Based on first-principle calculations, we show that a family of nonmagnetic materials including TaAs, TaP, NbAs, and NbP are Weyl semimetals (WSM) without inversion centers. We find twelve pairs of Weyl points in the whole Brillouin zone (BZ) for each of them. In the absence of spin-orbit coupling (SOC), band inversions in mirror-invariant planes lead to gapless nodal rings in the energy-momentum dispersion. The strong SOC in these materials then opens full gaps in the mirror planes, generating nonzero mirror Chern numbers and Weyl points off the mirror planes. The resulting surface-state Fermi arc structures on both (001) and (100) surfaces are also obtained, and they show interesting shapes, pointing to fascinating playgrounds for future experimental studies.

Graphene has received widespread attention due to its unique electronic properties. Much of the research conducted so far has focused on electron mobility, which is determined by scattering from charged impurities and other inhomogeneities. However, another important quantity, the quantum capacitance, has been largely overlooked. Here, we report a direct measurement of the quantum capacitance of graphene as a function of gate potential using a three-electrode electrochemical configuration. The quantum capacitance has a non-zero minimum at the Dirac point and a linear increase on both sides of the minimum with relatively small slopes. Our findings -- which are not predicted by theory for ideal graphene -- suggest that charged impurities also influences the quantum capacitance. We also measured the capacitance in aqueous solutions at different ionic concentrations, and our results strongly indicate that the long-standing puzzle about the interfacial capacitance in carbon-based electrodes has a quantum origin.

We theoretically study three-dimensional topological semimetals (TSMs) with nodal lines protected by crystalline symmetries. Compared with TSMs with point nodes, e.g., Weyl semimetals and Dirac semimetals, where the conduction and the valence bands touch at discrete points, in these new TSMs the two bands cross at closed lines in the Brillouin zone. We propose two new classes of symmetry protected nodal lines in the absence and in the presence of spin-orbital coupling (SOC), respectively. In the former, we discuss nodal lines that are protected by the combination of inversion symmetry and time-reversal symmetry; yet unlike any previously studied nodal lines in the same symmetry class, each nodal line has a Z2 monopole charge and can only be created (annihilated) in pairs. In the second class, with SOC, we show that a nonsymmorphic symmetry (screw axis) protects a four-band crossing nodal line in systems having both inversion and time-reversal symmetries.

We study a spin S quantum Heisenberg model on the Fe lattice of the rare-earth oxypnictide superconductors. Using both large S and large N methods, we show that this model exhibits a sequence of two phase transitions: from a high temperature symmetric phase to a narrow region of intermediate "nematic" phase, and then to a low temperature spin ordered phase. Identifying phases by their broken symmetries, these phases correspond precisely to the sequence of structural (tetragonal to monoclinic) and magnetic transitions that have been recently revealed in neutron scattering studies of LaOFeAs. The structural transition can thus be identified with the existence of incipient ("fluctuating") magnetic order.

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 study fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions (d=2, 3). We show that in both cases nontrivial topology is manifested by the presence of the (d-2)-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states.

We review the recent, mainly theoretical, progress in the study of topological nodal line semimetals in three dimensions. In these semimetals, the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone, and any perturbation that preserves a certain symmetry group (generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands. The nodal line(s) is hence topologically protected by the symmetry group, and can be associated with a topological invariant. In this Review, (i) we enumerate the symmetry groups that may protect a topological nodal line; (ii) we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface, establishing a topological classification; (iii) for certain classes, we review the proposals for the realization of these semimetals in real materials and (iv) we discuss different scenarios that when the protecting symmetry is broken, how a topological nodal line semimetal becomes Weyl semimetals, Dirac semimetals and other topological phases and (v) we discuss the possible physical effects accessible to experimental probes in these materials.

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