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

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.

11 pages, 5 figures. Typos corrected, modified figures and added references

References in corpus (7)

• Quantum critical scaling of the geometric tensors
• Observation of topological transitions in interacting quantum circuits
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• Fidelity in topological quantum phases of matter
• Oscillating fidelity susceptibility near a quantum multicritical point
• Geometric Critical Exponents in Classical and Quantum Phase Transitions

Cited by in corpus (5)

• Dynamical obstruction to localization in a disordered spin chain
• Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems
• Topological properties of a non-Hermitian quasi-1D chain with a flat band
• The Loschmidt Index
• Quantum geometry in many-body systems with precursors of criticality