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Supersolid Devil's Staircases of Spin-Orbit-Coupled Bosons in Optical Lattices

We study the emergence of supersolid Devil's staircases of spin-orbit coupled bosons loaded in optical lattices. We consider two- and three-dimensional systems of pseudo-spin-$1/2$ bosons interacting via local spin-dependent interactions. These interactions together with spin-orbit coupling produce length scales that are commensurate to the lattice spacing. This commensurability leads to Devil's staircases of supersolids, with fractal Hausdorff dimensions, which arise from uniform superfluid phases. We show that umklapp processes are essential for the existence of commensurate supersolids, and that without them the Devil's staircase does not exist. Lastly, we emphasize the generality of our results, suggest experiments that can unveil these unusual predictions, and discuss potential applications to the case of $^{87}$Rb.