Topological quantum chemistry

The past decade's apparent success in predicting and experimentally discovering distinct classes of topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000 stoichiometric compounds extant in material databases, only several hundred of them are topologically nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? To address this, we propose a new and complete electronic band theory that highlights the link between topology and local chemical bonding, and combines this with the conventional band theory of electrons. Topological Quantum Chemistry is a description of the universal global properties of all possible band structures and materials, comprised of a graph theoretical description of momentum space and a dual group theoretical description in real space. We classify the possible band structures for all 230 crystal symmetry groups that arise from local atomic orbitals, and show which are topologically nontrivial. We show how our topological band theory sheds new light on known TIs, and demonstrate the power of our method to predict a plethora of new TIs.Comment: v1: 8 pages + 40 pages supplemenetary material. Previously submitted v2: ~ Published version. 11 pages + 79 pages supplementary material. Descriptions of the data used in this paper can be found in arXiv:1706.08529 and arXiv:1706.09272. All data can be accessed via the Bilbao Crystallographic Server (http://cryst.ehu.es). Two additional papers elaborating on the general theory currently in pre

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

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

Short-Range B-site Ordering in Inverse Spinel Ferrite NiFe2O4

The Raman spectra of single crystals of NiFe2O4 were studied in various scattering configurations in close comparison with the corresponding spectra of Ni0.7Zn0.3Fe2O4 and Fe3O4. The number of experimentally observed Raman modes exceeds significantly that expected for a normal spinel structure and the polarization properties of most of the Raman lines provide evidence for a microscopic symmetry lower than that given by the Fd-3m space group. We argue that the experimental results can be explained by considering the short range 1:1 ordering of Ni2+ and Fe3+ at the B-sites of inverse spinel structure, most probably of tetragonal P4_122/P4_322 symmetry.Comment: 10 pages, 5 figures, 6 table

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPATIBILITY RELATIONS provides compatibility relations between the irreducible representations of double space groups at different k-vectors of the Brillouin zone when there is a group-subgroup relation between the corresponding little groups. The program DSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description and the search of topological insulators.Comment: 32 pages, 20 figures. Two extra figures and minor typo mistakes fixed. Published versio

Audiovisual annotation procedure for multi-view field recordings

Audio and video parts of an audiovisual document interact to produce an audiovisual, or multi-modal, perception. Yet, automatic analysis on these documents are usually based on separate audio and video annotations. Regarding the audiovisual content, these annotations could be incomplete, or not relevant. Besides, the expanding possibilities of creating audiovisual documents lead to consider different kinds of contents, including videos filmed in uncontrolled conditions (i.e. fields recordings), or scenes filmed from different points of view (multi-view). In this paper we propose an original procedure to produce manual annotations in different contexts, including multi-modal and multi-view documents. This procedure, based on using both audio and video annotations, ensures consistency considering audio or video only, and provides additionally audiovisual information at a richer level. Finally, different applications are made possible when considering such annotated data. In particular, we present an example application in a network of recordings in which our annotations allow multi-source retrieval using mono or multi-modal queries

The Raman spectrum of CaCO3 polymorphs calcite and aragonite: A combined experimental and computational study

Powder and single crystal Raman spectra of the two most common phases of calcium carbonate are calculated with ab initio techniques (using a “hybrid” functional and a Gaussian-type basis set) and measured both at 80 K and room temperature. Frequencies of the Raman modes are in very good agreement between calculations and experiments: the mean absolute deviation at 80 K is 4 and 8 cm−1 for calcite and aragonite, respectively. As regards intensities, the agreement is in general good, although the computed values overestimate the measured ones in many cases. The combined analysis permits to identify almost all the fundamental experimental Raman peaks of the two compounds, with the exception of either modes with zero computed intensity or modes overlapping with more intense peaks. Additional peaks have been identified in both calcite and aragonite, which have been assigned to 18O satellite modes or overtones. The agreement between the computed and measured spectra is quite satisfactory; in particular, simulation permits to clearly distinguish between calcite and aragonite in the case of powder spectra, and among different polarization directions of each compound in the case of single crystal spectra