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We generalize Hedin equations to a system of superconducting electrons coupled with a system of phonons. The electrons are described by an electronic Pauli Hamiltonian which includes the Coulomb interaction among electrons and an external vector and scalar potential. We derive the continuity equation in the presence of the superconducting condensate and point out how to cast vertex corrections in the form of a non-local effective interaction that can be used to describe both fluctuations of spin and superconducting phase beyond the screened Coulomb self-energy diagram.
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Taking the competition and the mutual screening of various bosonic fluctuations in correlated electron systems into account requires an unbiased approach to the many-body problem. One such approach is the self-consistent solution of the parquet equations, whose numerical treatment in lattice systems is however prohibitively expensive. In a recent article it was shown that there exists an alternative to the parquet decomposition of the four-point vertex function, which classifies the vertex diagrams according to the principle of single-boson exchange (SBE) [F. Krien, A. Valli, and M. Capone, arXiv:1907.03581 (2019)]. Here we show that the SBE decomposition leads to a closed set of equations for the Hedin three-leg vertex, the polarization, and the electronic self-energy, which sums self-consistently the diagrams of the Maki-Thompson type. This circumvents the calculation of four-point vertex functions and the inversion of the Bethe-Salpeter equations, which are the two major bottlenecks of the parquet equations. The convergence of the calculation scheme starting from a fully irreducible vertex is demonstrated for the Anderson impurity model.
Understanding the interaction of vortices with inclusions in type-II superconductors is a major outstanding challenge both for fundamental science and energy applications. At application-relevant scales, the long-range interactions between a dense configuration of vortices and the dependence of their behavior on external parameters, such as temperature and an applied magnetic field, are all important to the net response of the superconductor. Capturing these features, in general, precludes analytical description of vortex dynamics and has also made numerical simulation prohibitively expensive. Here we report on a highly optimized iterative implicit solver for the time-dependent Ginzburg-Landau equations suitable for investigations of type-II superconductors on massively parallel architectures. Its main purpose is to study vortex dynamics in disordered or geometrically confined mesoscopic systems. In this work, we present the discretization and time integration scheme in detail for two types of boundary conditions. We describe the necessary conditions for a stable and physically accurate integration of the equations of motion. Using an inclusion pattern generator, we can simulate complex pinning landscapes and the effect of geometric confinement. We show that our algorithm, implemented on a GPU, can provide static and dynamic solutions of the Ginzburg-Landau equations for mesoscopically large systems over thousands of time steps in a matter of hours. Using our formulation, studying scientifically-relevant problems is a computationally reasonable task.
Historically, the GW approach was put forward by Hedin as the simplest approximation to the so-called Hedin equations. In Section 2, we will derive these Hedin equations from a Feynman-diagrammatical point of view. Section 3.1 shows how GW arises as an approximation to the Hedin equations. In Section 3.2, we briefly present some typical GW results for materials, including quasiparticle renormalizations, lifetimes, and band gap enhancements. In Section 4, the combination of GW and DMFT is summarized. Finally, as a prospective outlook, ab initio dynamical vertex approximation D$Gamma$A is introduced in Section 5 as a unifying scheme for all that: GW, DMFT and non-local vertex correlations beyond.
We present a first-principles approach to describe magnetic and superconducting systems and the phenomena of competition between these electronic effects. We develop a density functional theory: SpinSCDFT, by extending the Hohenberg-Kohn theorem and constructing the non-interacting Kohn- Sham system. An exchange-correlation functional for SpinSCDFT is derived from the Sham Schluter connection between the SpinSCDFT Kohn-Sham and a self-energy in Eliashberg approximation. The reference Eliashberg equations for superconductors in the presence of magnetism are also derived and discussed.
Since the announcement in 2011 of the Materials Genome Initiative by the Obama administration, much attention has been given to the subject of materials design to accelerate the discovery of new materials that could have technological implications. Although having its biggest impact for more applied materials like batteries, there is increasing interest in applying these ideas to predict new superconductors. This is obviously a challenge, given that superconductivity is a many body phenomenon, with whole classes of known superconductors lacking a quantitative theory. Given this caveat, various efforts to formulate materials design principles for superconductors are reviewed here, with a focus on surveying the periodic table in an attempt to identify cuprate analogues.
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