On an Analytic Expression for the One-Magnon Dispersion Relation in a Heisenberg Antiferromagnet

Using the Dyson-equation approach to spin-spin correlation functions we derive self-consistent mean-field equations for the antiferromagnetic spin-(1/2) Heisenberg model. An analytic expression for the one-magnon dispersion valid in D = 1, 2, 3 is obtained. For the linear chain it gives the exact result ω k = (π/2) |sin k | known from the Bethe ansatz. For the square lattice the same procedure leads to ω k = 1.188 × 2 √1 - ((cos k x + cos k y )/2) 2 in agreement with recent Monte Carlo results. Values for the simple cubic lattice in three dimensions are also given.