A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries

Abstract:

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 $\tau$ of a finite group $G$. For each element $\omega$ in $SPG(\mathcal {A})$ we can associate a second cohomology class $c_{\omega ,R}$ of $G$. We consider a classification of $SPG(\mathcal {A})$ whose criterion is given as follows: $\omega _{0}$ and $\omega _{1}$ in $SPG(\mathcal {A})$ are equivalent if there are automorphisms $\Xi _{R}$, $\Xi _L$ on $\mathcal {A}_{R}$, $\mathcal {A}_{L}$ (right and left half infinite chains) preserving the symmetry $\tau$, such that $\omega _{1}$ and $\omega _{0}\circ \left ( \Xi _{L}\otimes \Xi _{R}\right )$ are quasi-equivalent. It means that we can move $\omega _{0}$ close to $\omega _{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{\omega ,R}$ is the complete invariant of this classification.