Classifying and measuring geometry of a quantum ground state manifold

From the Aharonov-Bohm effect to general relativity, geometry plays a central\nrole in modern physics. In quantum mechanics many physical processes depend on\nthe Berry curvature. However, recent advances in quantum information theory\nhave highlighted the role of its symmetric counterpart, the quantum metric\ntensor. In this paper, we perform a detailed analysis of the ground state\nRiemannian geometry induced by the metric tensor, using the quantum XY chain in\na transverse field as our primary example. We focus on a particular geometric\ninvariant -- the Gaussian curvature -- and show how both integrals of the\ncurvature within a given phase and singularities of the curvature near phase\ntransitions are protected by critical scaling theory. For cases where the\ncurvature is integrable, we show that the integrated curvature provides a new\ngeometric invariant, which like the Chern number characterizes individual\nphases of matter. For cases where the curvature is singular, we classify three\ntypes -- integrable, conical, and curvature singularities -- and detail\nsituations where each type of singularity should arise. Finally, to connect\nthis abstract geometry to experiment, we discuss three different methods for\nmeasuring the metric tensor, namely via integrating a properly weighted noise\nspectral function and by using leading order responses of the work distribution\nto ramps and quenches in quantum many-body systems.\n