Inelastic X-Ray Scattering in Correlated Mott Insulators

We calculate the inelastic light scattering from x rays, which allows the photon to transfer both energy and momentum to the strongly correlated charge excitations. We find that the charge-transfer peak and the low-energy peak both broaden and disperse through the Brillouin zone similar to what is seen in experiments in materials such as $\mathrm{C}{\mathrm{a}}_{\mathrm{2}}\mathrm{C}\mathrm{u}{\mathrm{O}}_{\mathrm{2}}\mathrm{C}{\mathrm{l}}_{\mathrm{2}}$.

Figure 1

Experimental data for $\mathrm{C}{\mathrm{a}}_{\mathrm{2}}\mathrm{C}\mathrm{u}{\mathrm{O}}_{\mathrm{2}}\mathrm{C}{\mathrm{l}}_{\mathrm{2}}$ [3] with momentum transfers along the BZ diagonal (a) and BZ edge (b), respectively. The values of the parameter $X=[\cos (q_{x}a)+\cos (q_{y}a)]/2$ indicate the values of the momentum transfer $(q_x,q_y)$. Note that for panel (b) the momentum transfer runs from $(0,0)$ to $(\pi ,0)$ only, $X=1$ to $0$.

Figure 2

Coupled Dyson equations for the inelastic x-ray scattering correlation functions described by the scattering amplitude ${\gamma }_a$. Panel (a) depicts the equation for the interacting correlation function, while panel (b) is the supplemental equation needed to solve for the correlation function. The symbol $\Gamma$ stands for the local dynamical irreducible charge vertex given in Eq. (5). In situations where there are no charge vertex corrections, the correlation function is simply given by the bare-bubble diagram in (a).

Figure 3

Inelastic x-ray scattering response in the $B_{\mathrm{1}\mathrm{g}}$ channel along (a) the Brillouin zone diagonal and (b) along the zone edge for the half-filled Falicov-Kimball model on a hypercubic lattice. The solid, dotted, short-dashed, and long-dashed curves correspond to temperatures $T=0.1,$ 0.25 (partially obscured by the 0.1 line), 0.5, 1.0, respectively.

Figure 4

Inelastic x-ray scattering response in the $A_{\mathrm{1}\mathrm{g}}$ channel along (a) the Brillouin zone diagonal and (b) along the zone edge for the half-filled Falicov-Kimball model on a hypercubic lattice. The solid, dotted, short-dashed, and long-dashed curves correspond to temperatures $T=0.1,$ 0.25 (partially obscured by the 0.1 line), 0.5, 1.0, respectively.

Figure 5

Plots of the low-energy (panel a) and charge-transfer (panel b) peak positions (solid symbols) and broadening (full width at half maximum, open symbols) for $T=t^{*}$ determined from Figs. 3, 4 for $A_{1g},B_{1g}$ (circles, squares), respectively.