Position representation of effective electron-electron interactions in solids

An essential ingredient in many model Hamiltonians, such as the Hubbard model, is the effective electron-electron interaction $U$, which enters as matrix elements in some localized basis. These matrix elements provide the necessary information in the model, but the localized basis is incomplete for describing $U$. We present a systematic scheme for computing the manifestly basis-independent dynamical interaction in position representation, $U(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$, and its Fourier transform to time domain, $U(\mathbf{r},{\mathbf{r}}^{\prime };\tau )$. These functions can serve as an unbiased tool for the construction of model Hamiltonians. For illustration we apply the scheme within the constrained random-phase approximation to the cuprate parent compounds ${\mathrm{La}}_2{\mathrm{CuO}}_4$ and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$ within the commonly used one- and three-band models, and to nonsuperconducting ${\mathrm{SrVO}}_3$ within the $t_{2g}$ model. Our method is used to investigate the shape and strength of screening channels in the compounds. We show that the $\mathrm{O}2p_{x,y}-\mathrm{Cu}3d_{x^2-y^2}$ screening gives rise to regions with strong attractive static interaction in the minimal (one-band) model in both cuprates. On the other hand, in the minimal ($t_{2g}$) model of ${\mathrm{SrVO}}_3$ only regions with a minute attractive interaction are found. The temporal interaction exhibits generic damped oscillations in all compounds, and its time integral is shown to be the potential caused by inserting a frozen point charge at $\tau =0$. When studying the latter within the three-band model for the cuprates, short time intervals are found to produce a negative potential.

Figure 1

Qualitative illustration of the screened interaction $W=v+W^c$ and its constituents $v$ and $W^c$ in the static limit. (a) Shallow screening hole. (b) Deep screening hole.

Figure 2

Schematics outlining the generation of matrix elements of $\chi$ (green box) used for the computation of $W$ and $U$ in position representation (red box). $W\left[\delta \right], U\left[\delta \right]$ and $W\left[\mathrm{\Theta }\right], U\left[\mathrm{\Theta }\right]$ are defined in Sec. 3c.

Figure 3

Effective one- and three-band interactions, $U_1(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ (solid lines) and $U_3(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ (dashed lines), of the cuprates, and $t_{2g}$ (three-band) interaction of ${\mathrm{SrVO}}_3$ (black dashed lines) along different paths in the ${\mathrm{CuO}}_2$ and ${\mathrm{VO}}_2$ sheets, respectively. These paths are indicated in each graph.

Figure 4

Effective one-band interaction $U_1(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ of the cuprates in the ${\mathrm{CuO}}_2$ sheet.

Figure 5

Effective three-band interaction $U_3(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ of the cuprates and ${\mathrm{SrVO}}_3$ ($t_{2g}$) in the ${\mathrm{CuO}}_2$ and ${\mathrm{VO}}_2$ sheets, respectively.

Figure 6

$W(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ of the cuprates and ${\mathrm{SrVO}}_3$ along different paths in the ${\mathrm{CuO}}_2$ and ${\mathrm{VO}}_2$ sheets respectively. These paths are indicated in each graph.

Figure 7

$W(\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ of the cuprates and ${\mathrm{SrVO}}_3$ in the ${\mathrm{CuO}}_2$ and ${\mathrm{VO}}_2$ sheets, respectively.

Figure 8

$[W-U_1](\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ (solid lines) and $[U_1-U_3](\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ (dashed lines) in the cuprates along different paths in the ${\mathrm{CuO}}_2$ sheet, which are indicated in each graph.

Figure 9

$[U_1-U_3](\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ in the ${\mathrm{CuO}}_2$ sheet of the cuprates.

Figure 10

$[W-U_1](\mathbf{r},{\mathbf{r}}^{\prime };\omega =0)$ in the ${\mathrm{CuO}}_2$ sheet of the cuprates.

Figure 11

$W^{\mathrm{c}}(\mathbf{r},{\mathbf{r}}^{\prime };\omega )$ and $W(\mathbf{r},{\mathbf{r}}^{\prime };\tau )\left[\delta \right]$ of the cuprates and ${\mathrm{SrVO}}_3$ with $\mathbf{r}={\mathbf{r}}^{\prime }$ at the Cu/V nucleus.

Figure 12

$W(\mathbf{r},{\mathbf{r}}^{\prime };\tau )\left[\mathrm{\Theta }\right]$ of the cuprates with $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ at the same Cu nucleus as well as $U_3(\mathbf{r},{\mathbf{r}}^{\prime };\tau )\left[\mathrm{\Theta }\right]$ with $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$ at the same Cu nucleus or at neighboring Cu nuclei.

Figure 13

LDA Band structures ($\mu$ at 0) and crystal structure data of ${\mathrm{SrVO}}_3$, LCO, and HBCO. $\mathrm{\Gamma }=(0,0,0), X=(\pi /a,0,0)$, and $K=(\pi /a,\pi /a,0)$.