Classifying and measuring the geometry of the quantum ground state manifold

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 XY 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, namely via integrating a properly weighted noise spectral function and by using leading order responses of the work distribution to ramps and quenches in quantum many-body systems.