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

Figure 1

(Color online) The ordered $120°$ spin structure of the triangular lattice HAF.

Figure 2

(Color online) Left panel: the Brillouin zone of a triangular lattice, lines are representative cuts. Central panel: 3D plot of the linear spin-wave energy ${\omega }_{\mathbf{k}}$ in the triangular lattice HAF. Right panel: intensity plot of ${\omega }_{\mathbf{k}}$. Note different velocities of the Goldstone modes and different symmetries of the dispersion near $\Gamma$ and $K$ $\left(K^{\prime }\right)$ points and the saddle point at $M$ $\left(M^{\prime }\right)$.

Figure 3

The lowest-order vertices that yield $1/S$ corrections to the spectrum and $1/S^2$ contributions to the static properties.

Figure 4

The lowest order normal [(a) and (b)] and anomalous [(c) and (d)] magnon self-energies generated by cubic vertices.

Figure 5

(Color online) Spin-wave decay rates in the vicinity of $\Gamma$ and $K$ points in the triangular lattice HAF. Dots are numerical solutions of Eq. (55) and dashed lines are the asymptotic results (see text).

Figure 6

(Color online) Spin-wave energy for the $S=1/2$ triangular lattice HAF along the symmetry directions in the BZ. Dashed and solid lines are the results obtained in the LSWT approximation $\left({\epsilon }_{\mathbf{k}}\right)$ and with the first-order $1/S$ correction $\left({\overline{\epsilon }}_{\mathbf{k}}\right)$, respectively. Vertical arrows indicate singularities and intersection points with the two-magnon continuum. Gray areas show the width of the spectral peaks due to damping ${\Gamma }_{\mathbf{k}}$ from Fig. 7. Dotted lines represent the minimum of the two-magnon continuum obtained from the LSWT spectrum.

Figure 7

(Color online) The spin-wave damping of the $S=1/2$ triangular lattice HAF calculated in the first order of the $1/S$ expansion along the same lines as in Fig. 6. Arrows indicate singularities and intersection points with the two-magnon continuum.

Figure 8

(Color online) The decay contours in $\mathbf{q}$ space for magnons with selected $\mathbf{k}$’s along the $\Gamma K$ direction; $k=k^{\prime }$ and $k=k^{\ast }$ are the same as in Fig. 7, left panel. The corresponding $\mathbf{q}$ contours undergo topological transitions at $k^{\prime }$ and $k^{\ast }$.

Figure 9

(Color online) The Brillouin zone of a triangular lattice. Shaded area shows the region where spontaneous two-magnon decays are allowed. Lines correspond to the extrema in the two-magnon continuum described in text.

Figure 10

(Color online) Renormalization of the decay region for the spin-1/2 triangular lattice HAF. Left and middle panels: Dashed and dotted lines are the LSWT predictions for the magnon dispersion ${\epsilon }_{\mathbf{k}}$ (dashed) and for the minimum of the two-magnon continuum (dotted); solid and dotted-dashed lines represent same results for the spectrum ${\overline{\epsilon }}_{\mathbf{k}}$ with the first-order quantum correction, respectively. Right panel: Modified decay region; lightly shaded regions highlight the area of uncertainty due to the re-entrant behavior seen in the middle panel.

Figure 11

(Color online) Comparison of the magnon spectrum (upper row) and the decay rates (lower row) obtained in the harmonic approximation (dotted lines), with the first-order $1/S$ corrections (dashed lines), and by solving the Dyson equation (solid lines), respectively.

Figure 12

(Color online) (Upper panel) Magnon spectral function $A\left(\mathbf{k},\omega \right)$ and (lower panel) real and imaginary parts of one-loop $G_{11}^{-1}\left(\mathbf{k},\omega \right)$ together with the two-magnon DoS (dotted line), all for $\mathbf{k}=\left(\pi ,\pi /\sqrt{3}\right)$ ($M$ point). Dashed arrow in the upper panel indicates position of the “bare” magnon peak. Solid arrows denote position of the renormalized quasiparticle peak, zero of $\text{Re}\left[G_{11}{\left(\mathbf{k},\omega \right)}^{-1}\right]$, and the van Hove singularities.

Figure 13

(Color online) Same as in Fig. 12, for momenta on the $\Gamma K$ line, (a) $\mathbf{k}={\mathbf{k}}^{\ast }=\pi \left(0.6085,0\right)$ and (b) $\mathbf{k}=\pi \left(0.2,0\right)$. Solid vertical arrows indicate position of the quasiparticle peak, zero of $\text{Re}\left[G_{11}^{-1}\left(\mathbf{k},\omega \right)\right]$, and van Hove and “edge” singularities (see text).

Figure 14

(Color online) Magnon energy for the $XY$ spin-1/2 triangular lattice AF along representative directions. Dashed line is the linear spin-wave dispersion and solid line is including the first $1/S$ correction.

Figure 15

(Color online) Decay region and singularity lines for the $XXZ$ triangular lattice antiferromagnet with $\alpha =0.99$ (a) and $\alpha =0.96$ (b). Definition of lines is the same as in Fig. 9.

Figure 16

(Color online) Left panel: the 3D shape of the linear spin-wave frequency ${\omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/2JS$ for the $J-J^{\prime }$ antiferromagnet with $J^{\prime }/J=0.34$. Central panel: the decay region and the singularity lines at $J^{\prime }/J=0.34$ (definitions are the same as in Fig. 9). Right panel: same for $J^{\prime }/J=0.7$.

Figure 17

(Color online) Linear spin-wave frequencies of the excitations in the $XXZ$ model on the kagomé lattice, Eq. (99) for $\alpha =0.95$, along $\Gamma XY$ cuts. Notations for the BZ are the same as in Ref. 51. Lines and shaded area are the decay boundary (dotted), strong singularity line (solid), and the decay region, respectively (see text).

Figure 18

(Color online) Schematic picture of all topologically inequivalent diagrams of order (a) $1/S$ and (b)–(d) $1/S^2$, respectively. The leading divergence in $1/S$ diagrams is $\text{ln}\left|\Delta \omega \right|$ [first diagram in (a)] and in $1/S^2$ order it is ${\left(\text{ln}\left|\Delta \omega \right|\right)}^2$ [all diagrams in (d)]. First two diagrams in (d) are identical.