  We investigate the quantum phases of ultracold atoms trapped in a vortex
lattice using a mixture of two bosonic species (A and B), in the presence of an
artificial gauge field. Heavy atoms of species B are confined in the array of
vortices generated in species A, and they are described through a Bose-Hubbard
model. In contrast to the optical-lattice setups, the vortex lattice has an
intrinsic dynamics, given by its Tkachenko modes. Including these quantum
fluctuations in the effective model for B atoms yields an extended Bose-Hubbard
model, with an additional "phonon"-mediated long-range attraction. The
ground-state phase diagram of this model is computed through a variational
ansatz and the quantum Monte Carlo technique. When compared with the ordinary
Bose-Hubbard case, the long-range interatomic attraction causes a shift and
resizing of the Mott-insulator regions. Finally, we discuss the experimental
feasibility of the proposed scheme, which relies on the proper choice of the
atomic species and on a large control of physical parameters, like the
scattering lengths and the vorticity.
