The matrix product representations for all valence bond states

We introduce a simple representation for irreducible spherical tensor operators of the rotation group of arbitrary integer or half integer rank and use these tensor operators to construct matrix product states corresponding to all the variety of valence-bond states proposed in the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction. These include the fully dimerized states of arbitrary spins, the partially dimerized states of different spins, which are ground states of Hamiltonians with nearest and next-nearest neighbor interactions, and also non-dimerized states which are constructed from these by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions. PACS: 03.67.-a, 75.10 Jm. Corresponding author:[email protected] email:[email protected]