Spiral spin state in a zigzag spin-chain system

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