Double crystallographic groups and their representations on the Bilbao Crystallographic Server
01 Oct 2017-Journal of Applied Crystallography (International Union of Crystallography)-Vol. 50, Iss: 5, pp 1457-1477
TL;DR: In this article, 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).
Abstract: 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 DCOMPREL 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 of and the search for topological insulators.
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TL;DR: A complete electronic band theory is proposed, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding and can be used to predict many more topological insulators.
Abstract: Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.
1,150 citations
TL;DR: Using a recently developed formalism called topological quantum chemistry, a high-throughput search of ‘high-quality’ materials in the Inorganic Crystal Structure Database is performed and it is found that more than 27 per cent of all materials in nature are topological.
Abstract: Using a recently developed formalism called topological quantum chemistry, we perform a high-throughput search of 'high-quality' materials (for which the atomic positions and structure have been measured very accurately) in the Inorganic Crystal Structure Database in order to identify new topological phases. We develop codes to compute all characters of all symmetries of 26,938 stoichiometric materials, and find 3,307 topological insulators, 4,078 topological semimetals and no fragile phases. For these 7,385 materials we provide the electronic band structure, including some electronic properties (bandgap and number of electrons), symmetry indicators, and other topological information. Our results show that more than 27 per cent of all materials in nature are topological. We provide an open-source code that checks the topology of any material and allows other researchers to reproduce our results.
782 citations
TL;DR: In this paper, the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulkboundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes.
Abstract: The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.
457 citations
TL;DR: It is established that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk–boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes.
Abstract: The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional surface states. This constitutes the topological bulk-boundary correspondence. Here, we establish that the electronic structure of bismuth, an element consistently described as bulk topologically trivial, is in fact topological and follows a generalized bulk-boundary correspondence of higher-order: not the surfaces of the crystal, but its hinges host topologically protected conducting modes. These hinge modes are protected against localization by time-reversal symmetry locally, and globally by the three-fold rotational symmetry and inversion symmetry of the bismuth crystal. We support our claim theoretically and experimentally. Our theoretical analysis is based on symmetry arguments, topological indices, first-principle calculations, and the recently introduced framework of topological quantum chemistry. We provide supporting evidence from two complementary experimental techniques. With scanning-tunneling spectroscopy, we probe the unique signatures of the rotational symmetry of the one-dimensional states located at step edges of the crystal surface. With Josephson interferometry, we demonstrate their universal topological contribution to the electronic transport. Our work establishes bismuth as a higher-order topological insulator.
440 citations
TL;DR: An effective, efficient and fully automated algorithm that diagnoses the nontrivial band topology in a large fraction of nonmagnetic materials is introduced, based on recently developed exhaustive mappings between the symmetry representations of occupied bands and topological invariants.
Abstract: Topological electronic materials such as bismuth selenide, tantalum arsenide and sodium bismuthide show unconventional linear response in the bulk, as well as anomalous gapless states at their boundaries. They are of both fundamental and applied interest, with the potential for use in high-performance electronics and quantum computing. But their detection has so far been hindered by the difficulty of calculating topological invariant properties (or topological nodes), which requires both experience with materials and expertise with advanced theoretical tools. Here we introduce an effective, efficient and fully automated algorithm that diagnoses the nontrivial band topology in a large fraction of nonmagnetic materials. Our algorithm is based on recently developed exhaustive mappings between the symmetry representations of occupied bands and topological invariants. We sweep through a total of 39,519 materials available in a crystal database, and find that as many as 8,056 of them are topologically nontrivial. All results are available and searchable in a database with an interactive user interface. Topological materials are thought to be scarce, but an algorithm that diagnoses nontrivial topology in nonmagnetic materials finds the opposite: more than 30 per cent of the 26,688 materials studied are topological.
429 citations
References
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TL;DR: A complete electronic band theory is proposed, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding and can be used to predict many more topological insulators.
Abstract: Since the discovery of topological insulators and semimetals, there has been much research into predicting and experimentally discovering distinct classes of these materials, in which the topology of electronic states leads to robust surface states and electromagnetic responses. This apparent success, however, masks a fundamental shortcoming: topological insulators represent only a few hundred of the 200,000 stoichiometric compounds in material databases. However, it is unclear whether this low number is indicative of the esoteric nature of topological insulators or of a fundamental problem with the current approaches to finding them. Here we propose a complete electronic band theory, which builds on the conventional band theory of electrons, highlighting the link between the topology and local chemical bonding. This theory of topological quantum chemistry provides a description of the universal (across materials), global properties of all possible band structures and (weakly correlated) materials, consisting of a graph-theoretic description of momentum (reciprocal) space and a complementary group-theoretic description in real space. For all 230 crystal symmetry groups, we classify the possible band structures that arise from local atomic orbitals, and show which are topologically non-trivial. Our electronic band theory sheds new light on known topological insulators, and can be used to predict many more.
1,150 citations
TL;DR: The aim of the article is to report on the current state of the Bilbao Crystallographic Server and to provide a brief description of the accessible databases and crystallographic computing programs.
Abstract: The Bilbao Crystallographic Server is a web site with crystallographic databases and programs available on-line at www.cryst.ehu.es. It has been operating for about six years and new applications are being added reg- ularly. The programs available on the server do not need a local installation and can be used free of charge. The only requirement is an Internet connection and a web browser. The server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The k-vector database in- cludes the Brillouin zones and the wave-vector types for all space groups. As a part of the server one can find also the database of incommensurate structures. The second shell contains applications which are essential for prob- lems involving group-subgroup relations between space groups (e.g. subgroups and supergroups of space groups, splittings of Wyckoff positions), while the third shell con- tains more sophisticated programs for the computation of space-group representations and their correlations for group-subgroup related space groups. There are also pro- grams for calculations focused on specific problems of so- lid-state physics. The aim of the article is to report on the current state of the server and to provide a brief descrip- tion of the accessible databases and crystallographic com- puting programs. The use of the programs is demonstrated by illustrative examples.
893 citations
TL;DR: In this paper, a quantitative Mas fur die Abweichung von der kubischen Symmetrie definiert werden, welches eindeutig die stabilste Elektronenanordnung im Kristall bestimmt.
Abstract: Der Einflus eines elektrischen Feldes von vorgegebener Symmetrie (Kristallfeld) auf ein Atom wird wellenmechanisch behandelt. Die Terme des Atoms spalten auf in einer Weise, die von der Symmetrie des Feldes und vom Drehimpuls l (bzw. j) des Atoms abhangt. s-Terme spalten allgemein, p-Terme in Feldern von kubischer Symmetrie nicht auf. Fur den Fall, das die einzelnen Elektronen des Atoms separat behandelt werden durfen (aufgehobene Wechselwirkung im Atom), werden zu jedem Term im Kristall die Eigenfunktionen nullter Naherung angegeben; aus diesen ergibt sich eine fur den Term charakteristische Gruppierung der Elektronendichte nach den Symmetrieachsen des Kristalls. — Die Grose der Termaufspaltung bewegt sich in der Ordnung einiger hundert cm−1. — Bei tetragonaler Symmetrie kann ein quantitatives Mas fur die Abweichung von der kubischen Symmetrie definiert werden, welches eindeutig die stabilste Elektronenanordnung im Kristall bestimmt.
871 citations