We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of Ktheory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band inversions at high symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure, and identify whether or not they give rise to intermediate Weyl semimetallic phases. arXiv:1612.02007v1 [cond-mat.mes-hall]

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...

We consider conditions for the existence of boundary modes in non-Hermitian systems with edges of arbitrary co-dimension. Through a universal formulation of formation criteria for boundary modes in terms of local Green functions, we outline a generic perspective on the appearance of such modes and generate corresponding dispersion relations. In the process, we explain the skin effect in both topological and non-topological systems, exhaustively generalizing bulk-boundary correspondence in the presence of non-Hermiticity. This is accomplished via a doubled Green's function, inspired by doubled Hamiltonian methods used to classify Floquet and, more recently, non-Hermitian topological phases. Our work constitutes a general tool, as well as, a unifying perspective for this rapidly evolving field. Indeed, as a concrete application we find that our method can expose novel non-Hermitian topological regimes beyond the reach of previous methods.arXiv :1902.07217v2 [cond-mat.mes-hall]

We illustrate a procedure that defines and converts non-Abelian charges of Weyl nodes via braid phase factors, which arise upon exchange inside the reciprocal momentum space. This phenomenon derives from intrinsic symmetry properties of topological materials, which are increasingly becoming available due to recent cataloguing insights. Specifically, we demonstrate that band nodes in systems with C2T symmetry exhibit such braiding properties, requiring no particular fine-tuning. We further present observables in the form of generalized Berry phases, calculated via a mathematical object known as Euler form. We demonstrate our findings with explicit models and a protocol involving three bands, for which the braid factors mimic quaternion charges. This protocol is implementable in cold atoms setups and in photonic systems, where observing the proposed braid factors relates to readily available experimental techniques. The required C2T symmetry is also omnipresent in graphene van-der-Waals heterostructures, which might provide an alternative route towards realizing the non-Abelian conversion of band nodes.

We present a self-contained review of the theory of dislocation-mediated quantum melting at zero temperature in two spatial dimensions. The theory describes the liquid-crystalline phases with spatial symmetries in between a quantum crystalline solid and an isotropic superfluid: quantum nematics and smectics. It is based on an Abelian-Higgs-type duality mapping of phonons onto gauge bosons ("stress photons"), which encode for the capacity of the crystal to propagate stresses. Dislocations and disclinations, the topological defects of the crystal, are sources for the gauge fields and the melting of the crystal can be understood as the proliferation (condensation) of these defects, giving rise to the Anderson-Higgs mechanism on the dual side. For the liquid crystal phases, the shear sector of the gauge bosons becomes massive signaling that shear rigidity is lost. After providing the necessary background knowledge, including the order parameter theory of two-dimensional quantum liquid crystals and the dual theory of stress gauge bosons in bosonic crystals, the theory of melting is developed step-by-step via the disorder theory of dislocation-mediated melting. Resting on symmetry principles, we derive the phenomenological imaginary time actions of quantum nematics and smectics and analyze the full spectrum of collective modes. The quantum nematic is a superfluid having a true rotational Goldstone mode due to rotational symmetry breaking, and the origin of this 'deconfined' mode is traced back to the crystalline phase. The two-dimensional quantum smectic turns out to be a dizzyingly anisotropic phase with the collective modes interpolating between the solid and nematic in a non-trivial way. We also consider electrically charged bosonic crystals and liquid crystals, and carefully analyze the electromagnetic response of the quantum liquid crystal phases. In particular, the quantum nematic is a real superconductor and shows the Meissner effect. Their special properties inherited from spatial symmetry breaking show up mostly at finite momentum, and should be accessible by momentum-sensitive spectroscopy.

We present a general methodology towards the systematic characterization of crystalline topological insulating phases with time reversal symmetry (TRS). In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed "topological quantum chemistry" (TQC) concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele Z2 classification, there is C2T -symmetry protected Z classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e. forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parameterization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial", i.e. necessarily non-trivial in the non-stable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele Z2 invariant implying that further stable topological phases must belong to the class of higher-order topological insulators. arXiv:1804.09719v4 [cond-mat.mes-hall]

We show that the local in-gap Green's function of a band insulator G 0 ( ,k ,r ⊥ = 0), with r ⊥ the position perpendicular to a codimension-1 or codimension-2 impurity, reveals the topological nature of the phase. For a topological insulator, the eigenvalues of this Green's function attain zeros in the gap, whereas for a trivial insulator the eigenvalues remain nonzero. This topological classification is related to the existence of in-gap bound states along codimension-1 and codimension-2 impurities. Whereas codimension-1 impurities can be viewed as soft edges, the result for codimension-2 impurities is nontrivial and allows for a direct experimental measurement of the topological nature of two-dimensional insulators. Introduction. The topological characterization of condensed states of matter has emerged as a prominent research interest over the last few decades. The flourishing of the quantum Hall effect (QHE) [1] in particular elucidated many connections between physical signatures and topological invariants [2], which supplement the order parameters of the usual symmetry-breaking Landau-Ginzburg paradigm. More recently, however, it became apparent that topological order can also arise by virtue of symmetry, and in particular the very common and robust time-reversal (TR) symmetry is sufficient to establish the existence and stability of topological insulators [3][4][5]. This is quantified via a Z 2 invariant, and results in gapless helical edge states or chiral Dirac fermions localized at the perimeter of the sample in two and three dimensions, respectively. The topological insulator has proven extremely rich both experimentally and theoretically [6,7]. The concept has been generalized to a periodic table describing various discrete symmetries and dimensions [8][9][10]. Lattice symmetries can similarly lead to further topological distinctions, resulting in crystalline topological insulators [11], for which a general classification has been provided [12].One may ask whether the topology of band insulators has some local signature. In fact, in this paper, we will show that even the fully local in-gap Green's function contains information about the band topology, which is then directly accessible by experiments. The natural way this insight arises is through the study of impurities [14][15][16], similar to how the space group classification can be probed using lattice defects [17][18][19][20][21][22][23][24]. Consider a codimension-1 impurity line or surface in an insulator. In the limit where the impurity strength diverges, V → ∞, such an impurity acts like a real edge, which, following the bulk-boundary correspondence, should host zero gap metallic bound states in the topological phase. For finite V the codimension-1 impurity surface can thus be viewed as a soft edge. The codimension-2 impurity lines or points do not host gapless states in the strong V limit, so a priori there is no reason to expect they probe topology. However, we will see that they in fact inherit the topological structure of the soft edges....

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on C 2 T-symmetric systems that have gained recent attention, for example, in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Plücker embedding, to induce windings in the band structure necessary to facilitate nontrivial topology. Specifically, this directly relates to the parametrization of the Grassmannian, which describes partitioning of an arbitrary band structure and is embedded in a better manageable exterior product space. From a physical perspective, our construction encapsulates and elucidates the concepts of fragile topological phases beyond symmetry indicators as well as non-Abelian reciprocal braiding of band nodes that arises when the multiple gaps are taken into account. The adopted geometric viewpoint most importantly culminates in a direct and easily implementable method to construct model Hamiltonians to study such phases, constituting a versatile theoretical tool.