident with the Bohr radius of the hydrogenic phosphorus donors, reflecting their atomic-scale quantum geometry. Approaching the transition, ℓ is enhanced, but does not diverge continuously like a correlation length; it jumps discontinuously to infinity at the critical point. This reflects the UV domination of the sum rule in three dimensions that renders it insensitive to the critical fluctuations driving the diverging dielectric constant and correlation length. Its enhancement demonstrates a ``puffing" of the donor polarizability volume of quantum geometric origin, which yields a quantum geometric corrected Clausius-Mossotti description in closer agreement with the diverging dielectric response and provides a quantum mechanical foundation for the century-old Herzfeld metallization criterion.