Topologically distinct classes of valence bond solid states with their parent Hamiltonians

TL;DR

This paper presents a method to construct and analyze topologically distinct valence bond solid states with specific symmetries, including new states and their parent Hamiltonians, in one-dimensional quantum spin chains.

Contribution

The paper introduces a general construction method for valence bond solid states with Lie group symmetries and derives their parent Hamiltonians, including novel spin-1 fermionic states.

Findings

Identified two topologically distinct classes of valence bond solid states.

Constructed a new spin-1 fermionic valence bond solid state and its parent Hamiltonian.

Generalized valence bond states with SO(5) symmetry and analyzed their properties.

Abstract

We introduce 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(2) spin-$J$ variables in each site and another is formed by using two $SO(2S+1)$ spinors. Among them, a new 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) symmetry are further generalized and their respective properties are analyzed as well.