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arXiv:0712.2018 · Physical Review B

Matrix product representations for all valence bond states

Vahid Karimipour, Laleh Memarzadeh

Published Mar 17, 2008• 31 citations
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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/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.

DOI
10.1103/physrevb.77.094416

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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.
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A brief note on the G2_22​ Affleck-Kennedy-Lieb-Tasaki chain
We present a general method to construct one-dimensional translationally invariant valence-bond solid states with a built-in Lie group GGG and derive their matrix product representations. The general strategies to find their parent Hamiltonians are provided so that the valence-bond solid states are their unique ground states. For quantum integer-spin-SSS chains, we discuss two topologically distinct classes of valence-bond solid states: one consists of two virtual SU(2)SU(2)SU(2) spin-JJJ variables in each site and another is formed by using two SO(2S+1)SO(2S+1)SO(2S+1) spinors. Among them, a spin-1 fermionic valence-bond solid state, its parent Hamiltonian, and its properties are discussed in detail. Moreover, two types of valence-bond solid states with SO(5)SO(5)SO(5) symmetries are further generalized and their respective properties are analyzed as well.