Hidden Symmetries, the Bianchi Classification and Geodesics of the Quantum Geometric Ground-State Manifolds

TL;DR

This paper investigates hidden symmetries in quantum ground-state manifolds, classifies them using Bianchi types, and explores geodesic paths, revealing insights into quantum phase transitions and adiabatic protocols.

Contribution

It introduces a Bianchi classification of quantum ground-state manifolds based on Killing vector symmetries and analyzes geodesics crossing critical points.

Findings

Different quantum phases exhibit distinct hidden symmetries.

Analytic solutions for geodesic equations in specific models.

Symmetries can be exploited to understand adiabatic state preparation.

Abstract

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.