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.

中文翻译：

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量子几何基态流形的隐藏对称性，Bianchi分类和测地线

我们研究了依赖参数的哈密顿量的量子基态流形的杀死矢量。我们发现，流形可能具有在哈密顿量级上不可见的对称性，并且物质的不同量子相表现出不同的对称性。我们提出使用Killing向量场的李代数对各种基态流形进行基于Bianchi的分类。此外，我们解释了如何利用这些对称性找到测地线，并在跨越临界线时探索其行为。我们简要讨论了测地线，能量波动和绝热准备方案之间的关系。我们的主要例子是各向异性横向场伊辛模型。我们还分析了Ising极限，并找到了两种情况下测地线方程的解析解。