We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains $\bigotimes_{ \mathbb Z}\mathop{ \mathrm{ M}}\nolimits_n$, 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.
Categories: math-ph math.MP