Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by the terms of 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras, on 1-dimensional linear and circular lattice structures. We find 17 unique dynamical Lie algebras. Our classification includes some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that appear new. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of variational quantum computing, quantum control and the spin chain literature.

关于一维自旋链，在纠缠特性、物理相和可积性方面的了解很多。然而，描述这些系统的哈密顿量的李代数性质在很大程度上仍未得到探索。在这项工作中，我们提供了由二维线性和圆形晶格结构上的 2 局部自旋链哈密顿量或所谓的动态 Lie 代数项生成的所有 Lie 代数的分类。我们找到了 17 个独特的动态 Lie 代数。我们的分类包括一些众所周知的模型，例如横场 Ising 模型和 Heisenberg 链，我们还发现了更多出现新的奇特哈密顿量类。除了封闭和开放的自旋链之外，我们还考虑了具有完全连接拓扑的系统，这可能与量子机器学习方法有关。我们讨论了我们在变分量子计算、量子控制和自旋链文献背景下工作的实际意义。