Magnon damping in the two-dimensional quantum Heisenberg antiferromagnet at short wavelengths

Dyson-Maleev boson formalism is used to calculate magnon damping ${\mathrm{\Gamma }}_{\mathbf{k}}$ in the two-dimensional quantum Heisenberg antiferromagnet at low temperatures T and for wavelengths short compared to the thermal de Broglie wavelength. From the evaluation of the second-order self-energy it is found that ${\mathrm{\Gamma }}_{\mathbf{k}}$∝${\mathit{T}}^3$Z(‖${\mathbf{v}}_{\mathbf{k}}$‖) as T→0, where ${\mathbf{v}}_{\mathbf{k}}$ is the gradient of dispersion relation of free magnons. For k close to the boundary of the Brillouin zone, where ‖${\mathbf{v}}_{\mathbf{k}}$‖ is small, the function Z has the expansion Z(‖${\mathbf{v}}_{\mathbf{k}}$‖)=1+O(‖${\mathbf{v}}_{\mathbf{k}}$ ${\mathrm{\Vert }}^2$). For general ‖${\mathbf{v}}_{\mathbf{k}}$‖, we have calculated Z numerically. Although there is no long-range order at any T≠0, the staggered correlation length ξ(T) is exponentially large as T→0. It is shown explicitly in the present work that, at low temperatures, magnons with momentum k in the regime ‖k‖a≫(T/${\mathit{E}}_{\mathrm{\pi }}$ ${)}^{1/3}$ (where a is the lattice spacing and ${\mathit{E}}_{\mathrm{\pi }}$ is the energy of zone-boundary magnons) are well-defined quasiparticles. We present evidence to support the argument that this remains true as long as ‖k‖ξ≫1.