Momentum-dependent light scattering in a two-dimensional Heisenberg antiferromagnet: Analysis of x-ray scattering data

Motivated by the achievements of the x-ray scattering technique, we analyzed the profile of the light scattering cross section $R(q,\omega )$ at a finite $q$ in a 2D Heisenberg antiferromagnet. Previous Raman scattering studies at $q=0$ identified the two-magnon peak in $B_{1g}$ scattering geometry. We found that the $B_{1g}$ peak disperses downwards at a finite $q$, and its cross section increases, reaching its maximum at $q=q_0=(0,\pi )$ and symmetry related points. In addition, the cross-section in the $A_{1g}$ geometry becomes nonzero at a finite $q$, and also displays a two-magnon peak which gains strength and disperses to larger frequencies with increasing $q$, and reaches its highest cross section at $q_0$. We found that the profile of $R(q_0,\omega )$ is equivalent in $A_{1g}$ and $B_{1g}$ geometries.

Figure 1

The diagrams for the Raman matrix element $M_R(k,q)$. Solid and dashed lines represent conduction and valence electrons. The dashed-dotted lines represent incoming and outgoing photons, and the solid wavy lines represent magnons. The full set includes an extra diagram (not shown) which cancels out a parasitic contribution from (b), see Ref. 5

Figure 2

(Color online) The scattering cross section in the $A_{1g}$ geometry along $\Gamma -M-X-\Gamma$ direction in momentum space, where $\Gamma =(0,0)$, $M=(0,\pi )$, and $X=(\pi ,\pi )$. Top panel: without final state magnon-magnon interaction. Middle panel: with the final state interaction but assuming that magnon-magnon interaction factorizes between $A_{1g}$ and $B_{1g}$ channels. Lower panel: the full cross section. Observe that the nonfactorization of the interaction (due to nonorthogonality of $M_R^{A_{1g}}$ and $M_R^{B_{1g}}$ at a nonzero $q$) does not much affect the cross-section profile. The largest cross section for the full $\mathrm{Im}{R}^{A_{1g}}(q,\omega )$ is at $q_0=(0,\pi )$.

Figure 3

(Color online) Same as in Fig. 2, but for $B_{1g}$ geometry. The cross section for the full $\mathrm{Im}{R}^{B_{1g}}(q,\omega )$ is again the largest for $q_0=(0,\pi )$.

Figure 4

(Color online) Top panel: the scattering intensities at $q_0=(0,\pi )$ without final state interaction (dashed line), with final state interaction, neglecting the mixing between $A_{1g}$ and $B_{1g}$ channels (solid line), and including the mixing (dashed-dotted line). Note that (i) due to a final state interaction, the two-magnon peak gets sharper, becomes more symmetric and shifts to a lower frequency, and (ii) the effect of the mixing is very minor. Bottom panel: 2D constant energy contours for the total energy of two magnons $\Omega (k+q_0∕2)+\Omega (-k+q_0∕2)$. The extended van Hove singularity is at $\Omega (k+q_0∕2)+\Omega (-k+q_0∕2)=2\sqrt{3}J$.