Level spacing statistics of two unitary equivalent models with distinct local symmetries

Abstract: The full spectrum and integrability of unitary equivalent models are the same. A standard diagnostic tool of integrability is level spacing statistics which requires separating the full spectrum into sectors according to the symmetry. When two unitary equivalent models have different symmetries, it is interesting to know how their level spacing statistics show consistent conclusions. In this work, we examine the level spacing statistics of two unitary equivalent models with distinct local symmetries. The first model is spin-1 XXZ chain $H$, the second model is obtained by Kennedy-Tasaki transformation $U_{KT}HU_{KT}$. We find that the level spacings of model $H$ follow the statistics of the Gaussian orthogonal ensemble (GOE). However, model $U_{KT}HU_{KT}$ only displays GOE statistics in some sectors after a "hidden non-local symmetry" is resolved, and the other sectors labeled by quantum numbers corresponding to the local symmetries exhibit non-GOE statistics. Additionally, a mapping relation between the levels in the sector ${Z,X,I}$ of model $H$ and the sector ${Z,X,I'}$ of model $U_{KT}HU_{KT}$ is found, where $Z,X,I$ correspond to the quantum numbers of $\pi$ rotation around $z,x$ axes and bond-centered inversion.