### Matrix Product States: Symmetries and Two-Body Hamiltonians

**M. Sanz, M. M. Wolf, D. Perez-Garcia, J. I. Cirac**

Phys. Rev. A 79, 042308 **, (2009)

**DOI**: 10.1103/PhysRevA.79.042308

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*arXiv.org*: 0901.2223

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**Abstract:**We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of SU(2)--invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.