Exact Localized and Ballistic Eigenstates in Disordered Chaotic Spin Ladders and the Fermi-Hubbard Model

We demonstrate the existence of exact atypical many-body eigenstates in a class of disordered, interacting one-dimensional quantum systems that includes the Fermi-Hubbard model as a special case. These atypical eigenstates, which generically have finite energy density and are exponentially many in number, are populated by noninteracting excitations. They can exhibit Anderson localization with area-law eigenstate entanglement or, surprisingly, ballistic transport at any disorder strength. These properties differ strikingly from those of typical eigenstates nearby in energy, which we show give rise to diffusive transport as expected in a chaotic quantum system. We discuss how to observe these atypical eigenstates in cold-atom experiments realizing the Fermi-Hubbard model, and comment on the robustness of their properties.

Figure 1

Level spacing distribution for a generic instance of the model (1), averaged over 100 disorder realizations. The distribution is well described by the Wigner-Dyson distribution (blue). Inset: Disorder-averaged density of states vs energy density $\epsilon$ (normalized to lie within the interval [0, 1]) for the same model. Eigenstates corresponding to energies between the black vertical lines were used to accumulate the statistics in the main figure. Dashed red vertical lines denote the bandwidth of a six-doublon invariant subspace, demonstrating the finite energy density of eigenstates within this subspace.

Figure 2

Diffusive NESS charge transport for weakly disordered models in the full Hilbert space (full triangles for disorder $h=0.5$, empty symbols for $h=1$). Both the Hubbard case (down triangles and circles) and a generic ladder (squares and triangles) are shown.

Figure 3

Ballistic (top) vs localized (bottom) charge dynamics in a two-doublon invariant subspace in the Fermi-Hubbard model from exact diagonalization at $L=100$. The charge density profile is shown for various times $t=0,\dots ,400$. Inset: Charge density vs $t$ on a single site initially populated by a doublon for $h=0$ (red) and $h=2$ (blue). The dynamics outside of the invariant subspaces is ergodic for $h=2$.

Figure 4

Evolution for a two-doublon [(a) and (b)], and a two-holon initial state [(c) and (d)] for weak 4% breaking of ${\mathbb{Z}}_2$ symmetry ($ϵ=0.04$). All data are for $J=0$, ${\mathrm{\Delta }}_j=h_j$, $L=200$, and the same disorder realization in all frames. Color (shown in a log-scale) shows the charge [(a) and (c)] and spin density [(b) and (d)].