Theory of the waterfall phenomenon in cuprate superconductors

Based on exact diagonalization and variational cluster approximation calculations we study the relationship between charge transfer models and the corresponding single band Hubbard models. We present an explanation for the waterfall phenomenon observed in angle resolved photoemission spectroscopy on cuprate superconductors. The phenomenon is due to the destructive interference between the phases of the O2$p$ orbitals belonging to a given Zhang-Rice singlet and the Bloch phases of the photohole which occurs in certain regions of $\mathbf{k}$ space. It therefore may be viewed as a direct experimental visualization of the Zhang-Rice construction of an effective single band model for the CuO${}_2$ plane.

Figure 1

Possible realization of the model (1).

Figure 2

Single-particle spectral function for the model (1) for the half-filled system (eight holes in eight unit cells). The figure combines spectra calculated with different boundary conditions to give an impression of larger systems. The left part of the spectra corresponds to hole creation (i.e., photoemission) the right part to hole annihilation (inverse photoemission). The upper figure shows the spectra for $U=0$, the lower one the spectra for $U=8$.

Figure 3

Spectral weight of the $p$-like band as a function of momentum. The weight is obtained by integrating the spectral weight over a finite window $\delta$ around the center of gravity of the band. The result for $U=0$ is also shown.

Figure 4

Top: Closeup of the photoemission part of the spectrum in Fig. 2. Bottom: Photoemission spectrum for a half-filled single-band Hubbard chain with eight sites and $U^{\text{'}}/t^{\text{'}}=10.8$ and $t^{\text{'}}=0.335\hspace{0.5em}t$. The single-band spectrum has been shifted by $0.87\hspace{0.5em}t$. The figure combines spectra calculated with periodic and antiperiodic boundary conditions.

Figure 5

Closeup of the photoemission part of the spectrum in Fig. 2. The left panel shows the $p$-like spectrum, the right panel the $d$-like spectrum.

Figure 6

Closeup of the photoemission spectrum in Fig. 2 (top) compared to the spectra of a single-band Hubbard model corrected according to Eq. (8) (bottom). The energy shift $\epsilon$ is $0.87\hspace{0.5em}t$ rather than $1.04\hspace{0.5em}t$ as would be obtained from the single-plaquette calculation.

Figure 7

Low-energy spectral function of the hole-doped two-band model (top) compared to the spectrum of the corresponding single-band model (bottom). In this figure, the chemical potential is the zero of energy, the part to the left (right) of the chemical potential corresponds to photoemission (inverse photoemission).

Figure 8

Top: Sum of $p$-like and $d$-like spectrum for the three-band model (9) with $U=0$. Center: $d$-like spectrum for $U=8$ as obtained by the VCA. The spectrum has been multiplied by $2.5$. Bottom: $p$-like weight for $U=8$. The spectra are computed for half-filling, that means one hole/unit cell, the black line denotes the respective chemical potential.

Figure 9

Right part: $d$-like spectrum of the three-band model (top) compared to the spectrum of the single band Hubbard model multiplied by $0.5$ (bottom). Left part: $p$-like spectral weight of the three-band model multiplied by $2$ (top) compared to the spectrum of the single band Hubbard model multiplied by $0.5\hspace{0.5em}f(\mathbf{k})$ in Eq. (15) (bottom).

Figure 10

Left: Spectral function of the the single band Hubbard model with $U=10$, $t=1$.Right: Spectral function of the the single band Hubbard model multiplied by $f(\mathbf{k})$ [see Eq. (15)].