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.

Figure 1

(a) The lattice structure of this model, with the coordinates of site, link and plaquette. The term $H_{v,\widehat{x}\mathrm{link}}$ in Eq. (4) involves all the sites and plaquettes denoted with solid circles in this figure. The $H_t$ term contains flavor-dependent boson hopping between sites and adjacent plaquettes (the $H_{sp}$ term in Eq. (4)), and also between nearest neighbor plaquettes (the $H_{pp}$ term). (b) the ring exchange term $-t_1\cos ({\mathcal{B}}_{zz})$ in Eq. (10), which includes eight boson hoppings.