Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities

We present a comprehensive study of the dynamical properties of the quantum Heisenberg antiferromagnet on a triangular lattice within the framework of spin-wave theory. The distinct features of spin-wave excitations in the triangular lattice antiferromagnet are (i) finite lifetime at zero temperature due to spontaneous two-magnon decays, (ii) strong renormalization of magnon energies εₖ with respect to the harmonic result, and (iii) logarithmic singularities in the decay rate Γₖ. Quantum corrections to the magnon spectrum are obtained using both the on-shell and off-shell solutions of the Dyson equation with the lowest-order magnon self-energy. At low-energies magnon excitations remain well defined albeit with the anomalous decay rate Γₖ∝k² at k→0 and Γₖ∝|k-QAF|7/2 at k→QAF. At high energies, magnons are heavily damped with the decay rate reaching (2Γₖ/εₖ)~0.3 for the case $S=1/2$. The on-shell solution shows logarithmic singularities in Γₖ with the concomitant jumplike discontinuities in Re[εₖ] along certain contours in the momentum space. Such singularities are even more prominent in the magnon spectral function A(k,ω). Although the off-shell solution removes such log singularities, the decay rates remain strongly enhanced. We also discuss the role of higher-order corrections and show that such singularities may lead to complete disappearance of the spectrum in the vicinity of certain k points. The kinematic conditions for two-magnon decays are analyzed for various generalizations of the triangular lattice antiferromagnet as well as for the $XXZ$ model on a kagom\'e lattice. Our results suggest that decays and singularities in the spin-wave spectra must be ubiquitous in all these systems. In addition, we give a detailed introduction in the spin-wave formalism for noncollinear Heisenberg antiferromagnets and calculate several quantities for the triangular lattice model including the ground-state energy and the sublattice magnetization.