Entanglement of Exact Excited States of AKLT Models: Exact Results, Many-Body Scars and the Violation of Strong ETH

We obtain multiple exact results on the entanglement of the exact excited states of non-integrable models we introduced in arXiv:1708.05021. We first discuss a general formalism to analytically compute the entanglement spectra of exact excited states using Matrix Product States and Matrix Product Operators and illustrate the method by reproducing a general result on single-mode excitations. We then apply this technique to analytically obtain the entanglement spectra of the infinite tower of states of the spin-$S$ AKLT models in the zero and finite energy density limits. We show that in the zero density limit, the entanglement spectra of the tower of states are multiple shifted copies of the ground state entanglement spectrum in the thermodynamic limit. We show that such a resemblance is destroyed at any non-zero energy density. Furthermore, the entanglement entropy $\mathcal{S}$ of the states of the tower that are in the bulk of the spectrum is sub-thermal $\mathcal{S} \propto \log L$, as opposed to a volume-law $\mathcal{S} \propto L$, thus indicating a violation of the strong Eigenstate Thermalization Hypothesis (ETH). These states are examples of what are now called many-body scars. Finally, we analytically study the finite-size effects and symmetry-protected degeneracies in the entanglement spectra of the excited states, extending the existing theory.