Phases of d-orbital bosons in optical lattices

We explore the properties of bosonic atoms loaded into the d bands of an isotropic square optical lattice. Following the recent experimental success reported in Zhai et al (2013 Phys. Rev. A 87 063638), in which populating d bands with a 99 $\%$ fidelity was demonstrated, we present a theoretical study of the possible phases that can appear in this system. Using the Gutzwiller ansatz for the three d band orbitals we map the boundaries of the Mott insulating phases. For not too large occupation, two of the orbitals are predominantly occupied, while the third, of a slightly higher energy, remains almost unpopulated. In this regime, in the superfluid phase we find the formation of a vortex lattice, where the vortices come in vortex/anti-vortex pairs with two pairs locked to every site. Due to the orientation of the vortices time-reversal symmetry is spontaneously broken. This state also breaks a discrete $Z_2$-symmetry. We further derive an effective spin-1/2 model that describe the relevant physics of the lowest Mott-phase with unit filling. We argue that the corresponding two dimensional phase diagram should be rich with several different phases. We also explain how to generate anti-symmetric spin interactions that can give rise to novel effects like spin canting. (paper)