  We calculate the effect of spin waves on the properties of finite size spin
chains with a chiral spin ground state observed on bi-atomic Fe chains
deposited on Iridium(001). The system is described with a Heisenberg model
supplemented with a Dzyaloshinskii-Moriya (DM) coupling and a uniaxial single
ion anisotropy that presents a chiral spin ground state. Spin waves are studied
using the Holstein-Primakoff (HP) boson representation of spin operators. Both
the renormalized ground state and the elementary excitations are found by means
of Bogoliubov transformation, as a function of the two variables that can be
controlled experimentally, the applied magnetic field and the chain length.
Three main results are found. First, because of the non-collinear nature of the
classical ground state, there is a significant zero point reduction of the
ground state magnetization of the spin spiral. Second, the two lowest energy
spin waves are edge modes in the spin spiral state that, above a critical field
the results into a collinear ferromagnetic ground state, become confined bulk
modes. Third, in the spin spiral state, the spin wave spectrum exhibits
oscillatory behavior as function of the chain length with the same period of
the spin helix.
