Itinerant electrons in a two-dimensional kagome lattice form a Dirac semi-metal, similar to graphene. When lattice and spin symmetries are broken by various periodic perturbations this semi-metal is shown to spawn interesting non-magnetic insulating phases. These include a twodimensional topological insulator with a non-trivial Z2 invariant and robust gapless edge states, as well as dimerized and trimerized 'Kekulé' insulators. The latter two are topologically trivial but the Kekulé phase possesses a complex order parameter with fractionally charged vortex excitations. A charge density wave is shown to couple to the Dirac fermions as an effective axial gauge field.Certain physical observables in solids, such as the magnetic flux in a superconductor or the Hall conductance in a quantum Hall liquid, are precisely quantized despite the fact that the host material may contain a significant amount of disorder. In all known cases this quantization phenomenon can be attributed to the notion of topological order. The bulk of such systems are characterized by a topological invariant that is insensitive to microscopic details and robust with respect to weak disorder. Recently, a new class of topological invariants has been established to characterize all time-reversal (T ) invariant band insulators in 2 and 3 spatial dimensions [1,2,3,4]. These new invariants are of the Z 2 variety and the precisely quantized physical observable is the number of gapless edge (surface) states modulo 2. Topological insulators (TI) exhibit an odd number of edge (surface) states while trivial insulators exhibit an even number, possibly zero. In many ways the edge (surface) states of a TI behave as a perfect metal and are predicted to exhibit various unusual properties [5,6,7]. They also show promise as possible components of future quantum computers [8].Experimentally, HgTe/(Hg,Cd)Te quantum wells of certain width and composition have been identified as 2D topological 'spin Hall' insulators with robust edge states [9]. In addition, several 3D compounds involving bismuth have been so identified [10,11] and several more have been predicted [12,13] as likely candidates. In view of these rapid developments it appears likely that TIs might be a fairly common occurrence in nature. Given their exotic properties and their potential for technological applications it is important to identify and study various model systems that exhibit this behavior. Such theoretical understanding will aid experimental searches for new materials and help understand their unusual properties.In this Brief Report we advance the above agenda by describing a new class of 2-dimensional topological insulators on the kagome lattice, Fig. 1a. Although the properties of spin systems on the kagome lattice have been extensively studied, relatively little attention has been paid to the non-magnetic insulating phases of itinerant electrons. In what follows we demonstrate, both analytically and numerically, that a simple tight-binding model of electrons on the kagome lattice at both ...

When the spin-orbit coupling generates a band inversion in a narrow-bandgap semiconductor such as Sb x Bi 1−x or Bi 2 Se 3 the resulting system becomes a strong topological insulator (STI) 1,2 . A key defining property of a STI are its topologically protected metallic surface states. These are immune to the effects of non-magnetic disorder and form a basis for numerous theoretically predicted exotic phenomena 3-7 as well as proposed practical applications 8,9 . Disorder, ubiquitously present in solids, is normally detrimental to the stability of ordered states of matter. In this letter we demonstrate that not only is STI robust to disorder but, remarkably, under certain conditions disorder can become fundamentally responsible for its existence. We show that disorder, when sufficiently strong, can transform an ordinary metal with strong spin-orbit coupling into a strong topological 'Anderson' insulator, a new topological phase of quantum matter in three dimensions.Disorder is well known to play a fundamental role in low-dimensional electronic systems, leading to electron localization and consequent insulating behavior in the time-reversal invariant systems 10 . Disorder also underlies much of the phenomenology of the integer quantum Hall effect 11 . Recently, in a remarkable development, it has been noted first by numerical simulations 12 and shortly thereafter by analytical studies 13 , that a phase similar to the two dimensional topological insulator (also known as the quantum spin-Hall insulator 14,15 ) can be brought about by introducing non-magnetic disorder into a 2D metal with strong spin orbit coupling. This new 2D topological phase, referred to as topological Anderson insulator (TAI), has a disordered insulating bulk with topologically protected gapless edge states that give rise to precisely quantized conductance e 2 /h per edge. In TAI, remarkably, conductance quantization owes its very existence to disorder.A question naturally arises whether such behavior can occur in three spatial dimensions. More precisely, one may inquire whether an inherently 3D topological phase analogous to the strong topological insulator 16-18 (STI) could be reached by disordering a clean system that is initially in a topologically trivial phase. This is a nontrivial question because just as the 3D STI cannot be reduced to the set of 2D topological insulators, the existence of a 3D 'strong' TAI presumably cannot be deduced from the physics of 2D TAI. Below, we show the answer to the above question to be affirmative. Employing a combination of analytical and numerical methods we construct an explicit example of a disorder-induced topological phase in three dimensions with physical properties analogous to those of the strong topological insulator. We propose to call this new phase a 'strong topological Anderson insulator' (STAI). We argue that some of the topologically trivial compounds with strong spin-orbit coupling discussed in the recent literature, such as e.g. Sb 2 Se 3 , could become STAI upon introducing disorder. In ot...

Electrons hopping on the sites of a three-dimensional pyrochlore lattice are shown to form topologically non-trivial insulating phases when the spin-orbit (SO) coupling and lattice distortions are present. Of 16 possible topological classes 9 are realized for various parameters in this model. Specifically, at half-filling undistorted pyrochlore lattice with SO term yields a 'pristine' strong topological insulator with Z2 index (1;000). At quarter filling various strong and weak topological phases are obtained provided that both SO coupling and uniaxial lattice distortion are present. Our analysis suggests that many of the non-magnetic insulating pyrochlores could be topological insulators. 72.25.Hg, According to recent pioneering theoretical studies [1,2] all time-reversal (T ) invariant (non-magnetic) band insulators in three spatial dimensions can be classified into 16 topological classes distinguished by a four-component topological index (ν 0 ; ν 1 ν 2 ν 3 ) with ν α = 0, 1. Ordinary 'trivial' band insulators have index (0;000) and, in general, possess no robust surface states. When some of the νs differ from zero then the insulator is said to be topologically non-trivial and, as a result, possesses topologically protected surface states on at least some of its surfaces. When ν 0 = 1 surface states exist on all surfaces and are in addition robust with respect to weak non-magnetic disorder. This is referred to as a strong topological insulator (STI). Strong topological insulators are predicted to exhibit a host of unusual phenomena associated with their non-trivial surface states. These include proximity-induced exotic superconducting state with Majorana fermions bound to a vortex [3], spincharge separated solitonic excitations [4,5], and, in a thin film geometry, an unconventional excitonic state with fractionally charged vortices [6]. There are also interesting bulk manifestations of STI physics such the 'axion' electromagnetic response [7,8] and the topologically protected fermion modes localized along the core of a crystal dislocation [9].Topologically nontrivial insulating phases have been predicted to occur [10,11,12] and subsequently experimentally discovered [13,14,15] in several 2 and 3-dimensional crystalline solids. Vigorous search for new materials in this class is ongoing. With the goal of enlarging the space of candidate crystalline structures that can potentially support topologically non-trivial insulating phases we study in this Letter a class of tight-binding models with SO coupling for electrons moving on the pyrochlore lattice displayed in Fig. 1a. Our model belongs to the class of 3D 'frustrated hopping' models [16] and the motivation for this study comes in part from our recent finding that electrons on the kagome lattice, a canonical example of the frustrated structure in 2D, form a 2D topological insulator when SO coupling is present [17].Our main finding here is that, quite generically, whenever electrons hopping on the pyrochlore lattice acquire a band gap from SO interactions the r...

Electrons on the surface of a strong topological insulator, such as Bi2Te3 or Bi1−xSbx, form a topologically protected helical liquid whose excitation spectrum contains an odd number of massless Dirac fermions. A theoretical survey and classification is given of the universal features, observable by the ordinary and spin-polarized scanning tunneling spectroscopy, in the interference patterns resulting from the quasiparticle scattering by magnetic and non-magnetic impurities in such a helical liquid. Our results confirm the absence of backscattering from non-magnetic impurities observed in recent experiments and predict new interference features, uniquely characteristic of the helical liquid, when the scatterers are magnetic.