Linear spin wave theory for the frustrated ferromagnetic spin chain in its incommensurate singlet phase

The ground state energy and the low-energy excitation of the frustrated ferromagnetic chain model in its incommensurate singlet phase are investigated by linear spin wave theory (LSWT). Differing from the previous works also using LSWT, we diagonalize the bosonic Hamiltonian and obtain the spin wave spectrum (${\omega }_k=A_k\sqrt{1-{\Gamma }_k^2}$) analytically by Bogoliubov transformation. Through numerical calculation, we find that the pitch angle $\phi$ and the frustration parameter $\alpha$ must have the relation $\cos \phi =1/\left(4\alpha \right)$ if we require that $1-{\Gamma }_k^2$ is non-zero for all wave vector $k$, that gives a new way to determine the critical point ($\alpha =1/4$) between the ferromagnetic phase and the incommensurate singlet phase. The quantum correction to the classical ground state energy as a function of $\alpha$ is also calculated and we find it will be zero as $\alpha \to {(1/4)}^{+}$, that means the spin fluctuations near the critical point are very small.