To obtain an effective many-body model of graphene and related materials from first principles we calculate the partially screened frequency dependent Coulomb interaction. In graphene, the effective on-site (Hubbard) interaction is $U_{00}=9.3\mathrm{eV}$ in close vicinity to the critical value separating conducting graphene from an insulating phase emphasizing the importance of nonlocal Coulomb terms. The nearest-neighbor Coulomb interaction strength is computed to $U_{01}=5.5\mathrm{eV}$. In the long-wavelength limit, we find the effective background dielectric constant of graphite to be $ϵ=2.5$ in very good agreement with experiment.

The effective on-site Coulomb interaction (Hubbard U) between localized d electrons in 3d, 4d, and 5d transition metals is calculated employing a parameter-free realization of the constrained random-phase approximation using Wannier functions within the full-potential linearized augmented-plane-wave method. The U values lie between 1.5 and 5.7 eV and depend on the crystal structure, spin polarization, d electron number, and d orbital filling. On the basis of the calculated U parameters, we discuss the strength of the electronic correlations and the instability of the paramagnetic state toward the ferromagnetic one for 3d metals.

We present an implementation of the GW approximation for the electronic self-energy within the full-potential linearized augmented-plane-wave (FLAPW) method. The algorithm uses an all-electron mixed product basis for the representation of response matrices and related quantities. This basis is derived from the FLAPW basis and is exact for wave-function products. The correlation part of the self-energy is calculated on the imaginary-frequency axis with a subsequent analytic continuation to the real axis. As an alternative we can perform the frequency convolution of the Green function G and the dynamically screened Coulomb interaction W explicitly by a contour integration. The singularity of the bare and screened interaction potentials gives rise to a numerically important self-energy contribution, which we treat analytically to achieve good convergence with respect to the k-point sampling. As numerical realizations of the GW approximation typically suffer from the high computational expense required for the evaluation of the nonlocal and frequency-dependent self-energy, we demonstrate how the algorithm can be made very efficient by exploiting spatial and time-reversal symmetry as well as by applying an optimization of the mixed product basis that retains only the numerically important contributions of the electron-electron interaction. This optimization step reduces the basis size without compromising the accuracy and accelerates the code considerably. Furthermore, we demonstrate that one can employ an extrapolar approximation for high-lying states to reduce the number of empty states that must be taken into account explicitly in the construction of the polarization function and the self-energy. We show convergence tests, CPU timings, and results for prototype semiconductors and insulators as well as ferromagnetic nickel.

Recently, Shih et al. [Phys. Rev. Lett. 105, 146401 (2010)] published a theoretical band gap for wurtzite ZnO, calculated with the non-self-consistent GW approximation, that agreed surprisingly well with experiment while deviating strongly from previous studies. They showed that a very large number of empty bands is necessary to converge the gap. We reexamine the GW calculation with the full-potential linearized augmented-plane-wave method and find that even with 3000 bands the band gap is not completely converged. A hyperbolical fit is used to extrapolate to infinite bands. Furthermore, we eliminate the linearization error for high-lying states with local orbitals. In fact, our calculated band gap is considerably larger than in previous studies, but somewhat smaller than that of Shih et al..

Using angle-resolved photoelectron spectroscopy and ab initio $GW$ calculations, we unambiguously show that the widely investigated three-dimensional topological insulator Bi${}_2$Se${}_3$ has a direct band gap at the $\Gamma$ point. Experimentally, this is shown by a three-dimensional band mapping in large fractions of the Brillouin zone. Theoretically, we demonstrate that the valence-band maximum is located at the $\Gamma$ point only if many-body effects are included in the calculation. Otherwise, it is found in a high-symmetry mirror plane away from the zone center.

We present quasiparticle self-consistent $GW$ $(\mathrm{QS}GW)$ calculations of semimetallic bulk Bi. We go beyond the conventional $\mathrm{QS}GW$ method by including the spin-orbit coupling throughout the self-consistency cycle. This approach improves the description of the electron and the hole pockets considerably with respect to standard density functional theory (DFT), leading to excellent agreement with experiment. We employ this relativistic $\mathrm{QS}GW$ approach to conduct a study of the semimetal-to-semiconductor and the trivial-to-topological transitions that Bi experiences under strain. DFT predicts that an unphysically large strain is needed for such transitions. We show, by means of the relativistic $\mathrm{QS}GW$ description of the electronic structure, that an in-plane tensile strain of only 0.3% and a compressive strain of 0.4% are sufficient to cause the semimetal-to-semiconductor and the trivial-to-topological phase transitions, respectively. Thus, the required strain moves into a regime that is likely to be realizable in experiment, which opens up the possibility to explore bulklike topological behavior of pure Bi.