The spiral spin state in a zigzag spin chain system

We considered a spin chain with nearest neighbor and next nearest neighbor exchange interactions, anisotropic exchange interaction and Dzyaloshinskii-Moriya interaction. The conditions of the spiral spin state as the ground state were analyzed. Our method was to build the connection between the spiral state and the fully polarized state with a unitary transformation. Under this transformation, anisotropic exchange interaction and Dzyaloshinskii-Moriya interaction can be transformed to each other. Then we used positive semi-definite matrix theorem to identify the region of fully polarized state as the ground state for the transformed Hamiltonian, and it is the region of spiral spin state as the ground state of the original Hamiltonian. We also found that the effect of Dzyaloshinskii-Moriya interaction is important. Its strength is related to the pitch angle of spiral spins. Our method can be applied to coupled spin chains and two dimensional triangular lattice systems. The result can be compared with experiment data.