We perform a complete classification of two-band k·p theories at band crossing points in 3D semimetals with n-fold rotation symmetry and broken time-reversal symmetry. Using this classification, we show the existence of new 3D topological semimetals characterized by C(4,6)-protected double-Weyl nodes with quadratic in-plane (along k(x,y)) dispersion or C(6)-protected triple-Weyl nodes with cubic in-plane dispersion. We apply this theory to the 3D ferromagnet HgCr(2)Se(4) and confirm it is a double-Weyl metal protected by C(4) symmetry. Furthermore, if the direction of the ferromagnetism is shifted away from the [001] axis to the [111] axis, the double-Weyl node splits into four single Weyl nodes, as dictated by the point group S(6) of that phase. Finally, we discuss experimentally relevant effects including the splitting of multi-Weyl nodes by applying a C(n) breaking strain and the surface Fermi arcs in these new semimetals.

We survey various quantized bulk physical observables in two-and three-dimensional topological band insulators invariant under translational symmetry and crystallographic point group symmetries (PGS). In two-dimensional insulators, we show that: (i) the Chern number of a Cn-invariant insulator can be determined, up to a multiple of n, by evaluating the eigenvalues of symmetry operators at high-symmetry points in the Brillouin zone; (ii) the Chern number of a Cn-invariant insulator is also determined, up to a multiple of n, by the Cn eigenvalue of the Slater determinant of a noninteracting many-body system and (iii) the Chern number vanishes in insulators with dihedral point groups Dn, and the quantized electric polarization is a topological invariant for these insulators. In three-dimensional insulators, we show that: (i) only insulators with point groups Cn, C nh and Sn PGS can have nonzero 3D quantum Hall coefficient and (ii) only insulators with improper rotation symmetries can have quantized magnetoelectric polarization P3 in the term P3E • B, the axion term in the electrodynamics of the insulator (medium).The study of novel topological phases of matter has become one of the most active fields in condensed matter physics. These phases are interesting because while deviating qualitatively from the conventional insulating phase, they cannot be described by any local order parameter in the Ginzberg-Landau-Wilson spontaneous symmetry breaking paradigm. Heuristically, the word 'topological' implies the presence of some global property, i.e., contributed by all electrons in the system, that distinguishes this special phase. Such a property is usually marked by a global observable that takes different values in a topological phase and in the conventional insulating phase (normal phase) adiabatically continuable to the atomic limit. In addition, 'topological' also implies that this global observable is quantized, or discretized, so that a topological phase cannot be adiabatically connected to the normal phase. Any quantized global observable characteristic to a topological phase is called a bulk topological invariant.

We study the topological features of non-interacting insulators subject to an antiferromangetic (AFM) Zeeman field, or AFM insulators, the period of which is commensurate with the lattice period. These insulators can be classified by the presence/absence of an emergent anti-unitary symmetry: the combined operation of time-reversal and a lattice translation by vector D. For AFM insulators that preserve this combined symmetry, regardless of any details in lattice structure or magnetic structure, we show that (i) there is a new type of Kramers' degeneracy protected by the combined symmetry; (ii) a new Z2 index may be defined for 3D AFM insulators, but not for those in lower dimensions and (iii) in 3D AFM insulators with a non-trivial Z2 index, there are odd number of gapless surface modes if and only if the surface termination also preserves the combined symmetry, but the dispersion of surface states becomes highly anisotropic if the AFM propagation vector becomes small compared with the reciprocal lattice vectors. We numerically demonstrate the theory by calculating the spectral weight of the surface states of a 3D TI in the presence of AFM fields with different propagation vectors, which may be observed by ARPES in Bi2Se3 or Bi2Te3 with induced antiferromagnetism.

Superconductivity involving topological Dirac electrons has recently been proposed as a platform between concepts in high-energy and condensed-matter physics. It has been predicted that supersymmetry and Majorana fermions, both of which remain elusive in particle physics, may be realized through emergent particles in these particular superconducting systems. Using artificially fabricated topological-insulator-superconductor heterostructures, we present direct spectroscopic evidence for the existence of Cooper pairing in a weakly interacting half Dirac gas. Our studies reveal that two dimensional topological superconductivity in a helical Dirac gas is distinctly di erent from that in an ordinary two-dimensional superconductor in terms of the spin degrees of freedom of electrons. We further show that the pairing of Dirac electrons can be suppressed by timereversal symmetry-breaking impurities, thereby removing the distinction. Our demonstration and momentum-space imaging of Cooper pairing in a half-Dirac-gas two-dimensional topological superconductor serve as a critically important platform for future testing of fundamental physics predictions such as emergent supersymmetry and topological quantum criticality. R ealization of novel superconductivity is one of the central themes in condensed matter physics in general 1-24 . Superconductivity is a collective phenomenon, where electrons moving to the opposite directions (±k) form dynamically bound pairs, resulting in a Cooper pair gas. In an ordinary superconductor, the conduction electrons that move along a certain direction have both spin-up and spin-down electrons available for the Cooper pairing. The superconductivity observed so far, including in the conventional s-wave BCS superconductors as well as the cuprate or heavy fermion d-wave superconductors, all share this property. Recently, the discovery of 3D topological insulators (TIs) in bismuth-based semiconducting compounds has attracted much interest in condensed matter physics. In these TI materials, the bulk has a full energy gap whereas the surface exhibits an odd number of Dirac-cone electronic states, where the spin of the surface electrons is uniquely locked to their momentum 1,2 . Therefore, at any given surface of a TI, the surface electrons moving in one direction (for example, +k) will have only spin-up electrons available whereas those of moving to −k have only spin-down electrons available. This is in contrast to the Fermi level electronic states in an ordinary superconductor. This distinction can give rise to a wide range of exotic physics. Recently, a number of theories have highlighted these possibilities from both the fundamental physics and applications point of view 4-10 . For example, both supersymmetry and Majorana fermions are interesting physics phenomena predicted in high-energy theories that remain unobserved in particle physics experiments. And it has been theoretically predicted, very recently, that such new physics can be realized in a condensed matter setting 4,6 , if superconductiv...

We explore the 32 crystallographic point groups and identify topological phases of matter with robust surface modes. For n=3,4, and 6 of the C_{nv} groups, we find the first-known 3D topological insulators without spin-orbit coupling, and with surface modes that are protected only by point groups; i.e., the relevant symmetries are purely crystalline and do not include time reversal. To describe these C_{nv} systems, we introduce the notions of (a) a halved mirror chirality, an integer invariant which characterizes half-mirror-planes in the 3D Brillouin zone, and (b) a bent Chern number, the traditional Thouless-Kohmoto-Nightingale-den Nijs invariant generalized to bent 2D manifolds. We find that a Weyl semimetallic phase intermediates two gapped phases with distinct halved chiralities. In addition to electronic systems without spin-orbit coupling, our findings also apply to intrinsically spinless systems such as photonic crystals and ultracold atoms.