Charge ordering in three-band models of the cuprates

We examine trends in the wave vectors and form factors of charge density wave instabilities of three-band models of the underdoped cuprates. For instabilities from a high-temperature state with a large Fermi surface, we extend a study by Bulut et al. [Phys. Rev. B 88, 155132 (2013)] to include a direct antiferromagnetic exchange coupling between the Cu sites. As in previous work, we invariably find that the primary instability has a diagonal wave vector $(\pm Q_0,\pm Q_0)$ and a $d$-form factor. The experimentally observed wave vectors along the principal axes $(\pm Q_0,0)$, $(0,\pm Q_0)$ have higher energy, but they also have a predominantly $d$-form factor. Next, we gap out the Fermi surface in the antinodal regions of the Brillouin zone by including static, long-range antiferromagnetic order at the wave vector $(\pi ,\pi )$: this is a simple model of the pseudogap in which we assume the antiferromagnetic order averages to zero by “renormalized classical” thermal fluctuations in its orientation, valid when the antiferromagnetic correlation length is large. The charge density wave instabilities of this pseudogap state are found to have the optimal wave vector $(\pm Q_0,0)$, $(0,\pm Q_0)$, with the magnitude of the $d$-form factor decreasing with increasing magnetic order.

Figure 1

(a) Diagram of the unit cell of the lattice showing the hopping amplitudes corresponding to the tight-binding Hamiltonian in Eq. (3). The onsite energies ${\epsilon }_d$ and ${\epsilon }_p$ and the interorbital hopping parameters $t_{pd}$ and $t_{pp}$ are displayed next to the corresponding orbitals and bonds, respectively. Note how the sign of $t_{pd}$ changes depending on the relative phases of the wave-function lobes closest to one another. (b) Diagram showing the interactions of the Hamiltonian within the unit cell and between two nearest-neighbor copper atoms. The corresponding expressions are given in Eqs. (7) and (8).

Figure 2

Feynman diagrams used in the $T$-matrix calculation discussed in Sec. 2a. (a), (b) give the exchange and direct interactions, respectively. Together, they compose the bare vertex shown in (c) as per Eq. (15). (d) shows the full interaction vertex, which is determined by the Bethe-Salpeter equation given diagrammatically in (e). Further details are presented in the main text.

Figure 3

Plot of the minimum eigenvalue of the matrix $A_{lm}\left(\mathbf{Q}\right)$ in Eq. (23) for $\mathbf{Q}$ in the first quadrant at different values of $J$ and $V_{pp}$. The temperature is $T=0.015$ and the filling $p=5-n=0.1643$. All other parameters are as given in Table 1. The diagonal point $\mathbf{Q}=Q_m(1,1)$ for $Q_m=1.19381$ is very consistently the point of greatest instability.

Figure 4

Real-space representation of hopping amplitudes for diagonal $\mathbf{Q}=Q_m(1,1)$ evaluated with $J=0.5$, $V_{pp}=1.25$, and $V_{pd}$ at $T=0.015$ and $p=0.1643$. For clarity, the lattice has been divided into two separate pictures. (a) and (c) display the onsite copper, the onsite oxygen, and the copper-copper hopping amplitudes whereas (b) and (d) give the $dx$, $dy$, and $xy$ bond amplitudes. (a) and (b) plot the full functions given in Table 5 while (c) and (d) simply display the $\mathbf{r}=0$ part. (c) and (d) also indicate which orbitals and bonds the symbols represent. The modulations of the order parameter in the diagonal direction can be seen in (a) and (b) by tracking the colors of the O $p_x$, O $p_y$ orbitals and the $xy$ bonds, respectively, over several unit cells. The (c) and (d) pictures make the $d$-form factor evident.

Figure 5

Just as in Fig. 4. Real-space representation of hopping amplitudes for axial $\mathbf{Q}=Q_m(1,0)$ evaluated with $J=0.5$, $V_{pp}=1.25$, and $V_{pd}$ at $T=0.015$ and $p=0.1643$.

Figure 6

Spectral functions and minimum eigenvalues for $M_d=0.73$, 1.60, and 3.24 (which corresponds to $U_d+2J=3.25$, 5.0, and 8.0, respectively). The chemical potential is adjusted so that $p\sim 0.10$, while $V_{pp}=1.5$ and $V_{pd}=1.0$. The second through fourth columns are for $J=0.0,1.0,$ and 1.5, respectively. Note that the minimum eigenvalues are mostly at $(Q_2,0)$ with $Q_2\approx 2\pi /3$; in some cases there are also well-formed minima at $(Q_1,0)$ with $Q_1\approx \pi /3$.

Figure 7

The sum of the squares of the $S=0$ components of the original eigenvectors corresponding to the minimum eigenvalues at $(Q_1,0)$ and $(Q_2,0)$. That is, we plot ${\sum }_{l=1}^{23}{\left|{\mathcal{P}}_l\left(\mathbf{Q}\right)\right|}^2$ at $\mathbf{Q}=(Q_{1,2},0)$ where $\left\{{\mathcal{P}}_l\left(\mathbf{Q}\right)\right\}$ are the components corresponding to the 46 basis functions given in Tables 6 and 7. We subsequently project out the $S=1$ components ($l=24\text{–}46$) and normalize.

Figure 8

$S=0$ form-factor dependence at $(Q_1,0)$ and $(Q_2,0)$, as a function of magnetization $M_d$ (from which $U_d+2J$ is determined). These values are determined by projecting out the $S=1$ components of the eigenvectors and subsequently normalizing. Only the most important contributions are displayed. On the first row, the legend is shown. In descending order, the remaining rows show the results for $J=0.0,1.0$, and $1.5$. The other interaction parameters are given in Table 1 and the chemical potential is chosen so that $p\sim 0.10$.

Figure 9

Real-space representation of hopping amplitudes. The signs written in the $dx$, $dy$, and $xy$ bonds in (b) and (c) indicate which sign the hopping term in the Hamiltonian corresponding to that bond has. The amplitudes given in Table 5 have been multiplied by this phase.