Hidden symmetries, the Bianchi classification and geodesics of the quantum geometric ground-state manifolds

We study the Killing vectors of quantum ground-state manifold a parameter-dependent Hamiltonian. find that may have symmetries are not visible at level Hamiltonian and different phases matter exhibit symmetries. propose Bianchi-based classification various manifolds using Lie algebra vector fields. Moreover, we explain how to exploit these geodesics explore their behaviour when crossing critical lines. briefly discuss relation between geodesics, energy fluctuations adiabatic preparation protocols. Our primary example is anisotropic transverse-field Ising model. also analyze limit analytic solutions geodesic equations for both cases.