The matrix product representations for all valence bond states

TL;DR

This paper presents a unified matrix product state representation for all valence-bond states in the AKLT framework, encompassing various dimerized and valence-bond solid states with different spin configurations.

Contribution

It introduces a simple tensor operator-based method to construct matrix product states for all AKLT valence-bond states, including dimerized and translation-invariant states.

Findings

Unified matrix product representation for all AKLT valence-bond states

Construction of ground states for Hamiltonians with various interaction ranges

Representation includes fully and partially dimerized states

Abstract

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/VBS (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.