Probing competing and intertwined orders with resonant inelastic x-ray scattering in the hole-doped cuprates

We develop a formalism to study indirect resonant inelastic x-ray scattering (RIXS) in systems with itinerant electrons, accounting for the attraction between valence electrons and the positively charged core hole exactly, and apply this formalism to the hole-doped cuprate superconductors. We focus on the relationship between RIXS line shapes and band structure, including broken symmetries. We show that RIXS is capable of distinguishing between competing order parameters, establishing it as a useful probe of the pseudogap phase.

Figure 1

Indirect RIXS intensity (arbitrary units) versus momentum over the entire Brillouin zone for underdoped Bi-2212 described by Hamiltonians $H_0$ (upper left) and mean-field Hamiltonians (clockwise from upper right): $H_0+H_{\mathrm{CDW}},H_0+H_{\mathrm{DDW}}$, and $H_0+H_{\mathrm{AF}}$, with mean-field perturbations of amplitude $V_{\mathrm{CDW},\mathrm{DDW},\mathrm{AF}}=100$ meV. Spectra are measured at energy transfer $\mathrm{\Delta }\omega =100$ meV and incident photon energy $\omega$ at the $4p$ threshold. Here and elsewhere in this paper black denotes zero intensity and bright red denotes maximal intensity.

Figure 2

Fourier transforms of RIXS intensities from Fig. 1, exhibiting clear period-4 periodicity of the CDW system and checkerboard periodicity in the DDW and AF systems.

Figure 3

One-dimensional plots of intensity along antinodal momenta for fixed $\mathrm{\Delta }\omega$ and various types of order.

Figure 4

Comparison of exact [Eqs. (2) and (5)] and approximate [Eq. (10)] formulas for momentum-dependent indirect RIXS intensity of an unordered system for doping $p=0.15$ and energy transfer $\mathrm{\Delta }\omega =100$ meV.

Figure 5

As Fig. 1 but with $\mathrm{\Delta }\omega$ integrated from 100 meV to 500 meV. Clockwise from top left: unordered state, CDW, AF, DDW.

Figure 6

Change in first moments integrated over $100\le \mathrm{\Delta }\omega \le$ 500 meV relative to unordered first moments as function of momenta across the entire Brillouin zone.