A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries
We consider a set $SPG(\mathcal{A})$ of pure split states on a quantum spin chain $\mathcal{A}$ which are invariant under the on-site action $τ$ of a finite group $G$. For each element $ω$ in $SPG(\mathcal{A})$ we can associate a second cohomology class $c_{ω,R}$of $G$. We consider a classification of $SPG(\mathcal{A})$ whose criterion is given as follows: $ω_{0}$ and $ω_{1}$ in $SPG(\mathcal{A})$ are equivalent if there are automorphisms $Ξ_{R}$, $Ξ_L$ on $\mathcal{A}_{R}$, $\mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $τ$, such that $ω_{1}$ and $ω_{0}\circ( Ξ_{L}\otimes Ξ_{R})$ are quasi-equivalent. It means that we can move $ω_{0}$ close to $ω_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{ω,R}$ is the complete invariant of this classification.