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arXiv:2608.26269

How quantum is quantum geometry?

Ady Stern, Felix von Oppen

cond-mat.mes-hallcond-mat.supr-conquant-phSubmitted Aug 26, 2026• Announced Aug 28, 2026
v1 · Aug 26, 2026v2 · Sep 15, 2026
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Abstract

The quantum geometric tensor - the Berry curvature together with the quantum metric - now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics. To answer this question, we study a purely classical system: a point particle carrying a classical magnetic moment $\boldsymbol{\ell}$ that precesses in a momentum-dependent magnetic field $\mathbf{B}(\mathbf{p})$. Within Hamiltonian classical mechanics, the component of $\boldsymbol{\ell}$ along the field reproduces the Berry-curvature phenomena, while its precessing transverse component reproduces the quantum-metric phenomena. The particle acquires a position spread whose second moment is the metric, an orbital magnetic moment, and - most strikingly - an inertial mass generated by a position-dependent force, and with it a nonzero Drude weight in a system that is nominally dispersionless.

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The quantum geometric tensor - the Berry curvature together with the quantum metric - now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics. To answer this question, we study a purely classical system: a point particle carrying a classical magnetic moment ℓ\boldsymbol{\ell}ℓ that precesses in a momentum-dependent magnetic field B(p)\mathbf{B}(\mathbf{p})B(p). Within Hamiltonian classical mechanics, the component of ℓ\boldsymbol{\ell}ℓ along the field reproduces the Berry-curvature phenomena, while its precessing transverse component reproduces the quantum-metric phenomena. The particle acquires a position spread whose second moment is the metric, an orbital magnetic moment, and - most strikingly - an inertial mass generated by a position-dependent force, and with it a nonzero Drude weight in a system that is nominally dispersionless.
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Abstract The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin‐half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.
From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum XYXYXY chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.
The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.
We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.
Geometrical concepts in quantum mechanics, such as the Berry phase or quantum metric, have been studied extensively in the case of pure quantum states, but their generalisation to density operators is still undergoing active research. It has become clear that more than one valid approach exists and different definitions of the quantum geometric tensor (QGT) may give inequivalent results. In this report, we cover some of the foundations of quantum geometry for pure states, motivate the need for mixed-state generalisations, and examine some approaches in that direction. The final section analyses more closely a recent proposal, deemed the “Sjöqvist QGT” and argues that its imaginary part does not produce a generalisation of the Berry curvature.