Magnon Decay in Noncollinear Quantum Antiferromagnets

Instability of the excitation spectrum of an ordered noncollinear Heisenberg antiferromagnet with respect to spontaneous two-magnon decays is investigated. We use a spin-$1/2$ antiferromagnet on a triangular lattice as an example and examine the characteristic long- and short-wavelength features of its zero-temperature spectrum within the $1/S$ approximation. The kinematic conditions are shown to be crucial for the existence of decays and for overall properties of the spectrum. The $XXZ$ and the $J\mathrm{\text{−}}J$ generalizations of the model, as well as the role of higher-order corrections, are discussed.

Figure 1

Brillouin zone of the TL. The shaded area corresponds to the region where spontaneous two-magnon decays are allowed. $\Gamma =(0,0)$, $K=(4\pi /3,0)$, $K^{\prime }=(2\pi /3,2\pi /\sqrt{3})$, $Y=(0,\pi /\sqrt{3})$, and $M=(\pi ,\pi /\sqrt{3})$ points are highlighted. The lines correspond to the extrema in the two-magnon continuum described in the text.

Figure 2

Upper row: Magnon dispersion along the lines $\Gamma K$ and $YM$ in the $BZ$; see Fig. 1. The dashed lines are the LSWT spectrum; the dotted line is the bottom of the LSWT two-magnon continuum, and the solid lines are the spectrum with the $1/S$ correction. Lower row: Imaginary part of the $1/S$ magnon energy along the same lines. $k_b$ is the intersection point of the one-magnon branch with the two-magnon continuum along the $YM$ line. $k^{*}$ and $k^{\prime }$ points correspond to the singularities discussed in text.

Figure 3

The upper right parts of the BZ in the $\mathbf{q}$ space. The decay contours for $k=k^{\prime }$ (left) and $k=k^{*}$ (right) along the $\Gamma K$ line are shown. Both $\mathbf{k}$ points correspond to the saddle points in the two-magnon continuum. The corresponding $\mathbf{q}$ contours undergo a topological transition.

Figure 4

Decay regions and singularity lines for the $XXZ$ model $\alpha =0.96$ (left) and the $J\mathrm{\text{−}}{J}^{\prime }$ model $J^{\prime }/J=0.9$ (right). The definition of lines is the same as in Fig. 1.