Correlated metallic two-particle bound states in Wannier--Stark flatbands

Tight-binding single-particle models on simple Bravais lattices in space dimension $d \geq 2$, when exposed to commensurate DC fields, result in the complete absence of transport due to the formation of Wannier--Stark flatbands [Phys. Rev. Res. $\textbf{3}$, 013174 (2021)]. The single-particle states localize in a factorial manner, i.e., faster than exponential. Here, we introduce interaction among two such particles that partially lifts the localization and results in metallic two-particle bound states that propagate in the directions perpendicular to the DC field. We demonstrate this effect using a square lattice with Hubbard interaction. We apply perturbation theory in the regime of interaction strength $(U)$ $\ll$ hopping strength $(t)$ $\ll$ field strength $(\mathcal{F})$, and obtain estimates for the group velocity of the bound states in the direction perpendicular to the field. The two-particle group velocity scales as $U {(t/\mathcal{F})}^\nu$. We calculate the dependence of the exponent $\nu$ on the DC field direction and on the dominant two-particle configurations related to the choices of unperturbed flatbands. Numerical simulations confirm our predictions from the perturbative analysis.