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Induced tunneling and localization for a quantum particle in tilted two-dimensional lattices

We consider a quantum particle in tilted two-dimensional lattices in the tight-binding approximations. We found that for some lattice geometries and certain orientations of the static force with respect to the lattice primary axes the particle can freely move across the lattice in the direction perpendicular to the vector of the static force. This effect is argued to be analogue of the photon-induced tunneling in driven one-dimensional lattices. The obtained dispersion relation for the transverse motion of the particle draws this analogy further by eventually showing band collapses when a control parameter is varied.