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Much is understood about 1-dimensional spin chains in terms of entanglementproperties, physical phases, and integrability. However, the Lie algebraicproperties of the Hamiltonians describing these systems remain largelyunexplored. In this work, we provide a classification of all Lie algebrasgenerated by translation-invariant 2-local spin chain Hamiltonians, orso-called dynamical Lie algebras. We consider chains with open and periodicboundary conditions and find 17 unique dynamical Lie algebras. Ourclassification covers some well-known models such as the transverse-field Isingmodel and the Heisenberg chain, and we also find more exotic classes ofHamiltonians that cannot be identified easily. In addition to the closed andopen spin chains, we consider systems with a fully connected topology, whichmay be relevant for quantum machine learning approaches. We discuss thepractical implications of our work in the context of quantum control,variational quantum computing, and the spin chain literature.

Classification of dynamical Lie algebras for translation-invariant 2-local spin systems in one dimension