Spatial symmetries in crystals may be distinguished by whether they preserve the spatial origin. Here we study spatial symmetries that translate the origin by a fraction of the lattice period, and find that these non-symmorphic symmetries protect an exotic surface fermion whose dispersion relation is shaped like an hourglass; surface bands connect one hourglass to the next in an unbreakable zigzag pattern. These 'hourglass' fermions are formed in the large-gap insulators, KHgX (X = As, Sb, Bi), which we propose as the first material class whose band topology relies on non-symmorphic symmetries. Besides the hourglass fermion, another surface of KHgX manifests a three-dimensional generalization of the quantum spin Hall effect, which has previously been observed only in two-dimensional crystals. To describe the bulk topology of non-symmorphic crystals, we propose a non-Abelian generalization of the geometric theory of polarization. Our non-trivial topology originates from an inversion of the rotational quantum numbers, which we propose as a criterion in the search for topological materials.

The ground state of translationally-invariant insulators comprise bands which can assume topologically distinct structures. There are few known examples where this distinction is enforced by a point-group symmetry alone. In this paper we show that 1D and 2D insulators with the simplest point-group symmetry -inversion -have a Z ≥ classification. In 2D, we identify a relative winding number that is solely protected by inversion symmetry. By analysis of Berry phases, we show that this invariant has similarities with the first Chern class (of time-reversal breaking insulators), but is more closely analogous to the Z2 invariant (of time-reversal invariant insulators). Implications of our work are discussed in holonomy, the geometric-phase theory of polarization, the theory of maximally-localized Wannier functions, and in the entanglement spectrum.There is strong evidence supporting the view that the topological properties of a condensed-matter system are encoded in its ground state alone. The ground state of translationally-invariant insulators comprise bands, which can assume topologically distinct structures. Bands are deemed distinct when they are not connected by continuous reparametrization of the Hamiltonian that preserves the energy gap. Some bands are distinct only because some reparametrizations are disallowed by a certain symmetry; in this sense we say that the topological distinction is protected by that symmetry. The symmetries which are ubiquitous in crystals are the point-group symmetries, which involve transformations that preserve a spatial point. Despite the large number of space groups in nature, there are few known examples in which the topological distinction is protected by a point-group symmetry alone. 46,47 In this paper we show that such distinction exists for insulators with arguably the simplest point-group symmetry -inversion (I). [48][49][50][51][52] In search for a tool to identify topological structure in bands, we note that the description of translationallyinvariant insulators has a local gauge redundancy -its ground state is invariant under a unitary transformation in the subspace of occupied bands. Since all topological quantities must be invariant under this transformation, the natural objects to investigate are the Berry phase factors acquired around a loop, which are known to be gauge-invariant quantites. [53][54][55][56][57][58][59][60][61][62] We are proposing that distinct bands can be distinguished by holonomy, i.e., parallel transport through certain non-contractible loops in the Brillouin zone. Holonomies are known to have diverse applications in physics. [63][64][65] The matrix representation of parallel transport is called a Wilson loop (W), and its eigenspectrum comprise the non-Abelian Berry phase factors. [66][67][68][69] A topological insulator cannot be continuously transformed to a direct-product state. This corresponds to a limit where all hoppings between atoms are turned off, so the ground state is a direct product of atomic wavefunctions. Such a limit is easily s...

The hallmark of a time-reversal symmetry protected topologically insulating state of matter in two-dimensions (2D) is the existence of chiral edge modes propagating along the perimeter of the system 1-5 . To date, evidence for such electronic modes has come from experiments on semiconducting heterostructures in the topological phase which showed approximately quantized values of the overall conductance 6-8 as well as edge-dominated current flow 9 . However, there have not been any spectroscopic measurements to demonstrate the one-dimensional (1D) nature of the edge modes. Among the first systems predicted to be a 2D topological insulator are bilayers of bismuth (Bi) 4 and there have been recent experimental indications of possible topological boundary states at their edges 10-13 . However, the experiments on such bilayers suffered from irregular structure of their edges or the coupling of the edge states to substrate's bulk states. Here we report scanning tunneling microscopy (STM) experiments which show that a subset of the predicted Bi-bilayers' edge states are decoupled from states of Bi substrate and provide direct spectroscopic evidence of their 1D nature. Moreover, by visualizing the quantum interference of edge mode quasi-particles in confined geometries, we demonstrate their remarkable coherent propagation along the edge with scattering properties that

We employed ab initio calculations to identify a class of crystalline materials of MSi (M=Fe, Co, Mn, Re, Ru) having double-Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed "spin-1 Weyl point" at the Brillouin zone center and "charge-2 Dirac point" at the zone corner. The corresponding gapless surface phonon dispersions are two helicoidal sheets whose isofrequency contours form a single noncontractible loop in the surface Brillouin zone. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions.