Motivated by the observation of spatially anisotropic exchange constants in the iron pnictide materials, we study the spin-wave spectra of the $J_{ 1a}-J_{ 1b}-J_{ 2}$ Heisenberg models on a square-lattice with nearest neighbor exchange $J_{ 1a}$ along x and $J_{ 1b}$ along y axis and a second neighbor exchange $J_2$. We focus on the regime, where the spins order at ($\pi,0$), and compute the spectra by systematic expansions around the Ising limit. We study both spin-half and spin-one Heisenberg models as well as a range of parameters to cover various cases proposed for the iron pnictide materials. The low-energy spectra have anisotropic spin-wave velocities and are renormalized with respect to linear spin-wave theory by up to 20 percent, depending on parameters. Extreme anisotropy, consisting of a ferromagnetic $J_{ 1b}=- |J_F|$, is best distinguished from a weak anisotropy ($J_{ 1a}\approx J_{ 1b}=J_1$, $J_2>J_1/2$) by the nature of the spin-waves near the wavevectors ($0,\pi$) or ($\pi,\pi$). The reported spectra for the pnictide material CaFe$_2$As$_2$ clearly imply such an extreme anisotropy.
Categories: cond-mat.str-el cond-mat.supr-con