$\eta$-pairing states as true scars in an extended Hubbard model

The $\eta$-pairing states are a set of exactly known eigenstates of the Hubbard model on hypercubic lattices, first discovered by Yang [C. N. Yang, Phys. Rev. Lett. 63, 2144 (1989)]. These states are not many-body scar states in the Hubbard model because they occupy unique symmetry sectors defined by the so-called $\eta$-pairing SU(2) symmetry. We study an extended Hubbard model with bond-charge interactions, popularized by Hirsch [J. E. Hirsch, Physica C 158, 326 (1989)], where the $\eta$-pairing states survive without the $\eta$-pairing symmetry and become true scar states. We also discuss similarities between the $\eta$-pairing states and exact scar towers in the spin-1 XY model found by Schecter and Iadecola [M. Schecter and T. Iadecola, Phys. Rev. Lett. 123, 147201 (2019)], and systematically arrive at all nearest-neighbor terms that preserve such scar towers in one dimension. We also generalize these terms to arbitrary bipartite lattices. Our study of the spin-1 XY model also leads us to several scarred models, including a spin-1/2 $J_1-J_2$ model with Dzyaloshinskii-Moriya interaction, in realistic quantum magnet settings in one and two dimensions.

Figure 1

(a) $r$ statistics of the Hirsch model with $U=t=1$ and $X=0.3$ on a periodic chain of length $L=12$, in the symmetry sector $N_{\uparrow }=N_{\downarrow }=4$, total spin $S=0$, momentum $k=0$, and site inversion $I_s=1$. The Hilbert-space dimension is 3072. The $r$ statistics are consistent with the Wigner-Dyson GOE prediction for quantum chaotic models. (b) Bipartite EE in the same symmetry sector. The $\eta$-pairing state $|{\psi }_4\rangle$ is a clear EE outlier. Each point is colored by the density of states at the corresponding energy, with colorbar (normalized to 1) on the right.