C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry

Abstract

We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains ${⨂}_{\mathbb{Z}}{\mathrm{M}}_n$, with local symmetry given by a group $G$. The $G$-symmetric gapped Hamiltonians are given by triples $(k,u,V)$, where $u$ is a projective unitary representation of $G$ on a finite dimensional space ${\mathbb{C}}^k$, and $V$ is an isometry from ${\mathbb{C}}^k$ to ${\mathbb{C}}^n\otimes {\mathbb{C}}^k$. We show that Hamiltonians $H_0,H_1$, given by the triples $(k^{(0)},u^{(0)},V^{(0)})$ and $(k^{(1)},u^{(1)},V^{(1)})$ are $C^1$-equivalent if the projective representations $u^{(0)}$ and $u^{(1)}$ are unitary equivalent.