Ab initio procedure for constructing effective models of correlated materials with entangled band structure

In a previous work [Phys. Rev. B 77, 085122 (2008)], a procedure for constructing low-energy models of electrons in solids was proposed. The procedure starts with dividing the Hilbert space into two subspaces: the low-energy part (“$d$ space”) and the rest of the space (“$r$ space”). The low-energy model is constructed for the $d$ space by eliminating the degrees of freedom of the $r$ space. The thus derived model contains the strength of electron correlation expressed by a partially screened Coulomb interaction, calculated in the constrained random-phase approximation (cRPA), where screening channels within the $d$ space, $P_d$, are subtracted. One conceptual problem of this established downfolding method is that for entangled bands it is not clear how to cut out the $d$ space and how to distinguish $P_d$ from the total polarization. Here, we propose a simple procedure to overcome this difficulty. The $d$ space is defined to be an isolated set of bands generated from a set of maximally localized Wannier basis, which consequently defines $P_d$. The $r$ subspace is constructed as the complementary space orthogonal to the $d$ subspace, resulting in two sets of completely disentangled bands. Using these disentangled bands, the effective parameters of the $d$ space are uniquely determined by the cRPA method. The method is successfully applied to $3d$ transition metals.

Figure 1

(Color) (a) Kohn-Sham band structure of nickel in the LDA. (b) Disentangled band structure with $d\text{−}r$ hybridization switched off. The red lines show the $d$ states obtained by the maximally localized Wannier scheme, while the blue lines are disentangled $r$ states. Energy is measured from the Fermi level.

Figure 2

(Color online) Fully screened Coulomb interaction of nickel as a function of frequency. The average of the diagonal terms in the Wannier basis is plotted. The present scheme using the disentangled bands is compared to the results obtained from the original band structure.

Figure 3

(Color online) Hubbard $U$ of nickel as a function of frequency. The diagonal term of the partially screened interaction in the Wannier basis is calculated by the present method and compared with the published data of Ref. 13.

Figure 4

(Color online) Static values of (a) fully screened Coulomb interaction $W$ and (b) Hubbard $U$ for $3d$ metals. We emphasize that there is no contradiction between the present and previous results. The difference arises from the choice of energy windows or the choice of bands defining the model Hamiltonian.