Understanding counter-rotating spiral ordering in three dimensional Kitaev materials

The counter-rotating spiral ordering found in three-dimensional (3D) iridate Kitaev candidates suggest frustration of spin interactions and raise a possibility of nearby spin liquid phases. Understanding the microscopic mechanism of this ordering may provide routes to 3D Kitaev spin liquids. We study a minimal 3D model including Kitaev K and a symmetric off-diagonal bond-dependent Γ interaction on the hyperhoneycomb lattice by using exact diagonalization (ED). An ED cluster was chosen to satisfy a hidden SU(2) symmetry. The magnetic ordering in the transformed basis then generically maps to the counter-rotating noncoplanar spiral order of the original spin when the moment is pinned along a certain direction. When K and Γ are negative, a small positive Heisenberg interaction favours the (1,1,0) direction consistent with the reported spiral order in β-Li₂IrO₃. Our findings offer a relevant set of microscopic parameters, which in turn guides a way to approach possible Kitaev spin liquids.