Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator, but protected conducting states on their edge or surface. The 2D topological insulator is quantum spin Hall which close cousin of the integer state. A 3D supports novel polarized Dirac fermions its In this Colloquium article we will review theoretical foundation for these and describe recent experiments in signatures been observed. We transport HgCdTe wells demonstrate existence predicted...

We study the effects of spin orbit interactions on low energy electronic structure a single plane graphene. find that in an experimentally accessible temperature regime symmetry allowed potential converts graphene from ideal two dimensional semimetallic state to quantum Hall insulator. This novel matter is gapped bulk and supports quantized transport charge gapless edge states propagate at sample boundaries. The are non chiral, but they insensitive disorder because their directionality...

The quantum spin Hall (QSH) phase is a time reversal invariant electronic state with bulk band gap that supports the transport of charge and in gapless edge states. We show this associated novel ${Z}_{2}$ topological invariant, which distinguishes it from an ordinary insulator. classification, defined for Hamiltonians, analogous to Chern number classification effect. establish order QSH two model graphene propose generalization formalism applicable multiband interacting systems.

We study the proximity effect between an $s$-wave superconductor and surface states of a strong topological insulator. The resulting two-dimensional state resembles spinless ${p}_{x}+i{p}_{y}$ superconductor, but does not break time reversal symmetry. This supports Majorana bound at vortices. show that linear junctions superconductors mediated by insulator form nonchiral one-dimensional wire for fermions, circuits formed from these provide method creating, manipulating, fusing states.

We study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single Z2 topological invariant governs effect, in three dimensions there are 4 invariants distinguishing 16 phases with general classes: weak (WTI) and strong (STI) insulators. The WTI like layered 2D QSH states, but destroyed by disorder. STI robust lead to novel "topological metal" surface states. introduce tight binding model which realizes phases, we discuss its relevance...

Topological insulators are materials with a bulk excitation gap generated by the spin-orbit interaction that different from conventional insulators. This distinction is characterized ${Z}_{2}$ topological invariants, which characterize ground state. In two dimensions, there single invariant distinguishes ordinary insulator quantum spin-Hall phase. three four invariants distinguish ``weak'' and ``strong'' These phases presence of gapless surface (or edge) states. two-dimensional phase...

We show that the pseudorelativistic physics of graphene near Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological normal insulators, we particular space groups also allow 3D Dirac points as symmetry protected degeneracies. provide criteria necessary identify these and, an example, present ab initio calculations β-cristobalite BiO(2) which exhibits at level. find is metastable, so it physically realized a analog graphene.

Using low-resistance electrical contacts, we have measured the intrinsic high-field transport properties of metallic single-wall carbon nanotubes. Individual nanotubes appear to be able carry currents with a density exceeding 10(9) A/cm(2). As bias voltage is increased, conductance drops dramatically due scattering electrons. We show that current-voltage characteristics can explained by considering optical or zone-boundary phonon emission as dominant mechanism at high field.

A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry–based classification condensed matter. Exotic spin-transport phenomena, such as dissipationless spin Hall effect, have been speculated to originate from topological order whose identification requires spin-sensitive measurement, which does not exist this date in any system. Using Mott polarimetry, we probed degrees freedom and demonstrated...

We study theoretically transport of a one-dimensional single-channel interacting electron gas through barriers or constrictions. find that electrons with repulsive interactions, incident upon single barrier, are completely reflected at zero temperature. At finite temperature (T), the conductance is shown to vanish as power T, and temperature, power-law current-voltage characteristics predicted. For attractive we predict perfect transmission similar corrections. also resonant tunneling...

We study theoretically the transport of a one-channel Luttinger liquid through weak link. For repulsive electron interactions, electrons are completely reflected by even smallest scatterer, leading to truly insulating link, in striking contrast that for noninteracting electrons. At finite temperature (T) conductance is nonzero, and predicted vanish as power T. T=0 power-law current-voltage characteristics predicted. attractive argued be perfectly transmitted largest barriers. The role...

A theory of the long wavelength low energy electronic structure graphite-derived nanotubules is presented. The propagating $\pi$ electrons are described by wrapping a massless two dimensional Dirac Hamiltonian onto curved surface. effects tubule size, shape and symmetry included through an effective vector potential which we derive for this model. rich gap all straight single wall cylindrical tubes obtained analytically in theory, inhomogeneous deformations on nominally metallic armchair analyzed.

We study junctions between superconductors mediated by the edge states of a quantum-spin-Hall insulator. show that such exhibit fractional Josephson effect, in which current phase relation has $4\ensuremath{\pi}$ rather than $2\ensuremath{\pi}$ periodicity. This effect is consequence conservation fermion parity---the number electron $\text{mod}\text{ }2$---in superconducting junction and closely related to ${Z}_{2}$ topological structure Inelastic processes, violate parity, lead telegraph...

We introduce and analyze a class of one-dimensional insulating Hamiltonians that, when adiabatically varied in an appropriate closed cycle, define ``${Z}_{2}$ pump.'' For isolated system, single cycle the pump changes expectation value spin at each end even spin-orbit interactions violate conservation spin. A second however, returns system to its original state. When coupled leads, we show that ${Z}_{2}$ functions as sense define, transmits finite, though nonquantized, cycle. is...

The correlation functions of the transmission coefficients for scalar wave propagation through disordered media are calculated by use both diagrammatic techniques and numerical simulations. calculation is valid in diffusive regime: multiple elastic scattering with negligible absorption or inelastic a length much longer than wavelength. In addition to familiar large local intensity fluctuations we find novel memory effect long-range correlations which decay positive background value....

We formulate a quasiparticle theory for single hole in quantum antiferromagnet the limit that Heisenberg exchange energy is much less than hopping matrix element, J\ensuremath{\ll}t. consider ground state of spins to be either N\'eel or d-wave resonating-valence-bond (RVB) state. show self-consistent perturbation spectrum strongly renormalized by interactions with spin excitations. The can described narrow band located at an order -t residue J/t and bandwidth J. Above incoherent width t. Our...

We propose and characterize a new ${\mathbb{Z}}_{2}$ class of topological semimetals with vanishing spin-orbit interaction. The proposed are characterized by the presence bulk one-dimensional (1D) Dirac line nodes (DLNs) two-dimensional (2D) nearly flat surface states, protected inversion time-reversal symmetries. develop invariants dictating DLNs based on parity eigenvalues at parity-invariant points in reciprocal space. Moreover, using first-principles calculations, we predict to occur...

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch Bogoliubov de Gennes Hamiltonians. consider Hamiltonians $\mathcal{H}(\mathbf{k},\mathbf{r})$ that vary slowly with adiabatic parameters $\mathbf{r}$ surrounding the defect belong any of ten symmetry classes defined time-reversal particle-hole symmetry. The for such are identified explicit formulas invariants presented. introduce generalization bulk-boundary...

We study the electronic surface states of semiconducting alloy bismuth antimony $({\text{Bi}}_{1\ensuremath{-}x}{\text{Sb}}_{x})$. Using a phenomenological tight-binding model, we show that Fermi for 111 encloses an odd number time-reversal-invariant momenta (TRIM) in Brillouin zone. This confirms is strong topological insulator (1;111) ${\mathbb{Z}}_{2}$ class. go on to develop general arguments which spatial symmetries lead additional structure bulk energy bands, and impose further...

The celebrated ``tenfold way'' provides a scheme for categorizing general topological states of matter, but it does not take into account the crystal symmetries that always exist in real materials. A new method extends this organization to allow categorization all topologically distinct electronic band structures materials with only any number physically relevant dimensions.

Graphene is famous for being a host of 2D Dirac fermions. However, spin-orbit coupling introduces small gap, so that graphene formally quantum spin hall insulator. Here we present symmetry-protected semimetals, which feature cones at high-symmetry points are \emph{not} gapped by interactions, and exhibit behavior distinct from both 3D semimetals. Using two-site tight-binding model, construct representatives three possible semimetal phases, show single impossible in two dimensions. An...

We argue that long-range Coulomb forces convert an isolated (N,N) armchair carbon nanotube into a strongly-renormalized *Luttinger liquid*. At high temperatures, we find anomalous temperature dependences for the interaction and impurity contributions to resistivity, similar power-law local tunneling density of states. low exhibits spin-charge separation, visible as extra energy scale in discrete states (for which give analytic form), signaling departure from orthodox theory blockade.

We propose two experiments to probe the Majorana fermion edge states that occur at a junction between superconductor and magnet deposited on surface of topological insulator. Combining fermions into single Dirac magnetic domain wall allows neutral be probed with charge transport. will discuss novel interferometer for fermions, which probes their Z2 phase. This setup also transmission through point contact measured. introduce formed by superconducting show its can controlled phase difference...

We study a class of Dirac semimetals that feature an eightfold-degenerate double point. show 7 the 230 space groups can host such points and argue they all generically display linear dispersion. introduce explicit tight-binding model for 130 135, showing 135 intrinsic semimetal -- one with no additional degeneracies at Fermi energy. consider symmetry-lowering perturbations uniaxial compressive strain in different directions leads to topologically distinct insulating phases. In addition,...