We present a general method to construct one-dimensional translationally invariant valence-bond solid states with a built-in Lie group $G$ 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-$S$ chains, we discuss two topologically distinct classes of valence-bond solid states: one consists of two virtual $SU\left(2\right)$ spin-$J$ variables in each site and another is formed by using two $SO\left(2S+1\right)$ 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 $SO\left(5\right)$ symmetries are further generalized and their respective properties are analyzed as well.

• Received 6 April 2009

DOI:https://doi.org/10.1103/PhysRevB.80.014401