Using RIXS to Uncover Elementary Charge and Spin Excitations

Despite significant progress in resonant inelastic x-ray scattering (RIXS) experiments on cuprates at the Cu $L$-edge, a theoretical understanding of the cross section remains incomplete in terms of elementary excitations and the connection to both charge and spin structure factors. Here, we use state-of-the-art, unbiased numerical calculations to study the low-energy excitations probed by RIXS in the Hubbard model, relevant to the cuprates. The results highlight the importance of scattering geometry, in particular, both the incident and scattered x-ray photon polarization, and they demonstrate that on a qualitative level the RIXS spectral shape in the cross-polarized channel approximates that of the spin dynamical structure factor. However, in the parallel-polarized channel, the complexity of the RIXS process beyond a simple two-particle response complicates the analysis and demonstrates that approximations and expansions that attempt to relate RIXS to less complex correlation functions cannot reproduce the full diversity of RIXS spectral features.

Popular Summary

X rays are ideal probes of microscopic properties in solid-state materials since the wavelength of this radiation is comparable to that of atomic spacing. X-ray scattering techniques have been used extensively to measure crystal structures, electronic excitations, and lattice elementary excitations. Tuning the incident x-ray energy to an atomic absorption edge results in resonance that boosts the scattered intensity, and this so-called resonant inelastic x-ray scattering (RIXS) technique probes fundamentally different excitations since the resonantly excited intermediate state creates new excitation pathways compared with nonresonant processes. A theoretical understanding of the properties of excitations remains far from complete, especially for strongly correlated materials. Here, we use state-of-the-art, unbiased numerical calculations to systematically study the RIXS cross section in a cuprate, which is an example of a high-temperature superconducting material.

Our simulations using a common experimental scattering geometry demonstrate the importance of both incident and scattered photon polarizations for characterizing elementary spin and charge excitations. With crossed incident and scattered photon polarizations, the RIXS cross section qualitatively approximates the two-particle spin response; with parallel polarizations, the RIXS cross section captures charge excitations. We show that less-complex two-particle correlation functions are unable to reproduce the full diversity of RIXS spectral features, especially for charge excitations. These results highlight the necessity of using RIXS to study excitations in a range of materials (e.g., iron pnictide superconductors).

We expect that our findings will motivate future experimental studies using RIXS to probe spin and charge excitations in a variety of interesting correlated materials.

Figure 1

A schematic that illustrates the RIXS process at the Cu $L$-edge appropriate for cuprates.

Figure 2

Exact RIXS cross sections and approximations calculated using exact diagonalization. Top (bottom) panels show spectra for the $\pi -\sigma$ ($\pi -\pi$) polarization geometries. Left, middle, and right panels show RIXS spectra calculated for $n=0.83$ (“hole-doping”), $n=1$ (“half-filling”), and $n=1.17$ (“electron-doping”) electron fillings, respectively. The spectra are summed over all momenta along the nodal direction and along the $k_y=0$ direction, which are accessible in RIXS on a 12-site cluster, weighted with the RIXS form factors, i.e., $S(\omega )=\sum _{\mathbf{q}}|W_{\pi -\sigma }{|}^{2}S(\mathbf{q},\omega )$, $\tilde{N}(\omega )=\sum _{\mathbf{q}}|W_{\pi -\pi }{|}^2\tilde{N}(\mathbf{q},\omega )$ and similarly for $S^{\prime }(\omega )$ and $\widetilde{N^{\prime }}(\omega )$. The elastic response has been removed in each panel.

Figure 3

Momentum and energy integrated spectral weight for full RIXS ${\int }_0^{1.6t}I(\omega )d\omega$ and the various approximations $S={\int }_0^{1.6t}S(\omega )d\omega$, $\tilde{N}={\int }_0^{1.6t}\tilde{N}(\omega )d\omega$, $S^{\prime }={\int }_0^{1.6t}{S}^{\prime }(\omega )d\omega$, $\widetilde{N^{\prime }}={\int }_0^{1.6t}\widetilde{N^{\prime }}(\omega )d\omega$ with $S(\omega )$, $\tilde{N}(\omega )$, $S^{\prime }(\omega )$, $\widetilde{N^{\prime }}(\omega )$ defined as in the caption of Fig. 2, RIXS spectra. The $\pi -\sigma$ ($\pi -\pi$) relative polarizations are shown on the left (right) panels, respectively. Note that the spectral weight $\tilde{N}=5\times {10}^{-5}(\mathrm{eV}{)}^{-2}$ in the half-filled case does not appear in the figure. All momenta along the nodal direction and along the $q_y=0$ direction accessible on a 12-site cluster are taken into account (elastic response excluded).

Figure 4

Cross sections at $\mathbf{q}=(2\pi /3,0)$ with the elastic response removed. Top (bottom) panels show spectra for the $\pi \text{−}\sigma$ ($\pi \text{−}\pi$) polarization. Left, middle, and right panels show RIXS spectra for $n=0.83$, $n=1$, and $n=1.17$ electron fillings, respectively. The approximate spectra are weighted with the RIXS form factors, i.e., $S(\mathbf{q},\omega )\to |W_{\pi \text{−}\sigma }{|}^{2}S(\mathbf{q},\omega )$, $\tilde{N}(\mathbf{q},\omega )\to |W_{\pi \text{−}\pi }{|}^2\tilde{N}(\mathbf{q},\omega )$ and similarly for $S^{\prime }(\mathbf{q},\omega )$ and $\widetilde{N^{\prime }}(\mathbf{q},\omega )$.

Figure 5

Cross sections at the $\mathbf{q}=(\pi /2,\pi /2)$ point with the elastic response removed. Top (bottom) panels show spectra for the $\pi \text{−}\sigma$ ($\pi \text{−}\pi$) polarization. Left, middle, and right panels show RIXS spectra calculated for $n=0.83$, $n=1$, and $n=1.17$ electron fillings, respectively. The approximate spectra are weighted with the RIXS form factors, i.e., $S(\mathbf{q},\omega )\to |W_{\pi \text{−}\sigma }{|}^{2}S(\mathbf{q},\omega )$, $\tilde{N}(\mathbf{q},\omega )\to |W_{\pi \text{−}\pi }{|}^2\tilde{N}(\mathbf{q},\omega )$ and similarly for $S^{\prime }(\mathbf{q},\omega )$ and $\widetilde{N^{\prime }}(\mathbf{q},\omega )$.

Figure 6

Comparison between $S^{\prime }$ and three-spin Green’s functions $S_3$ and between $\widetilde{N^{\prime }}$ and two-spin Green’s functions $S_2$ calculated using exact diagonalization and following Eqs. (8) and (11) from the main text of the paper and Eqs. (b4) and (b5). The top (bottom) panels show spectra for the $\pi -\sigma$ ($\pi -\pi$) polarization setups (see main text of the paper for further details). Left, middle, and right panels show RIXS spectra calculated for $n=0.83$, $n=1$, and $n=1.17$ electron fillings, respectively. The spectra are summed over all momenta along the nodal direction and along the $k_y=0$ direction that are accessible in RIXS and on a 12-site cluster used in the exact diagonalization calculations and weighted with the RIXS form factors, i.e., $S^{\prime }(\omega )=\sum _{\mathbf{q}}|W_{\pi -\sigma }{|}^2{S}^{\prime }(\mathbf{q},\omega )$, $\widetilde{N^{\prime }}(\omega )=\sum _{\mathbf{q}}|W_{\pi -\pi }{|}^2\widetilde{N^{\prime }}(\mathbf{q},\omega )$, $S_2(\omega )=\sum _{\mathbf{q}}|W_{\pi -\pi }{|}^2{S}_2(\mathbf{q},\omega )$, and $S_3(\omega )=\sum _{\mathbf{q}}|W_{\pi -\sigma }{|}^2{S}_3(\mathbf{q},\omega )$. The intensity scale is different on each panel (it is chosen in such a way that each of the six spectra can be quite visible), and the elastic response has been removed.

Figure 7

Comparison between the Cu $L$-edge XAS spectrum calculated using exact diagonalization on a single site (top three panels) and on a 12-site cluster with both hopping $t$ and $t^{\prime }$ taken into account in the Hubbard model (bottom three panels) for hole-doped, half-filling, and electron-doped cases (see text for more details). In this paper, RIXS is always calculated for the resonance corresponding to the XAS transition from the $2p^{0}3d^{*1}$ to the $2p^{1}3d^{*0}$ state and the $L_3$ edge (923 eV peak in the above spectra).

Figure 8

Exact RIXS cross section calculated using exact diagonalization for $n=1$ (half filling) and for three different values of the core-hole potential: $U_c=2t$, $4t$, and $8t$ [the model, its parameters (except for $U_c$), and the exact diagonalization method are the same as in the main text]. The left (right) panels show the RIXS spectra for the $\pi -\sigma$ ($\pi -\pi$) relative polarization. The top, middle, and bottom panels show spectra calculated at $\mathbf{q}=(\pi /2,\pi /2)$, $\mathbf{q}=(2\pi /3,0)$, and integrated over all momenta along the nodal direction and along the $q_y=0$ direction accessible on a 12-site cluster (“$\mathbf{q}$ integrated”). Elastic response is excluded, and each RIXS spectrum is calculated at the incident energy corresponding to the main (“$L_3$”) XAS peak at half filling.

Figure 9

A cartoon picture of the dominant Cu $L$-edge RIXS process in the cross-polarized “channel.”

Figure 10

A cartoon picture of the dominant Cu $L$-edge RIXS process in the parallel-polarized channel.