Resonant Inelastic X-Ray Scattering Response of the Kitaev Honeycomb Model

We calculate the resonant inelastic x-ray scattering (RIXS) response of the Kitaev honeycomb model, an exactly solvable quantum-spin-liquid model with fractionalized Majorana and flux excitations. We find that the fundamental RIXS channels, the spin-conserving (SC) and the non-spin-conserving (NSC) ones, do not interfere and give completely different responses. SC RIXS picks up exclusively the Majorana sector with a pronounced momentum dispersion, whereas NSC RIXS also creates immobile fluxes, thereby rendering the response only weakly momentum dependent, as in the spin structure factor measured by inelastic neutron scattering. RIXS can, therefore, pick up the fractionalized excitations of the Kitaev spin liquid separately, making it a sensitive probe to detect spin-liquid character in potential material incarnations of the Kitaev honeycomb model.

Figure 1

(a) Illustration of the honeycomb lattice. Sites in sublattice $A$ ($B$) are marked by white (black) circles, while $x$, $y$, and $z$ bonds are marked by dotted, dashed, and solid lines, respectively. (b) Flux excitations $\varphi$ around the photon-scattering site $\mathbf{r}$ (white circle) in the final states $|m⟩$ of the three fundamental NSC RIXS channels with amplitudes $\propto ⟨m|{\sigma }_{\mathbf{r}}^{x,y,z}{e}^{-it\tilde{H}(\mathbf{r})}|0⟩$ [see Eq. (3)], respectively. (c) Illustration of the standard Brillouin zone (BZ), and the extended Brillouin zone (EBZ) with respect to which the RIXS response is periodic. The SC RIXS response in Fig. 3 is plotted at special points (black squares) [4] and generic representative points (red squares).

Figure 2

Momentum-integrated RIXS intensities at (a) the isotropic point of the gapless phase [$J_{x,y,z}=J_0$] and (b) a representative point in the gapped phase [$J_{x,y}=J_0/2$ and $J_z=2J_0$]. The SC response $I_0(\omega )$ is plotted by a solid line, while the NSC responses $I_{x,y,z}(\omega )$ are plotted by dashed and/or dotted lines. Each response is normalized such that $\int d\omega I_{\eta }(\omega )=1$. At the gapped point, the NSC response $I_z(\omega )$ has a delta peak at low energies and is multiplied by 10 to be comparable with the other responses.

Figure 3

Momentum-resolved SC RIXS intensities at the isotropic point of the gapless phase [$J_{x,y,z}=J_0$] (a)–(c) and at a representative point in the gapped phase [$J_{x,y}=J_0/2$ and $J_z=2J_0$] (d)–(f). The response is plotted along the entire ${\tilde{\mathit{M}}}_z\text{−}{\mathit{K}}_z\text{−}\mathrm{\Gamma }\text{−}{\mathit{M}}_z\text{−}\tilde{\mathit{K}}\text{−}{\tilde{\mathit{M}}}_z$ cut (a) and (d), at specific points on the $\mathrm{\Gamma }\text{−}{\mathit{K}}_z\text{−}{\tilde{\mathit{M}}}_z$ cut (b) and (e), and at specific points on the $\mathrm{\Gamma }\text{−}{\mathit{M}}_z$ cut (c) and (f) [see Fig. 1]. The intensity is normalized such that $\int d\omega \int d^2\mathbf{q}{I}_0(\omega ,\mathbf{q})=1$, where $\mathbf{q}$ is measured in units of $a^{-1}\equiv |{\widehat{\mathbf{r}}}_{x,y,z}{|}^{-1}$ and is integrated over the entire EBZ. Some responses are multiplied by numerical factors (written next to them) to be comparable with other responses.