Raychaudhuri-Ehlers Kinematics on Quantum State-Space: Caustics and Extrinsic Quantum Geometry

Abstract

Quantum mechanics is inherently geometric in that the pure states form a curved complex manifold and the Schrödinger evolution defines a Hamiltonian flow on this space. Geometry identifies constraints on flows on quantum state-space that hold independently of their microscopic realization. We uncover such universal behavior in families of geodesic paths on the quantum state-space using the Raychaudhuri-Ehlers equations. We show that the complex structure leads to tidal anisotropy in the curvature, and for Hilbert space dimension $N\geq3$, complete point-emitted geodesic bundles necessarily develop shear and undergo direction-selective focusing at finite distance. We give these geometric results an information-theoretic interpretation in terms of fidelity susceptibility and quantum Fisher information. For ground-state manifolds constrained by physical parameters, we resolve the tidal curvature into Berry-curvature-dependent ambient tides and contributions from extrinsic bending within the full ambient state space. We further relate extrinsic bending to variations in the unitary generator of motion along intrinsic geodesics. We present exact many-body geometric results for the transverse-field XY spin chain. The product structure of the ground state leads to an exact cancellation of the ambient tide, including its Berry-curvature enhancement, by extrinsic curvature contributions. At the zero-field Ising point, we obtain exact results at finite size and show that the shear source for anisotropy-directed motion is entirely extrinsic relative to the full many-body state-space. For the metric in the thermodynamic limit, the geodesic families exhibit repeated $A_2$ fold catastrophes inside the ferromagnetic phase. Our framework thus separates universal state-space effects from those of physical restrictions and exposes their consequences in a paradigmatic many-body system.