Topological band insulators (TBIs) are bulk insulating materials which feature topologically protected metallic states on their boundary. The existing classification departs from time-reversal symmetry, but the role of the crystal lattice symmetries in the physics of these topological states remained elusive. Here we provide the classification of TBIs protected not only by time-reversal, but also by crystalline symmetries. We find three broad classes of topological states: (a) Γ states robust against general time-reversal invariant perturbations; (b) Translationally-active states protected from elastic scattering, but susceptible to topological crystalline disorder; (c) Valley topological insulators sensitive to the effects of non-topological and crystalline disorder. These three classes give rise to 18 different two-dimensional, and, at least 70 three-dimensional TBIs, opening up a route for the systematic search for new types of TBIs. 1 arXiv:1209.2610v2 [cond-mat.mes-hall] 19 Nov 2012 Topological phases of free fermionic matter are in general characterized by an insulating gap in the bulk and protected gapless modes on the boundary of the system[1, 2]. Integer quantum Hall states represent first examples of topologically protected phases in absence of any symmetries with the topological invariant directly related to the measured Hall conductance[3]. Recently, it became understood that even in the presence of fundamental symmetries such as time-reversal, topologically protected states of matter can, in principle, exist. In particular, it has been shown that time-reversal invariant (TRI) insulators in two dimensions (2D)[4] and three dimensions (3D) [5, 6, 7] are characterized by Z 2 topological invariants which pertain to the existence of the gapless boundary modes robust against time-reversal preserving perturbations, and may host Majorana quasiparticles[8], as well as provide the condensed-matter realization of the theta-vacuum[9]. The theoretical prediction [10, 11] and experimental realization of the Z 2 -invariant topological band insulators [12,13,14,15,16] gave a crucial boost in the understanding of these exotic phases of matter which culminated in the general classification of topological insulators and superconductors based on time-reversal symmetry (TRS) and particle-hole symmetry (PHS) within the so-called tenfold periodic table [17,18,19].The role of the crystal lattice in this classification is to provide a unit cell in the momentum space, the Brillouin zone (BZ), topologically equivalent to the d-dimensional torus, over which the electronic Bloch wavefunctions are defined. The tenfold classification follows then assuming that all the unitary symmetries of the corresponding Bloch Hamiltonian have been exhausted and therefore the only remaining symmetries are, according to the Wigner's theorem, antiunitary TRS and PHS. In three dimensions (3D), however, by considering a Z 2 TBI as a stack of two-dimensional (2D) ones, thus assuming a layered 3D lattice, additional three "weak" invariants associated...

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this paper we present a non-trivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave-functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wavefunc-tions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the non-approximate study of topological one-dimensional parafermionic chains with spatial-inversion and time-reversal symmetry in the absence of strong edge modes. Introduction. The study of topological order (TO) is currently one of the most active research fields in condensed-matter physics. From the AKLT model [1] to the Laughlin wavefunction [2], from the Kitaev chain [3] to the Toric code [4], this study has always benefited from the development of exactly-solvable models and of paradigmatic wavefunctions, whose detailed analysis permits the formation of a clear physical intuition, to be used in the understanding of complex experimental setups. In this letter we focus on parafermions, a generalization of Majorana fermions [5]. After the experimental clarification that two zero-energy Majorana modes can be localized at the edges of a one-dimensional fermionic wire [6, 7], the possibility of localizing parafermionic modes, and letting them interact, is currently under deep investigation. These excitations cannot appear in strictly one-dimensional spinless fermionic systems [8, 9], but may emerge at the edge of a two-dimensional fractional topological insulator coupled to alternating ferro-magnetic and superconducting materials [5, 10-15], as well as in other nanostructures or models [16-24]. In these setups, one-dimensional chains of interacting parafermions arise, which, in certain circumstances, display TO and edge Z N parafermionic modes [5, 25-34]. Such edge modes are called strong when they commute with the Hamiltonian [35] and thereby generate a N-fold degeneracy in the entire spectrum, and weak when the commutation property and associated degeneracy are restricted to the ground state manifold. TO survives weak perturbations and hosts indistinguishably weak or strong modes [36]. The importance of parafermionic zero-modes for topological quantum computation [37] motivates further investigations of these fractionalized systems. In this letter we provide a non-trivial family of parafermionic models for which the properties of the ground states can be exactly characterized. These models are gapped, display TO, have spatial-inversion and time-reversal symmetries, and feature weak edge modes; they thus belong to the same symmetry class for which weak edge modes have been discussed so far with numerical and perturbative analytical methods [28, 31, 36], with the advantage of being easy to handle. We analytically establish ...

We study the coexisting smectic modulations and intra-unit-cell nematicity in the pseudogap states of underdoped Bi(2)Sr(2)CaCu(2)O(8+δ). By visualizing their spatial components separately, we identified 2π topological defects throughout the phase-fluctuating smectic states. Imaging the locations of large numbers of these topological defects simultaneously with the fluctuations in the intra-unit-cell nematicity revealed strong empirical evidence for a coupling between them. From these observations, we propose a Ginzburg-Landau functional describing this coupling and demonstrate how it can explain the coexistence of the smectic and intra-unit-cell broken symmetries and also correctly predict their interplay at the atomic scale. This theoretical perspective can lead to unraveling the complexities of the phase diagram of cuprate high-critical-temperature superconductors.

In topologically ordered quantum states of matter in ð2 þ 1ÞD (spacetime dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property that is directly related to global transformations of the ground-state wave functions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in ð3 þ 1ÞD, which exhibit both pointlike and looplike excitations, but systematic understanding of the fundamental physical distinctions between phases, and how these distinctions are connected to quantum statistics of excitations, is still lacking. One main result of this work is that the three-dimensional generalization of modular transformations, when applied to topologically ordered ground states, is directly related to a certain braiding process of looplike excitations. This specific braiding surprisingly involves three loops simultaneously, and can distinguish different topologically ordered states. Our second main result is the identification of the three-loop braiding as a process in which the worldsheets of the three loops have a nontrivial triple linking number, which is a topological invariant characterizing closed two-dimensional surfaces in four dimensions. In this work, we consider realizations of topological order in ð3 þ 1ÞD using cohomological gauge theory in which the loops have Abelian statistics and explicitly demonstrate our results on examples with Z 2 × Z 2 topological order.

Essentials of the scientific discovery process have remained largely unchanged for centuries 1 : systematic human observation of natural phenomena is used to form hypotheses that, when validated through experimentation, are generalized into established scientific theory. Today, however, we face major challenges because automated instrumentation and large-scale data acquisition are generating data sets of such volume and complexity as to defy human analysis. Radically different scientific approaches are needed, with machine learning (ML) showing great promise, not least for materials science research 2-5 . Hence, given recent advances in ML analysis of synthetic data representing electronic quantum matter (EQM) 6-16 , the next challenge is for ML to engage equivalently with experimental data. For example, atomic-scale visualization of EQM yields arrays of complex electronic structure images 17 , that frequently elude effective analyses. Here we report development and training of an array of artificial neural networks (ANN) designed to recognize different types of hypothesized order hidden in EQM imagearrays. These ANNs are used to analyze an experimentally-derived EQM image archive from carrier-doped cuprate Mott insulators. Throughout these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unitcell periodic, translational-symmetry-breaking EQM state. Further, the ANNs find these phenomena to be unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals 18,19 are congruent with all these observations. 1Frontier research in EQM concentrates on exotic electronic phases that emerge when electrons interact so strongly that they lack a definite momentum. These electrons often self-organize into complex new states of EQM including, for example, electronic liquid crystals 18,19 , high temperature superconductors 20,21 , fractionalized electronic fluids and quantum spin liquids. In this field, vast experimental data sets have emerged, for example from real space (r-space) visualization of EQM using spectroscopic imaging scanning tunneling microscopy 17 (SISTM), from momentum space (k-space) visualization of EQM using angle resolved photoemission (ARPES), or from modern X-ray 22 and neutron scattering. The challenge is to develop ML strategies capable of scientific discovery using such large and complex experimental data structures from EQM experiments. 2An excellent example is the electronic structure of the CuO2 plane in the cuprate compounds supporting high temperature superconductivity 20 (Fig. 1a). With one electron per Cu site, strong Coulomb interactions produce charge localization in an antiferromagnetic Mott insulator (MI) state. Removing p electrons (adding p 'holes') perCuO2 plaquette generates the 'pseudogap' (PG) phase 20 . It exhibits strongly depleted density-of-electronic states ( ) for energies |E| < Δ ! , where Δ ! is the characteristic pseudogap energy scale that emerges for < * ( ) (Fig. 1a). Although the PG pha...

We show that the π flux and the dislocation represent topological observables that probe two-dimensional topological order through binding of the zero-energy modes. We analytically demonstrate that π flux hosts a Kramers pair of zero modes in the topological Γ (Berry phase Skyrmion at the zero momentum) and M (Berry phase Skyrmion at a finite momentum) phases of the M-B model introduced for the HgTe quantum spin Hall insulator. Furthermore, we analytically show that the dislocation acts as a π flux, but only so in the M phase. Our numerical analysis confirms this through a Kramers pair of zero modes bound to a dislocation appearing in the M phase only, and further demonstrates the robustness of the modes to disorder and the Rashba coupling. Finally, we conjecture that by studying the zero modes bound to dislocations all translationally distinguishable two-dimensional topological band insulators can be classified.

We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this K-b-t rule, the lattice topology, represented by dislocation lines oriented in direction t with Burgers vector b, combines with the electronic-band topology, characterized by the band-inversion momentum K inv , to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux = K inv · b(mod 2π ). Although it has already been discovered by Ran et al. [Nat. Phys. 5, 298 (2009)] that dislocation lines host propagating modes, the exact mechanism of their appearance in conjunction with the crystal symmetries of a topological state is provided by the K-b-t rule. Finally, we discuss possible experimentally consequential examples in which the modes are oblivious to the direction of propagation, such as the recently proposed topologically insulating state in electron-doped BaBiO 3 . Topological band insulators (TBIs) represent a new class of quantum materials that, due to the presence of time-reversal symmetry (TRS), feature an insulating bulk band gap together with metallic edge or surface states protected by a Z 2 topological invariant [1][2][3][4]. This Z 2 classification of TBIs is a part of the classification of free gapped fermion matter in the presence of the fundamental antiunitary time-reversal and particle-hole symmetries, the so-called tenfold way [5][6][7]. On the other hand, topologically insulating crystals break continuous translational and rotational symmetries down to discrete symmetries mathematically characterized by the space groups. By considering the crystal symmetries, an extra layer in this Z 2 classification of TBIs has been recently uncovered [8]. This space group classification of TBIs results in the enrichment of the tenfold way with extra phases, such as crystalline (or "valley") phases [9] and translationally active phases, the latter featuring an odd number of band inversions at non-points in the Brillouin zone (BZ) [8,10]. Dislocations are of central interest in this endeavor, being the topological defects exclusively related to the lattice translations. In two dimensions (2D), the role of these lattice defects has been recently elucidated in TBIs [10,11], as well as in topological superconductors [12,13] and interacting topological states [14][15][16]. In particular, it has been shown that these lattice defects in two-dimensional TBIs act as probes of distinct topological states through binding of the localized zero-energy modes [10].Although early on it was identified that in three-dimensional TBIs dislocation lines support propagating helical modes [17], the precise role of dislocations has not been explored thoroughly [18][19][20]. In particular, the relation between the lattice symmetry and the electronic topology, as well as the characterization of these topological states through the response of the dislocation lines, still needs to be addressed. Dislocations in thr...

We introduce a model for amorphous grain boundaries in graphene and find that stable structures can exist along the boundary that are responsible for local density of states enhancements both at zero and finite ͑ϳ0.5 eV͒ energies. Such zero-energy peaks, in particular, were identified in STS measurements ͓J. Červenka, M. I. Katsnelson, and C. F. J. Flipse, Nat. Phys. 5, 840 ͑2009͔͒ but are not present in the simplest pentagonheptagon dislocation array model ͓O. V. Yazyev and S. G. Louie, Phys. Rev. B 81, 195420 ͑2010͔͒. We consider the low-energy continuum theory of arrays of dislocations in graphene and show that it predicts localized zero-energy states. Since the continuum theory is based on an idealized lattice scale physics it is a priori not literally applicable. However, we identify stable dislocation cores, different from the pentagonheptagon pairs that do carry zero-energy states. These might be responsible for the enhanced magnetism seen experimentally at graphite grain boundaries.