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

The spectrum of short-wavelength magnons in two-dimensional quantum Heisenberg antiferromagnet on a square lattice is calculated in the third order in $1/S$ expansion. It is shown that $1/S$ series for $S=1/2$ converges fast in the whole Brillouin zone except for the neighborhood of the point ${\bf k}=(\pi,0)$, 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 ${\bf k}=(\pi,0)$ improving the agreement with the recent experiments and numerical results in the neighborhood of this point. It is suggested that $1/S$ series converges slowly near ${\bf k}=(\pi,0)$ also for $S=1$ although the spectrum renormalization would be small in this case due to very small values of high-order $1/S$ corrections.