Quantized Nonlinear Conductance in Ballistic Metals.

We introduce a nonlinear frequency-dependent D+1 terminal conductance that characterizes a D-dimensional Fermi gas, generalizing the Landauer conductance in D=1. For a 2D ballistic conductor, we show that this conductance is quantized and probes the Euler characteristic of the Fermi sea. We critically address the roles of electrical contacts and Fermi liquid interactions, and we propose experiments on 2D Dirac materials, such as graphene, using a triple point contact geometry.

Fibonacci Topological Superconductor.

We introduce a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory, which is based on a SO(7)_{1}/(G_{2})_{1} coset factorization, leads to a solvable one-dimensional model that is extended to two dimensions using a network construction. In addition to providing a description of the Fibonacci phase without parafermions, our theory predicts a closely related "anti-Fibonacci" phase, whose topological order is characterized by the tricritical Ising model. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that generalizes the Z_{2} Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.

Fractional Excitonic Insulator.

We argue that a correlated fluid of electrons and holes can exhibit a fractional quantum Hall effect at zero magnetic field analogous to the Laughlin state at filling 1/m. We introduce a variant of the Laughlin wave function for electrons and holes and show that for m=1 it is the exact ground state of a free fermion model that describes p_{x}+ip_{y} excitonic pairing. For m>1 we develop a simple composite fermion mean field theory, and we present evidence that our wave function correctly describes this phase. We derive an interacting Hamiltonian for which our wave function is the exact ground state, and we present physical arguments that the m=3 state can be realized in a system in which energy bands with angular momentum that differ by 3 cross at the Fermi energy. This leads to a gapless state with (p_{x}+ip_{y})^{3} excitonic pairing, which we argue is conducive to forming the fractional excitonic insulator in the presence of interactions. Prospects for numerics on model systems and band structure engineering to realize this phase in real materials are discussed.

Dirac-Weyl Semimetal: Coexistence of Dirac and Weyl Fermions in Polar Hexagonal ABC Crystals.

We propose that the noncentrosymmetric LiGaGe-type hexagonal ABC crystal SrHgPb realizes a new type of topological semimetal that hosts both Dirac and Weyl points in momentum space. The symmetry-protected Dirac points arise due to a band inversion and are located on the sixfold rotation z axis, whereas the six pairs of Weyl points related by sixfold symmetry are located on the perpendicular k_{z}=0 plane. By studying the electronic structure as a function of the buckling of the HgPb layer, which is the origin of inversion symmetry breaking, we establish that the coexistence of Dirac and Weyl fermions defines a phase separating two topologically distinct Dirac semimetals. These two Dirac semimetals are distinguished by the Z_{2} index of the k_{z}=0 plane and the corresponding presence or absence of 2D Dirac fermions on side surfaces. We formalize our first-principles calculations by deriving and studying a low-energy model Hamiltonian describing the Dirac-Weyl semimetal phase. We conclude by proposing several other materials in the noncentrosymmetric ABC material class, in particular SrHgSn and CaHgSn, as candidates for realizing the Dirac-Weyl semimetal.

Topological Phonons and Weyl Lines in Three Dimensions.

Topological mechanics and phononics have recently emerged as an exciting field of study. Here we introduce and study generalizations of the three-dimensional pyrochlore lattice that have topologically protected edge states and Weyl lines in their bulk phonon spectra, which lead to zero surface modes that flip from one edge to the opposite as a function of surface wave number.

Double Dirac Semimetals in Three Dimensions.

We study a class of Dirac semimetals that feature an eightfold-degenerate double Dirac point. We show that 7 of the 230 space groups can host such Dirac points and argue that they all generically display linear dispersion. We introduce an explicit tight-binding model for space groups 130 and 135. Space group 135 can host an intrinsic double Dirac semimetal with no additional states at the Fermi energy. This defines a symmetry-protected topological critical point, and we show that a uniaxial compressive strain applied in different directions leads to topologically distinct insulating phases. In addition, the double Dirac semimetal can accommodate topological line defects that bind helical modes. Connections are made to theories of strongly interacting filling-enforced semimetals, and potential materials realizations are discussed.

Observation of neutral modes in the fractional quantum Hall regime.

The quantum Hall effect takes place in a two-dimensional electron gas under a strong magnetic field and involves current flow along the edges of the sample. For some particle-hole conjugate states of the fractional regime (for example, with fillings between 1/2 and 1 of the lowest Landau level), early predictions suggested the presence of counter-propagating edge currents in addition to the expected ones. When this did not agree with the measured conductance, it was suggested that disorder and interactions will lead to counter-propagating modes that carry only energy--the so called neutral modes. In addition, a neutral upstream mode (the Majorana mode) was expected for selected wavefunctions proposed for the even-denominator filling 5/2. Here we report the direct observation of counter-propagating neutral modes for fillings of 2/3, 3/5 and 5/2. The basis of our approach is that, if such modes impinge on a narrow constriction, the neutral quasiparticles will be partly reflected and fragmented into charge carriers, which can be detected through shot noise measurements. We find that the resultant shot noise is proportional to the injected current. Moreover, when we simultaneously inject a charge mode, the presence of the neutral mode was found to significantly affect the Fano factor and the temperature of the backscattered charge mode. In particular, such observations for filling 5/2 may single out the non-Abelian wavefunctions for the state.

Phonons and elasticity in critically coordinated lattices.

Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site z. When z < zc ≈ 2d, where d is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on the properties of frames with z at or near zc, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, N0 -NS = dN -NB relating the number of sites, N, and number of bonds, NB, to the number, N0, of modes of zero energy and the number, NS, of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which z = zc and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.

Layered Topological Crystalline Insulators.

Topological crystalline insulators (TCIs) are insulating materials whose topological property relies on generic crystalline symmetries. Based on first-principles calculations, we study a three-dimensional (3D) crystal constructed by stacking two-dimensional TCI layers. Depending on the interlayer interaction, the layered crystal can realize diverse 3D topological phases characterized by two mirror Chern numbers (MCNs) (µ1,µ2) defined on inequivalent mirror-invariant planes in the Brillouin zone. As an example, we demonstrate that new TCI phases can be realized in layered materials such as a PbSe (001) monolayer/h-BN heterostructure and can be tuned by mechanical strain. Our results shed light on the role of the MCNs on inequivalent mirror-symmetric planes in reciprocal space and open new possibilities for finding new topological materials.

Dirac Line Nodes in Inversion-Symmetric Crystals.

We propose and characterize a new Z2 class of topological semimetals with a vanishing spin-orbit interaction. The proposed topological semimetals are characterized by the presence of bulk one-dimensional (1D) Dirac line nodes (DLNs) and two-dimensional (2D) nearly flat surface states, protected by inversion and time-reversal symmetries. We develop the Z2 invariants dictating the presence of DLNs based on parity eigenvalues at the parity-invariant points in reciprocal space. Moreover, using first-principles calculations, we predict DLNs to occur in Cu_{3}N near the Fermi energy by doping nonmagnetic transition metal atoms, such as Zn and Pd, with the 2D surface states emerging in the projected interior of the DLNs. This Letter includes a brief discussion of the effects of spin-orbit interactions and symmetry breaking as well as comments on experimental implications.