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

Liska, Diego; Gritsev, Vladimir

doi:10.21468/SciPostPhys.10.1.020

SciPost Physics

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

Diego Liska, Vladimir Gritsev

SciPost Phys. 10, 020 (2021) · published 28 January 2021

doi: 10.21468/SciPostPhys.10.1.020
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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.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.10.1.020
TI  - Hidden symmetries, the Bianchi classification and geodesics of the quantum geometric ground-state manifolds
PY  - 2021/01/28
UR  - https://scipost.org/SciPostPhys.10.1.020
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 10
IS  - 1
SP  - 020
A1  - Liska, Diego
AU  - Gritsev, Vladimir
AB  - 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.
ER  -

@Article{10.21468/SciPostPhys.10.1.020,
	title={{Hidden symmetries, the Bianchi classification and geodesics of the quantum geometric ground-state manifolds}},
	author={Diego Liska and  Vladimir Gritsev},
	journal={SciPost Phys.},
	volume={10},
	pages={020},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.10.1.020},
	url={https://scipost.org/10.21468/SciPostPhys.10.1.020},
}

Cited by 4

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¹ Diego Liska,
¹ ² Vladimir Gritsev

Funders for the research work leading to this publication