Spectrum of short-wavelength magnons in a two-dimensional quantum Heisenberg antiferromagnet on a square lattice: third-order expansion in 1/S

The spectrum of short-wavelength magnons in a two-dimensional quantum Heisenberg antiferromagnet on a square lattice is calculated to the third order in a 1/S expansion. It is shown that a 1/S series for S = 1/2 converges quickly in the whole Brillouin zone except in the neighborhood of the point k = (π, 0), at which absolute values of the third-and the second-order 1/S-corrections are approximately equal to each other. It is shown that the third-order corrections make deeper the roton-like local minimum at k = (π, 0), improving the agreement with recent experiments and numerical results in the neighborhood of this point. It is suggested that the 1/S series converges slowly near k = (π, 0) also for S = 1 although the spectrum renormalization would be small in this case due to the very small values of high-order 1/S corrections.