A wide range of materials, like d-wave superconductors, graphene, and topological insulators, share a fundamental similarity: their low-energy fermionic excitations behave as massless Dirac particles rather than fermions obeying the usual Schrödinger Hamiltonian. This emergent behavior of Dirac fermions in condensed matter systems defines the unifying framework for a class of materials we call "Dirac materials". In order to establish this class of materials, we illustrate how Dirac fermions emerge in multiple entirely different condensed matter systems and we discuss how Dirac fermions have been identified experimentally using electron spectroscopy techniques (angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy). As a consequence of their common low-energy excitations, this diverse set of materials shares a significant number of universal properties in the lowenergy (infrared) limit. We review these common properties including nodal points in the excitation spectrum, density of states, specific heat, transport, thermodynamic properties, impurity resonances, and magnetic field responses, as well as discuss many-body interaction effects. We further review how the emergence of Dirac excitations is controlled by specific symmetries of the material, such as time-reversal, gauge, and spin-orbit symmetries, and how by breaking these symmetries a finite Dirac mass is generated. We give examples of how the interaction of Dirac fermions with their distinct real material background leads to rich novel physics with common fingerprints such as the suppression of back scattering and impurity-induced resonant states. PACS

We investigate the possibility of inducing superconductivity in a graphite layer by electronic correlation effects. We use a phenomenological microscopic Hamiltonian which includes nearest-neighbor hopping and an interaction term which explicitly favors nearest-neighbor spin singlets through the well-known resonance valence bond ͑RVB͒ character of planar organic molecules. Treating this Hamiltonian in mean-field theory, allowing for bond-dependent variation of the RVB order parameter, we show that both s-and d-wave superconducting states are possible. The d-wave solution belongs to a two-dimensional representation and breaks time-reversal symmetry. At zero doping there exists a quantum critical point at the dimensionless coupling J / t = 1.91 and the s-and d-wave solutions are degenerate for low temperatures. At finite doping the d-wave solution has a significantly higher T c than the s-wave solution. By using density functional theory we show that the doping induced from sulfur absorption on a graphite layer is enough to cause an electronically driven d-wave superconductivity at graphite-sulfur interfaces. We also discuss applying our results to the case of the intercalated graphites, as well as the validity of a mean-field approach.

We study the carrier-mediated exchange interaction, the so-called RKKY coupling, between two magnetic moments in graphene using exact diagonalization on the honeycomb lattice. By using the tight-binding nearest neighbor band structure of graphene we avoid the use of a momentum cut-off which plagues results in the Dirac continuum model formulation. We extract both the short and long impurity-impurity distance behavior and show several corrections to earlier long distance results. In the bulk the RKKY coupling is proportional to $1/|{\bf R}|^3$ and displays $(1+\cos(2{\bf k}_D\cdot {\bf R})$-type oscillations. A-A sublattice coupling is always ferromagnetic whereas A-B subattice coupling is always antiferromagnetic and three times as large. We also study the effect of edges in zigzag graphene nanoribbons (ZGNRs) and find that for impurities on the edge, the RKKY coupling decays exponentially because of the localized zero energy edge states. For impurities inside a ZGNR the bulk characteristics are quickly regained.Comment: 8 pages, 6 figure

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS). In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed "topological quantum chemistry" (TQC) concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele Z2 classification, there is C2T -symmetry protected Z classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e. forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parameterization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele Z2 invariant implying that further stable topological phases must belong to the class of higher-order topological insulators. arXiv:1804.09719v4 [cond-mat.mes-hall]

We point out that essentially all multi-band superconductors have an odd-frequency pairing component, as follows from a general symmetry analysis of even-and odd-frequency pairing states. We show that odd-frequency superconducting pairing requires only a finite band hybridization, or scattering, and non-identical intraband order parameters, of which only one band needs to be superconducting. Under these conditions odd-frequency odd-interband pairing is always present. From a symmetry analysis we establish a complete reciprocity between parity in band-index and frequency.