Lifetimes of Shockley electrons and holes at the Cu(111) surface

A theoretical many-body analysis is presented of the electron-electron inelastic lifetimes of Shockley electrons and holes at the (111) surface of Cu. For a description of the decay of Shockley states both below and above the Fermi level, single-particle wave functions have been obtained by solving the Schr\"odinger equation with the use of an approximate one-dimensional pseudopotential fitted to reproduce the correct bulk energy bands and surface-state dispersion. A comparison with previous calculations and experiment indicates that inelastic lifetimes are very sensitive to the actual shape of the surface-state single-particle orbitals beyond the $\bar\Gamma$ (${\bf k}_\parallel=0$) point, which controls the coupling between the Shockley electrons and holes.Comment: 4 pages, 3 figures, to appear in Phys. Rev.

Non-Dirac topological surface states in (SnTe)n≥2_{n\geq2}(Bi2_2Te3_3)m=1_{m=1}

A new type of topological spin-helical surface states was discovered in layered van der Waals bonded (SnTe)$_{n=2,3}$(Bi$_2$Te$_3$)$_{m=1}$ compounds which comprise two covalently bonded band inverted subsystems, SnTe and Bi$_2$Te$_3$, within a building block. This novel topological states demonstrate non-Dirac dispersion within the band gap. The dispersion of the surface state has two linear sections of different slope with shoulder feature between them. Such a dispersion of the topological surface state enables effective switch of the velocity of topological carriers by means of applying an external electric field

Self-energy and lifetime of Shockley and image states on Cu(100) and Cu(111): Beyond the GW approximation of many-body theory

We report many-body calculations of the self-energy and lifetime of Shockley and image states on the (100) and (111) surfaces of Cu that go beyond the $GW$ approximation of many-body theory. The self-energy is computed in the framework of the GW\Gamma approximation by including short-range exchange-correlation (XC) effects both in the screened interaction W (beyond the random-phase approximation) and in the expansion of the self-energy in terms of W (beyond the GW approximation). Exchange-correlation effects are described within time-dependent density-functional theory from the knowledge of an adiabatic nonlocal XC kernel that goes beyond the local-density approximation.Comment: 8 pages, 5 figures, to appear in Phys. Rev.

Graph Theory Data for Topological Quantum Chemistry

Topological phases of noninteracting particles are distinguished by global properties of their band structure and eigenfunctions in momentum space. On the other hand, group theory as conventionally applied to solid-state physics focuses only on properties which are local (at high symmetry points, lines, and planes) in the Brillouin zone. To bridge this gap, we have previously [B. Bradlyn et al., Nature 547, 298--305 (2017)] mapped the problem of constructing global band structures out of local data to a graph construction problem. In this paper, we provide the explicit data and formulate the necessary algorithms to produce all topologically distinct graphs. Furthermore, we show how to apply these algorithms to certain "elementary" band structures highlighted in the aforementioned reference, and so identified and tabulated all orbital types and lattices that can give rise to topologically disconnected band structures. Finally, we show how to use the newly developed BANDREP program on the Bilbao Crystallographic Server to access the results of our computation.Comment: v1: 29 Pages, 13 Figures. Explains how to access the data presented in arXiv:1703.02050 v2: Accepted version. References updated, figures improve

Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem

The conventional theory of solids is well suited to describing band structures locally near isolated points in momentum space, but struggles to capture the full, global picture necessary for understanding topological phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298 (2017)], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local "k-dot-p" band structures across the Brillouin zone in terms of graph theory. In the current manuscript we give the details of our full theoretical construction. We show that crystal symmetries strongly constrain the allowed connectivities of energy bands, and we employ graph-theoretic techniques such as graph connectivity to enumerate all the solutions to these constraints. The tools of graph theory allow us to identify disconnected groups of bands in these solutions, and so identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529 v2: Accepted version, minor typos corrected and references added. Now 19+epsilon page

Building Blocks of Topological Quantum Chemistry: Elementary Band Representations

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In Nature 547, 298-305 (2017), we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, which we introduced in Nature 547, 298-305 (2017), we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.Comment: 15+9 pages, 4 figures; v2: minor corrections; v3: updated references (published version

Higher-Order Topological Insulators

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation symmetry. Their hinge states are chiral modes and the bulk topology is $\mathbb{Z}_2$-classified. (2) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs and the bulk topology is $\mathbb{Z}$-classified. We provide the topological invariants for both cases. Furthermore we show that SnTe as well as surface-modified Bi$_2$TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.Comment: 8 pages (4 figures) and 16 pages supplemental material (7 figures