The accuracy of the constrained random phase approximation(cRPA) method is examined in multi-orbital Hubbard models containing all possible on-site density-density interactions. Using DMFT, we show that the effective model constructed using cRPA fails to reproduce the spectral properties of the original full model in a wide parameter range. By comparing quantities such as the density of states and quasiparticle residues of the full and the effective models, we show that cRPA systematically overestimates the screening of Hubbard U for DMFT impurity solvers. We instead propose a new method to investigate the screening mechanism in the system using the local polarization, which is highly successful in reproducing spectra and also shows that the true screening is far less than that predicted by RPA. Furthermore, we compare the fully screened interaction
W given by RPA and our new method and show that the RPA
W is also overscreened and misses the signatures of local screening, which are clearly present in our new method.
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.
The effective interaction of downfolded low-energy models for electrons in solids can be obtained by integrating out the high energy bands away from the target band near the Fermi level. Here we apply the constrained random-phase approximation (cRPA) and constrained functional renormalization group (cfRG), which can go beyond cRPA by including all one-loop diagrams, to calculate and compare the effective interactions of the three-band Emery model, which is often used to investigate cuprate high-temperature superconductors. At half-filling we find that the effective interaction increases as the charge transfer energy (${\mathrm{\ensuremath{\Delta}}}_{dp}$) increases and similar behavior is obtained as a function of the interatomic $2p\text{\ensuremath{-}}3d$ interaction (
Udp). However, the effective interaction is more sensitive to ${\mathrm{\ensuremath{\Delta}}}_{dp}$ than
Udp. For most of the parameter sets, the effective static interaction is overscreened in cRPA compared to cfRG. The low-energy models at half-filling are solved within dynamical mean-field theory (DMFT). The results show that despite the different static interactions, the systems with cRPA and cfRG interaction exhibit a Mott transition at similar values of ${\mathrm{\ensuremath{\Delta}}}_{dp}$. We also investigate the effective interaction as a function of doping. The cfRG effective interaction decreases as the electron number increases and displays a trend opposite to that of cRPA. Antiscreening is observed for the hole-doped case. For all the cases studied, the near cancellation of the direct particle-hole channel is observed. This indicates that at least for the downfolding of the on-site interaction terms, methods beyond cRPA may be required.
In the derivation of low-energy effective models for solids targeting the bands near the Fermi level, the constrained random-phase approximation (cRPA) has become an appreciated tool to compute the effective interactions. The Wick-ordered constrained functional renormalization group (cfRG) generalizes the cRPA approach by including all interaction channels in an unbiased way. Here we present applications of the cfRG to two simple multiband systems and compare the resulting effective interactions to the cRPA. First, we consider a multiband model for monolayer graphene, where we integrate out the $\ensuremath{\sigma}$ bands to get an effective theory for $\ensuremath{\pi}$ bands. It turns out that terms beyond cRPA are strongly suppressed by the different
xy-plane reflection symmetry of the bands. In our model the cfRG corrections to cRPA become visible when one disturbs this symmetry difference slightly, however, without qualitative changes. This study shows that the embedding or layering of two-dimensional electronic systems can alter the effective interaction parameters beyond what is expected from screening considerations. The second example is a one-dimensional model for a diatomic system reminiscent of a CuO chain, where we consider an effective theory for Cu
3d-like orbitals. Here the fRG data shows relevant and qualitative corrections compared to the cRPA results. We argue that the new interaction terms affect the magnetic properties of the low-energy model.
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 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,
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
d space and how to distinguish
Pd 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
Pd. 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.