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 ${\omega }_0\circ ({\Xi }_L\otimes {\Xi }_R)$ 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.

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