Exact diagonalization results for resonant inelastic x-ray scattering spectra of one-dimensional Mott insulators

We examine the momentum-dependent excitation spectra of indirect as well as direct resonant inelastic x-ray scattering (RIXS) processes in half-filled (extended) Hubbard rings. We determine the fundamental features of the ground-state RIXS response and discuss the experimental conditions that can allow for the low-energy part of these features to be distinguished in one-dimensional copper-oxide materials, focusing particularly on the different magnetic excitations occurring in indirect and direct RIXS processes. We study the dependence of spin and charge excitations on the choice of and detuning from resonance. Moreover, final-state excitation weights are calculated as a function of the core-hole potential strength and lifetime. We show that these results can be used to determine material characteristics, such as the core-hole properties, from RIXS measurements.

Figure 1

Illustration of RIXS processes in half-filled copper oxides. (a) Indirect RIXS process. (b) Direct RIXS process without spin flip. (c) Direct RIXS process with spin flip. For a spin flip to occur, the core-hole spin must also be flipped (dotted arrow).

Figure 2

Overview of magnetic excitations in calculated RIXS spectra for the Hubbard model on a half-filled, 14-site ring. The parameters are $U/t=10$, $V_c/t=15$, ${\omega }_{\mathrm{in}}/t=-19$, $\Gamma /t=1$, unless otherwise noted. The peaks are Lorentzians of width $0.1t$. Note that RIXS spectra in (a) and (c) are rescaled by the factors in parentheses. First row: (a) Indirect RIXS spectra for input energies corresponding to well-screened (solid line) and poorly screened (dashed line) intermediate states. (b) $\Delta S=0$ direct RIXS spectrum (solid line) compared to indirect RIXS (dashed line). (c) $\Delta S=1$ direct RIXS spectrum (solid line) compared to $\Delta S=0$ direct RIXS (dashed line). Second row: Spin DSF for the same system (solid line) compared to $\Delta S=1$ direct RIXS spectrum (dashed line) for (d) $\Gamma /t=0.2$, (e) $\Gamma /t=0.5$, and (f) “instant” ($\Gamma /t=\infty$) RIXS process.

Figure 3

Total spin-excitation weight, summed over all inequivalent momenta $q$ in the first Brillouin zone, as a function of ${\omega }_{\mathrm{in}}$ in (a) indirect, (b) $\Delta S=0$, and (c) $\Delta S=1$ direct RIXS, respectively, for the same system as in Fig. 2. The thick (thin) solid lines are the corresponding XAS spectra with ${\Gamma }_{\mathrm{XAS}}$ set to $0.5t$ ($0.01t$). The arrows indicate the XAS peaks at which the magnetic RIXS signal is maximal.

Figure 4

Total spin-excitation weight percentage as a function of the core-hole potential in (a) indirect RIXS, (b) $\Delta S=0$ direct RIXS, and (c) $\Delta S=1$ direct RIXS for Hubbard (empty symbols) and extended Hubbard (full symbols) models, for three values of the inverse lifetime $\Gamma$. The value for the detuning ${\omega }_{\mathrm{in}}$ of the incoming photon energy from resonance at each point is defined by the procedure outlined in the text. All other relevant parameters are as in Fig. 2. (d) Core-hole lifetime dependence of magnetic excitations in all RIXS processes for $V_c=15t$.