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arXiv:0906.1344 · Physical Review B

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

Takashi Miyake, F. Aryasetiawan, Masatoshi Imada

Published Oct 27, 2009• 214 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.80.155134

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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 (rrr space''). The low-energy model is constructed for the ddd space by eliminating the degrees of freedom of the rrr 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 ddd space, Pd{P}_{d}Pd​, are subtracted. One conceptual problem of this established downfolding method is that for entangled bands it is not clear how to cut out the ddd space and how to distinguish Pd{P}_{d}Pd​ from the total polarization. Here, we propose a simple procedure to overcome this difficulty. The ddd space is defined to be an isolated set of bands generated from a set of maximally localized Wannier basis, which consequently defines Pd{P}_{d}Pd​. The rrr subspace is constructed as the complementary space orthogonal to the ddd subspace, resulting in two sets of completely disentangled bands. Using these disentangled bands, the effective parameters of the ddd space are uniquely determined by the cRPA method. The method is successfully applied to 3d3d3d transition metals.
In correlated electron materials, the application of many-body techniques for the study of interaction effects or unconventional superconductivity often requires the formulation of an effective low-energy model that contains only the relevant bands near the Fermi level. However, the bands away from the Fermi level are known to renormalize the low-energy interactions substantially. Here we compare different schemes to derive low-energy effective theories for interacting electrons in solids. The frequently used constrained random phase approximation (cRPA) is identified as a particular resummation of higher-order interaction terms that includes important virtual corrections. We then propose an adapted functional renormalization group scheme that includes the cPRA, but also allows one to go beyond the cRPA approximation. We study a simple two-band model in order to explore the differences between the approximations.
In this chapter, we discuss how low-energy models can be derived from first-principle calculations. The main quantity to be determined is the effective Coulomb interaction between the electrons in the low-energy subspace, the so-called Hubbard U. We begin by giving a brief historical introduction to different ways to determine U. However, the main focus in the chapter lies on the constrained random-phase approximation (cRPA), which is a method to compute U based on the concept of partial screening. At the end of the chapter, we put the cRPA in a broader perspective by introducing the multi-tier approach, which is a general approach for downfolding that naturally encompasses the cRPA in the first step.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
Recent trends of ab initio studies and progress in methodologies for electronic structure calculations of strongly correlated electron systems are discussed. The interest for developing efficient methods is motivated by recent discoveries and characterizations of strongly correlated electron materials and by requirements for understanding mechanisms of intriguing phenomena beyond a single-particle picture. A three-stage scheme is developed as renormalized multi-scale solvers (RMS) utilizing the hierarchical electronic structure in the energy space. It provides us with an ab initio downfolding of the global band structure into low-energy effective models followed by low-energy solvers for the models. The RMS method is illustrated with examples of several materials. In particular, we overview cases such as dynamics of semiconductors, transition metals and its compounds including iron-based superconductors and perovskite oxides, as well as organic conductors of kappa-ET type.
We check the accuracy of the constrained random phase approximation (cRPA) downfolding scheme by considering one-dimensional two- and three-orbital Hubbard models with a target band at the Fermi level and one or two screening bands away from the Fermi level. Using numerically exact quantum Monte Carlo simulations of the full and downfolded model, we demonstrate that depending on filling, the effective interaction in the low-energy theory is either barely screened or antiscreened, in contrast to the cRPA prediction. This observation is explained by a functional renormalization group analysis, which shows that the cRPA contribution to the screening is to a large extent canceled by other diagrams in the direct particle-hole channel. We comment on the implications of this finding for the ab initio estimation of interaction parameters in low-energy descriptions of solids.
Starting from the full many-body Hamiltonian of interacting electrons the effective self-energy acting on electrons residing in a subspace of the full Hilbert space is derived. This subspace may correspond to, for example, partially filled narrow bands, which often characterize strongly correlated materials. The formalism delivers naturally the frequency-dependent effective interaction (the Hubbard U) and provides a general framework for constructing theoretical models based on the Green's function language. It also furnishes a general scheme for first-principles calculations of complex systems in which the main correlation effects are concentrated on a small subspace of the full Hilbert space.