The magnetic properties of two-dimensional altermagnets can be obtained from a square lattice Heisenberg model with antiferromagetic nearest-neighbor interaction and two types of next-nearest-neighbor interactions arranged in a checkerboard pattern. Using nonlinear spin-wave theory, we calculate for this model the corrections to the renormalized magnon spectrum and the staggered magnetization to first order in the inverse spin quantum number $1/S$. We also show that to order $1/S^2$ the ground state energy is not sensitive to the component of the interaction which is responsible for altermagnetism. At the $\mathrm{\Gamma }$ point, the $1/S$ correction to the magnon dispersion vanishes so that quantum fluctuations do not induce a gap in the magnon spectrum of altermagnets, as expected by Goldstone's theorem. We extract the leading $1/S$ corrections to the spin-wave velocity and the effective mass characterizing the curvature of the magnon dispersion in altermagnets.

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• Altermagnetism
• Magnons
• Altermagnets
• Heisenberg model
• Spin wave theory