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

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

Automated summary

In this paper, we study the Killing vectors of the quantum ground-state manifold of a parameter-dependent Hamiltonian, uncovering hidden symmetries and proposing a classification method based on Bianchi. Utilizing these symmetries to find geodesics and explore their behavior when crossing critical lines, we discuss the relation between geodesics, energy fluctuations, and adiabatic preparation protocols. Our main focus is on the anisotropic transverse-field Ising model, where we also provide analytic solutions to the geodesic equations for the Ising limit.