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

Abstract

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 ${\ensuremath{\epsilon}}_{\mathbf{k}}$ with respect to the harmonic result, and (iii) logarithmic singularities in the decay rate ${\ensuremath{\Gamma}}_{\mathbf{k}}$. 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 ${\ensuremath{\Gamma}}_{\mathbf{k}}\ensuremath{\propto}{k}^{2}$ at $\mathbf{k}\ensuremath{\rightarrow}0$ and ${\ensuremath{\Gamma}}_{\mathbf{k}}\ensuremath{\propto}{|\mathbf{k}\ensuremath{-}{\mathbf{Q}}_{\text{AF}}|}^{7/2}$ at $\mathbf{k}\ensuremath{\rightarrow}{\mathbf{Q}}_{\text{AF}}$. At high energies, magnons are heavily damped with the decay rate reaching $(2{\ensuremath{\Gamma}}_{\mathbf{k}}/{\ensuremath{\epsilon}}_{\mathbf{k}})\ensuremath{\sim}0.3$ for the case $S=1/2$. The on-shell solution shows logarithmic singularities in ${\ensuremath{\Gamma}}_{\mathbf{k}}$ with the concomitant jumplike discontinuities in $\text{Re}[{\ensuremath{\epsilon}}_{\mathbf{k}}]$ along certain contours in the momentum space. Such singularities are even more prominent in the magnon spectral function $A(\mathbf{k},\ensuremath{\omega})$. 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 $\mathbf{k}$ 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.