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Register a new userRandom Phase Approximation for gapped systems: role of vertex corrections and applicability of the constrained random phase approximation
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The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the determination of electronic screening properties of the Coulomb interaction, the Random Phase Approximation (RPA) provides such a simplification. Here, we explicitly show that this approximation is justified for band structures with sizeable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equivalent to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our analysis of electron-electron interactions provides a real-space analogy to Migdals theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained Random Phase Approximation (cRPA) calculations of effective interactions. We find that their usage of Kohn-Sham energies already accounts for the leading local (excitonic) vertex correction in insulators.
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In the framework of a multiorbital Hubbard model description of superconductivity, a matrix formulation of the superconducting pairing interaction that has been widely used is designed to treat spin, charge and orbital fluctuations within a random phase approximation (RPA). In terms of Feynman diagrams, this takes into account particle-hole ladder and bubble contributions as expected. It turns out, however, that this matrix formulation also generates additional terms which have the diagrammatic structure of vertex corrections. Here we examine these terms and discuss the relationship between the matrix-RPA superconducting pairing interaction and the Feynman diagrams that it sums.
Recent experiments on twisted bilayer graphene show the urgent need for establishing a low-energy lattice model for the system. We use the constrained random phase approximation to study the interaction parameters of such models taking into account screening from the moire bands left outside the model space. Based on an atomic-scale tight-binding model, we develop a numerically tractable approximation to the polarization function and study its behavior for different twist angles. We find that the polarization has three different momentum regimes. For small momenta, the polarization is quadratic, leading to a linear dielectric function expected for a two-dimensional dielectric material. For large momenta, the polarization becomes independent of the twist angle and approaches that of uncoupled graphene layers. In the intermediate momentum regime, the dependence on the twist-angle is strong. Close to the largest magic angle the dielectric function peaks at a momentum of $~1/(4 : nm)$ attaining values of 25, meaning very strong screening at intermediate distances. We also calculate the effective screened Coulomb interaction in real space and give estimates for the on-site and extended interaction terms for the recently developed hexagonal-lattice model. For free-standing TBG the effective interaction decays slower than $1/r$ at intermediate distances $r$, while it remains essentially unscreened at large enough $r$.
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 band filling, we find that the effective interaction increases as the charge transfer energy ($Delta_{dp}$) increases and similar behavior is obtained as a function of the interatomic 2$p$-3$d$ interaction ($U_{dp}$). However, the effective interaction is more sensitive to $Delta_{dp}$ 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 $Delta_{dp}$. 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 onsite interaction terms, methods beyond cRPA may be required.
We describe a finite-field approach to compute density response functions, which allows for efficient $G_0W_0$ and $G_0W_0Gamma_0$ calculations beyond the random phase approximation. The method is easily applicable to density functional calculations performed with hybrid functionals. We present results for the electronic properties of molecules and solids and we discuss a general scheme to overcome slow convergence of quasiparticle energies obtained from $G_0W_0Gamma_0$ calculations, as a function of the basis set used to represent the dielectric matrix.
The optimized effective potential (OEP) method presents an unambiguous way to construct the Kohn-Sham potential corresponding to a given diagrammatic approximation for the exchange-correlation functional. The OEP from the random-phase approximation (RPA) has played an important role ever since the conception of the OEP formalism. However, the solution of the OEP equation is computationally fairly expensive and has to be done in a self-consistent way. So far, large scale solid state applications have therefore been performed only using the quasiparticle approximation (QPA), neglecting certain dynamical screening effects. We obtain the exact RPA-OEP for 15 semiconductors and insulators by direct solution of the linearized Sham-Schluter equation. We investigate the accuracy of the QPA on Kohn-Sham band gaps and dielectric constants, and comment on the issue of self-consistency.
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