Counter-rotating spiral order in three dimensional iridates: signature of hidden symmetry in Kitaev-\(\Gamma\) 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 \(\Gamma\) interaction on the hyperhoneycomb lattice using exact diagonalization. We first show that a 12-site unitary transformation unveils a Heisenberg model with hidden SU(2) symmetry when \(K = \Gamma\). The magnetic ordering in the transformed basis then generically maps to the counter-rotating noncoplanar spiral order of the original spin. The moment direction depends on perturbations away from the SU(2) point. When \(K\) and \(\Gamma\) are negative, a positive Heisenberg interaction favors the (110)-direction reported in the neutron scattering measurement on \(\beta\)-Li\(_2\)IrO\(_3\). Our findings offer a relevant set of microscopic parameters, which in turn guides a way to approach possible Kitaev spin liquids.