Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis

We obtain multiple exact results on the entanglement of the exact excited states of nonintegrable models we introduced in Phys. Rev. B 98, 235155 (2018). 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 energy 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 nonzero energy density. Furthermore, the entanglement entropy $\mathcal{S}$ of the states of the tower that are in the bulk of the spectrum is subthermal $\mathcal{S}\propto logL$, 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.

Figure 1

Ground state of the spin-1 AKLT model with open boundary conditions. Big and small circles represent physical spin-1 and spin-1/2 Schwinger bosons, respectively. The lines represent singlets between spin-1/2. The two edge spin-1/2's are free.

Figure 2

Spin-2 AKLT model ground state with two singlets between nearest neighbors. The four edge spin-1/2s are free. Spin-$S$ AKLT has $S$ singlets.

Figure 3

The normalized entanglement entropy $\mathcal{S}/\left[\right(L/2)ln3]$ for the eigenstates of the Hamiltonian with energy $E$ (141) in the quantum number sector $(s,S_z,k,I,P_z)=(6,0,\pi ,-1,+1)$, where the quantum numbers respectively correspond to the total spin, the projection of the total spin along the $z$ direction, momentum, inversion and spin-flip symmetries [42]. (a) and (b) show the entropy at the AKLT point. This sector has a single exact state $\left|S_6\right.〉$ that belongs to the tower of states, which exhibits a sharp dip at $E=6$. (c) and (d) show the entropy for $\alpha =-0.025$, where remnants of the low-entropy states are seen. (e) and (f) show the entropy in the same sector for $\alpha =-0.05$.

Figure 4

Apparent atypical eigenstates in the spin-1 pure Heisenberg model (i.e., $\alpha =-1/3$ in Eq. (141)) in the quantum number sector $(s,S_z,k,I,P_z)=(0,0,0,+1,+1)$ for system sizes (a) $L=14$ and (b) 16. We use the same conventions as Fig. 3 for the axis labels. Looking at the evolution between $L=14$ and 16 might suggest that those atypical states are finite-size artifacts.

Figure 5

Schematic depiction of the entanglement spectra of spin-1 AKLT tower of states $\left\{\left|S_{2N}\right.〉\right\}$ (left) and spin-2 AKLT tower of states $\left\{\left|2S_{2N}\right.〉\right\}$ (right). The almost-degenerate levels shown in red have an exponential finite-size splitting whereas the black doublets are exactly degenerate. Power-law finite-size splittings are depicted by black two-headed arrows and constants by blue two-headed arrows.

Figure 6

Labelling in the MPS construction of the spin-$S$ AKLT ground state. Big and small circles represent physical spin-$S$ and virtual spin-$S/2$ degrees of freedom, respectively.