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arXiv:2010.14045 · Physical review. B./Physical review. B

Investigation of the effective interactions for the Emery model by the constrained random-phase approximation and constrained functional renormalization group

Xing-Jie Han, Philipp Werner, Carsten Honerkamp A

Published Mar 15, 2021• 12 citations
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Abstract

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 (${U}_{dp}$). However, the effective interaction is more sensitive to ${\mathrm{\ensuremath{\Delta}}}_{dp}$ than ${U}_{dp}$. 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.

DOI
10.1103/physrevb.103.125130

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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 (\ensuremathΔdp{\mathrm{\ensuremath{\Delta}}}_{dp}\ensuremathΔdp​) increases and similar behavior is obtained as a function of the interatomic 2p\ensuremath-3d2p\text{\ensuremath{-}}3d2p\ensuremath-3d interaction (Udp{U}_{dp}Udp​). However, the effective interaction is more sensitive to \ensuremathΔdp{\mathrm{\ensuremath{\Delta}}}_{dp}\ensuremathΔdp​ than Udp{U}_{dp}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 \ensuremathΔdp{\mathrm{\ensuremath{\Delta}}}_{dp}\ensuremathΔdp​. 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.
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.
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 xyxyxy-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 3d3d3d-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.
We describe an efficient approximation for the electron-electron interaction in the determination of the low-energy effective interaction in multiband lattice systems. By using ideas for channel decomposition, form-factor expansion, and the truncated-unity technique we describe the interaction as arising from the nonlocal and orbital-dependent coupling of particle-hole and particle-particle bilinears formed by fields residing in the same one or two orbitals. This allows us to employ the constrained functional renormalization group (cfRG) with a suitable momentum and frequency discretization. The approach gives insights into the nonlocal screening of spin and charge interactions when bands away from the Fermi level are integrated out. Specifically, we compute the effective low-energy interactions in the low-energy target band of a three-band model with on-site and nonlocal bare interactions. We show that the cfRG adds important features to the effective target-band interaction that cannot be found using the constrained random phase approximation (cRPA).
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.
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.