Spin-wave analysis for Kagome-triangular spin system and coupled spin tubes: low-energy excitation for the cuboc order

The coupled spin tube system, which is equivalent to a stacked Kagome-triangular spin system, exhibits the cuboc order -- a non-coplanar spin order with a twelve-sublattice structure accompanying spontaneous breaking of the translational symmetry -- in the Kagome-triangular plane. On the basis of the spin-wave theory, we analyze spin-wave excitations of the planar Kagome-triangular spin system, where the geometric phase characteristic to the cuboc spin structure emerges. We further investigate spin-wave excitations and dynamical spin structure factors for the coupled spin tubes, assuming the staggered cuboc order.