Emergent gravity at a Lifshitz point from a Bose liquid on the lattice

We propose a model with quantum bosons on the fcc lattice, which has a stable algebraic Bose liquid phase at low energy. We show that this phase is described by emergent quantum gravity at the Gaussian $z=3$ Lifshitz fixed point in $3+1$ dimensions. The stability of this algebraic Bose liquid phase is guaranteed by the gauge symmetry of gravitons and self-duality of the low-energy field theory. By tuning one parameter in the lattice boson model we can drive a phase transition between the $z=3$ Lifshitz gravity and another algebraic Bose liquid phase, described by gravity at the $z=2$ Lifshitz point.