Counter-rotating spiral order in three-dimensional iridates: Signature of hidden symmetry in the Kitaev-Γ model

The unconventional magnetic orderings found in Kitaev spin liquid candidates suggest frustration of spin interactions and raise a possibility of nearby spin liquid phases. In particular, counter-rotating spiral ordering in three-dimensional (3D) iridates is striking, and understanding the microscopic mechanism of such 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 F interaction on the hyperhoneycomb lattice by using exact diagonalization (ED). An ED cluster was chosen to satisfy a hidden SU(2) symmetry when K = Gamma via a 12-site unitary transformation. 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 Gamma are negative, a small positive Heisenberg interaction favors the (1,1,0) direction consistent with the reported spiral order in beta-Li2IrO3. Two other ordered phases close to the spiral order and spin liquids are identified. Our findings offer a relevant set of microscopic parameters, which in turn guides a way to approach possible Kitaev spin liquids.