Accuracy of downfolding based on the constrained random-phase approximation

We study the reliability of the constrained random-phase approximation (cRPA) method for the calculation of low-energy effective Hamiltonians by considering multiorbital lattice models with one strongly correlated “target” band and two weakly correlated “screening” bands. The full multiorbital system and the effective model are solved within dynamical mean-field theory (DMFT) in a consistent way. By comparing the quasiparticle weights for the correlated bands, we examine how accurately the effective model describes the low-energy properties of the multiband system. We show that the violation of the Pauli principle in the cRPA method leads to overscreening effects when the interorbital interaction is small. This problem can be overcome by using a variant of the cRPA method which restores the Pauli principle.

Figure 1

Three-orbital model on a cubic lattice. The three levels are split by orbital dependent on-site energies. We include an orbital off-diagonal transfer $t^{\prime }$, but the highest and lowest orbitals are not connected by a hopping term. The on-site repulsion for the target orbital is denoted by $U_d$, while the highest and lowest orbitals are less correlated with an on-site repulsion of $U_d/2$. The interorbital repulsion $U^{\prime }$ acts between all pairs of orbitals.

Figure 2

Noninteracting band structure of the three-orbital model for $\mathrm{\Delta }=10$ and $t^{\prime }=4$. The half-filled target band is shown in red.

Figure 3

Schematic band structure. The solid line denotes the low-energy band in the target manifold of the low-energy model, while screening bands are denoted by broken lines. Solid and broken arrows show possible contributions to the polarization: excitations between (1) occupied screening bands and unoccupied screening bands, (2) occupied screening bands and unoccupied target bands, (3) occupied target bands and unoccupied screening bands, (4) occupied target bands and unoccupied target bands. In the cRPA method, we exclude the contribution (4) because this should be taken into account in solving the low-energy effective model.

Figure 4

Screened interactions obtained by the charge-cRPA (a) and the spin-cRPA (b) for the three-orbital model ($\mathrm{\Delta }=10$ and $t^{\prime }=4$). We show the on-site interaction $U\left(\omega \right)$ and nearest-neighbor interaction $V_{\mathrm{nn}}\left(\omega \right)$. The bare on-site interaction $U(\omega =\infty )$ is represented by a dashed solid line. For the nearest-neighbor interactions computed by the spin-cRPA (b), the solid and broken lines represent the spin-diagonal element of the interaction $\left[V_{\mathrm{nn}}^{\uparrow \uparrow }(=V_{\mathrm{nn}}^{\downarrow \downarrow })\right]$ and the spin off-diagonal one $(V_{\mathrm{nn}}^{\uparrow \downarrow }$), respectively.

Figure 5

Strength of the screening effect for the on-site interaction $\mathrm{\Lambda }$ computed by the charge-cRPA (a) and spin-cRPA (b) methods for the three-orbital model. The definition of $\mathrm{\Lambda }$ is given in Eq. (62).

Figure 6

Quasiparticle weights computed for the three-orbital model with $\mathrm{\Delta }=10$ and $t^{\prime }=4$. We compare the results of the effective models downfolded by the charge-cRPA and spin-cRPA methods, the unscreened model, and the full model.

Figure 7

Comparison of the local Green's function for the three-orbital model with $\mathrm{\Delta }=10$ and $t^{\prime }=4 \left(\beta =15\right)$. The triangles, squares, crosses and filled circles denote the data obtained by solving the charge-cRPA model, the spin-cRPA model, the unscreened model, and the full model, respectively.

Figure 8

(a) Two-orbital model with a lower target band and an upper screening band. (b) First four diagrams in the charge-cRPA series for the two-orbital model. The red diagrams with an even number of interaction lines violate the Pauli principle.

Figure 9

Spectral functions projected on the band basis. The data were obtained by solving the three-orbital model with $\mathrm{\Delta }=10$ and $t^{\prime }=4$ at $\beta =15$. The solid thick lines denote the spectral function obtained by analytical continuation of the DMFT data. The Hartree-Fock and noninteracting band structures are shown by broken lines and thin gray lines, respectively.

Figure 10

Phase diagram of the three-orbital model for $\mathrm{\Delta }=10$ and $t^{\prime }=4$ at $\beta =15$. The solid lines denote the Mott-transition lines for the charge-cRPA (triangle), spin-cRPA (square), unscreened (cross), and full models (filled circle).

Figure 11

Single-particle excitations in the atomic limit: (a) the ground state, (b) the excited state with an additional hole in the lowest orbital, and (c) the excited state with an additional spin in the highest orbital. We assume that half-filled target orbitals are paramagnetic.