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

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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 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 the recent experiments and numerical results in the neighborhood of this point. It is suggested that 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 very small values of high-order 1/S corrections.