Jörn W. F. Venderbos

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands. Different types of topological semimetals can be distinguished on the basis of the degeneracy of the band crossings, their codimension (e.g., point or line nodes), and the crystal space group symmetries on which the protection of stable band crossings relies. The dispersion near the band crossing is a further discriminating characteristic. These properties give rise to a wide range of distinct semimetal phases such as Dirac or Weyl semimetals, point or line node semimetals, and type I or type II semimetals. In this review we give a general description of various families of topological semimetals, with an emphasis on proposed material realizations from first-principles calculations. The conceptual framework for studying topological gapless electronic phases is reviewed, with a particular focus on the symmetry requirements of energy band crossings, and the relation between the different families of topological semimetals is elucidated. In addition to the paradigmatic Dirac and Weyl semimetals, we pay particular attention to more recent examples of topological semimetals, which include nodal line semimetals, multifold fermion semimetals, and triple-point semimetals. Less emphasis is placed on their surface state properties, their responses to external probes, and recent experimental developments.

The recent observation of superconductivity in proximity to an insulating phase in twisted bilayer graphene (TBG) at small ``magic'' twist angles has been linked to the existence of nearly flat bands, which make TBG a fresh playground to investigate the interplay between correlations and superconductivity. The low-energy narrow bands were shown to be well described by an effective tight-binding model on the honeycomb lattice (the dual of the triangular Moir\'e superlattice) with a local orbital degree of freedom. In this paper, we perform a strong-coupling analysis of the proposed $\left({p}_{x},\phantom{\rule{0.16em}{0ex}}{p}_{y}\right)$ two-orbital extended Hubbard model on the honeycomb lattice. By decomposing the interacting terms in the particle-particle and particle-hole channels, we classify the different possible superconducting, magnetic, and charge instabilities of the system. In the pairing case, we pay particular attention to the two-component ($d\text{\ensuremath{-}}\mathrm{wave}$) pairing channels, which admit vestigial phases with nematic or chiral orders, and study their phenomenology. Furthermore, we explore the strong-coupling regime by obtaining a simplified spin-orbital exchange model which may describe a putative Mott-type insulating state at quarter-filling. Our mean-field solution reveals a rich intertwinement between ferromagnetic and antiferromagnetic orders with different types of nematic and magnetic orbital orders. Overall, our work provides a solid framework for further investigations of the phase diagram of the two-orbital extended Hubbard model in both strong- and weak-coupling regimes.

The discovery of superconductivity in the $j$=$\frac{3}{2}$ spin-orbit coupled half-Heusler semimetals has given the field of low-density superconductivity a new twist. Exotic pairings are likely, yet the favored superconducting instabilities remain to be determined. The authors provide a method to analyze superconductivity in systems with symmetry-protected band touchings. They apply it to $j$=$\frac{3}{2}$ systems to find that polar phonons play an important role in the superconducting mechanism of the half-Heusler compounds. They also explicitly show a number of new features associated with such band structures, such as the strong dependence on hole or electron doping in the determination of the leading odd-parity superconducting instability.

Recently, there is widespread interest in odd-parity (e.g., $p$-wave) superconductivity in strongly spin-orbit-coupled materials, such as doped topological insulators. Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ is a prime example. A series of recent experiments, including NMR, specific heat, magnetoresistance, and torque magnetometry, has provided mounting evidence of odd-parity spin-triplet superconductivity in Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ as well as Sr${}_{x}$Bi${}_{2}$Se${}_{3}$ and Nb${}_{x}$Bi${}_{2}$Se${}_{3}$. Here, the authors construct a general theory of such degenerate superconducting components and the competition between chiral and nematic states. They derive a general criterion to establish which state is energetically favored, which highlights the crucial role of spin-orbit coupling. In particular, when applied to the specific case of Cu${}_{x}$Bi${}_{2}$Se${}_{3}$, the theory is consistent with three key experimental findings: a full pairing gap, triplet pairing, and spontaneous rotational symmetry breaking. Remarkably, both nematic and chiral superconductors with odd-parity pairing symmetry are topological superconductors, belonging respectively to time-reversal-invariant (class DIII) and time-reversal-breaking (class D) categories of the topological classification. The authors show that due to the nonunitary nature of chiral pairing in spin-orbit coupled materials, spin nondegenerate point nodes realize Majorana fermion quasiparticles in three dimensions.