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) pr
ovides 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.
Strong repulsive interactions between electrons can lead to a Mott metal-insulator transition. The Dynamical Mean-Field Theory (DMFT) explains the critical end-point and hysteresis region with single-particle concepts such as the spectral function an
d the quasiparticle weight. In this work, we reconsider the critical end point of the metal-insulator transition on the two-particle level. We show that the relevant eigenvalue and eigenvector of the non-local Bethe-Salpeter kernel provide a unified picture of the hysteresis region and of the critical end point of the Mott transition. In particular, they simultaneously explain the thermodynamics of the hysteresis region and the iterative stability of the DMFT equations. This analysis paves the way for a deeper understanding of phase transitions in correlated materials.
Two-dimensional materials can be strongly influenced by their surroundings. A dielectric environment screens and reduces the Coulomb interaction between electrons in the two-dimensional material. Since the Coulomb interaction is responsible for the i
nsulating state of Mott materials, dielectric screening provides direct access to the Mottness. Our many-body calculations reveal the spectroscopic fingerprints of Coulomb engineering. We demonstrate eV-scale changes to the position of the Hubbard bands and show a Coulomb engineered insulator-to-metal transition. Based on this theoretical analysis, we discuss prerequisites for an effective experimental realization of Coulomb engineering.
In this article we perform a critical assessment of different known methods for the analytical continuation of bosonic functions, namely the maximum entropy method, the non-negative least-square method, the non-negative Tikhonov method, the Pade appr
oximant method, and a stochastic sampling method. Three functions of different shape are investigated, corresponding to three physically relevant scenarios. They include a simple two-pole model function and two flavours of the non-interacting Hubbard model on a square lattice, i.e. a single-orbital metallic system and a two-orbitals insulating system. The effect of numerical noise in the input data on the analytical continuation is discussed in detail. Overall, the stochastic method by Mishchenko et al. [Phys. Rev. B textbf{62}, 6317 (2000)] is shown to be the most reliable tool for input data whose numerical precision is not known. For high precision input data, this approach is slightly outperformed by the Pade approximant method, which combines a good resolution power with a good numerical stability. Although none of the methods retrieves all features in the spectra in the presence of noise, our analysis provides a useful guideline for obtaining reliable information of the spectral function in cases of practical interest.
