Irreducible half-integer rank unit spherical tensors

Vahid Karimipour, Laleh Memarzadeh

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 ground states for quantum spin chains of arbitrary spin with nearest and next-nearest neighbor interactions. The ground states of these quantum spin chains are doubly degenerate and are in the form of juxtaposition of dimers (maximally entangled singlets) on adjacent sites. Hence each ground state breaks the translation invariance of the Hamiltonian. In this way, we unravel the matrix product structure of the well-known Majumdar-Ghosh model, with particular emphasis on its symmetry properties and generalize it to arbitrary integer or half integer spin models. We also make a comparison with the AKLT construction of such spin models.

D. Dubbers, H.‐J. Stöckmann

Vahid Karimipour, Laleh Memarzadeh

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, with uniform or alternating patterns of spins, which are ground states of Hamiltonians with nearest and next-nearest-neighbor interactions, and the partially dimerized or AKLT/valence bond solid states, which are constructed from them by projection. The latter states are translation-invariant ground states of Hamiltonians with nearest-neighbor interactions.

Edgard Elbaz

L L Boyle, L C Jones

A new method of derivation of operator equivalents is proposed, in which individual irreducible tensor components are expressed in terms of both real and complex unit tensors derived from real and complex (or bi) vectors. The method is extended to irreducible tensor operators which do not have full permutation symmetry and it is found that these are equivalent to totally symmetric tensor operators of rank equal to the angular momentum quantum number.

K. T. Hecht