Raman scattering in a Heisenberg $S=\frac{1}{2}$ antiferromagnet on the triangular lattice

We investigate the two-magnon Raman scattering from the $S=\frac{1}{2}$ Heisenberg antiferromagnet on the triangular lattice by considering both the effect of the renormalization of the one-magnon spectrum by $1/S$ corrections and the final-state magnon-magnon interactions. The bare Raman intensity displays two peaks related to one-magnon Van Hove singularities. We find that $1/S$ self-energy corrections to the one-magnon spectrum strongly modify this intensity profile. The central Raman peak is significantly enhanced due to plateaus in the magnon dispersion, the high frequency peak is suppressed due to magnon damping, and the overall spectral support narrows considerably. Additionally, we investigate final-state interactions by solving the Bethe-Salpeter equation to $O\left(1/S\right)$. In contrast to collinear antiferromagnets, the noncollinear nature of the magnetic ground state leads to an irreducible magnon scattering that is retarded and nonseparable already for the lowest order. We show that final-state interactions lead to a rather broad Raman continuum centered on approximately twice the “roton” energy. We also discuss the dependence on the scattering geometry.

Figure 1

(Color online) (a) Classical $120°$ noncollinear spin order on the triangular lattice. Basic vectors of triangular lattice: ${\vec{\delta }}_1=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$, ${\vec{\delta }}_2=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$, and ${\vec{\delta }}_3=\left(1,0\right)$. (b) Definition of scattering angles for the LF vertex.

Figure 2

(Color online) One magnon dispersion. Top: Linear spin-wave dispersion $E_{\mathbf{k}}$ from Eq. (14). Middle and bottom: Real and imaginary parts $\text{Re}\left(\text{Im}\right)E_{\mathbf{k}}^r$ of one magnon dispersion to $O\left(1/S\right)$ from a solution of Eqs. (10)–(12) of Ref. 17 on a lattice of $252\times 252$ $\mathbf{k}$ points with artificial line broadening of $\eta =0.05$.

Figure 3

Diagrams for the Raman intensity: (a) bare Raman vertex $R$ from Eq. (21); (b) Raman susceptibility [both bare, $G^0$, and dressed, $G$, magnon propagators are considered (see text)]. (c) The integral equation for the dressed Raman vertex $\Gamma$ in terms of the irreducible magnon particle-particle (IPP) vertex $\gamma$. (d) Leading order $1/S$ contributions to the IPP vertex.

Figure 4

Raman intensity neglecting final state interactions, i.e., replacing $\Gamma$ by $M$ in Eq. (33). Scattering geometry: $\varphi =\theta =0$. Number of $\mathbf{k}$ points: $N\times N$. Dashed curve (BB): bubble obtained by using bare magnon energies $E_{\mathbf{k}}$ from Eq. (14), as shown in top panel of Fig. 2. The imaginary broadening $\eta =0.003$ was chosen as such as to retain visible but small finite-size oscillations. Solid curve (RB): bubble obtained by using renormalized magnon energies $E_{\mathbf{k}}^r$ to $O\left(1/S\right)$ obtained from Eqs. (10)–(12) of Ref. 17 and shown in the middle and lower panels of Fig. 2. Finite-size oscillations are suppressed by $\text{Im}\left[E_{\mathbf{k}}^r\right]$. The absolute scale of $I\left(\Omega \right)$ is set to unity, but the relative scale of BB and RB is kept.

Figure 5

Effect of final state interactions on Raman intensity. Scattering geometry: $\varphi =\theta =0$. Number of $\mathbf{k}$ points: $N\times N$. Imaginary shift off real axis is $\eta =0.03$. Dashed line (RB): replacing $\Gamma$ by $M$ in Eq. (33) and using bubble with renormalized magnon energies $E_{\mathbf{k}}^r$ to $O\left(1/S\right)$ obtained from Eqs. (10)–(12) of Ref. 17. Solid line (RBV): using dressed vertex $\Gamma$ obtained from Eq. (30) in Eq. (33) and renormalized magnon energies $E_{\mathbf{k}}^r$. The absolute scale of $I\left(\Omega \right)$ is set to unity, but the relative scale of RB and RBV is kept.