Heisenberg-Kitaev model on the hyperhoneycomb lattice

Motivated by recent experiments on $\beta$-Li$_2$IrO$_3$, we study the phase diagram of the Heisenberg-Kitaev model on a three dimensional lattice of tri-coordinated Ir$^{4+}$, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir$^{4+}$ ions carrying $J_{eff}=1/2$ moments, suggests that the Heisenberg-Kitaev model may effectively capture the low energy spin physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean field theory, we find a spin-liquid and four different magnetically ordered phases---the Neel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag. The three dimensional $Z_2$ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi-circle (co-dimensions, $d_c=2$). We discuss the effect of magnetic field and finite temperature on various phases that may be relevant for future experiments.