Heisenberg-Kitaev model on 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 by Takagi et. al. The lattice geometry of this material, along with Ir$^{4+}$ ions carrying $J_{\rm eff}=1/2$ 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, in addition to the spin-liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases--the N\'{e}el, the polarized ferromagnet, the skew-stripy and the skew-zig-zag, have collinear spin ordering. 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 the magnetic field and finite temperature on different phases that may be relevant for future experiments.