Antiferromagnetic Magnon Dispersion Law and Bloch Wall Energies in Ferromagnets and Antiferromagnets

The exact eigenstates of the exchange Hamiltonian $\mathcal{H}=2J\Sigma l^{}({\mathrm{S}}_l·{\mathrm{S}}_{l+1\mathrm{}}-\frac{1}{4})$, are found for short chains of 4, 6, 8, and 10 atoms of spin ½. A linear dispersion law for magnons in an antiferromagnet is exhibited by the energy spectrum. The periodic boundary conditions (${\mathrm{S}}_{N+l}={\mathrm{S}}_l$ where $N$ is the number of spins in the chain) are then removed and the spins at the two ends of the chain are constrained to be either parallel or antiparallel to each other. The eigenstates for these arrangements are computed and give the exact energy of the 180° Bloch wall. This energy is compared with the semiclassical result. It is found that the semiclassical ferromagnetic Bloch wall energy is in good agreement with the exact wall energy. The energy of the semiclassical antiferromagnetic Bloch wall is not in good agreement with the exact wall energy but appears to have the correct dependence on the wall thickness.