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The spin excitations of the block-antiferromagnetic K$_{0.8}$Fe$_{1.6}$Se$_2$

We study the spin excitations of the newly discovered block-antiferromagnetic state in K$_{0.8}$Fe$_{1.6}$Se$_2$ using an effective spin Hamiltonian suggested in the literature. Interestingly in addition to the usual Goldstone mode, there exists three other "optical" spin wave branches. These spin excitations are the analog of "optical phonons" in crystals with more than one atom per unit cell. We compute the spin wave corrected ordering moment and the uniform spin susceptibilities.