J. Appl. Cryst.の発刊に際して

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.

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Higher-Order Topology in Bismuth

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Higher-Order Topology in Bismuth.

Band structure diagram paths based on crystallography

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semimetallic and topological phases. In particular, we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles have a nontrivial topology stabilized by a magnetic symmetry, provided band gaps appear at fillings of +/- 4 electrons per moire unit cell. The topological index is given as the winding number (a Z number) of the Wilson loop in the moire Brillouin zone. Furthermore, we also claim that, when the gapped bands are allowed to couple with higher-energy bands, the Z index collapses to a stable Z(2) index. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: When strongly broken, nontopological phases can appear. Our Letter underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a four-band short-range tight-binding model whose two lower bands have the same topology, symmetry, and flatness as those of the twisted bilayer graphene and which can be used as an effective low-energy model. We then perform large-scale (11000 atoms per unit cell, 40 days per k-point computing time) ab initio calculations of a series of small angles, from 3 degrees to 1 degrees, which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical moire model but which confirm the topological nature of the system.

https://link.aps.org/accepted/10.1103/PhysRevLett.123.036401

All Magic Angles in Twisted Bilayer Graphene are Topological

In recent years, two important advances have opened new doors for the characterization and determination of magnetic structures. Firstly, researchers have produced computer-readable listings of the magnetic or Shubnikov space groups. Secondly, they have extended and applied the superspace formalism, which is presently the standard approach for the description of nonmagnetic incommensurate structures and their symmetry, to magnetic structures. These breakthroughs have been the basis for the subsequent development of a series of computer tools that allow a more efficient and comprehensive application of magnetic symmetry, both commensurate and incommensurate. Here we briefly review the capabilities of these computation instruments and present the fundamental concepts on which they are based, providing various examples. We show how these tools facilitate the use of symmetry arguments expressed as either a magnetic space group or a magnetic superspace group and allow the exploration of the possible magnetic o...

Symmetry-Based Computational Tools for Magnetic Crystallography

Received January 7, 2011; Revised April 19, 2011 The Bilbao Crystallographic Server is a web site with crystallographic databases and programs available online (www.cryst.ehu.es). It has been operating for more than ten years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The server gives access to general information related to crystallographic symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones, etc.). Apart from the simple tools for retrieving the stored data, there are programs for the analysis of group-subgroup relations between space groups. There are also software package studying specific problems of solid-state physics, structural chemistry and crystallography.

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Crystallography online: Bilbao Crystallographic Server

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Crystallography online by the Bilbao Crystallographic Server

New ferroelectrics can be predicted by considering the existence of pseudosymmetry with respect to a higher symmetry structure using the so-called atomic displacement method and investigating the minimal super-groups of the given structure's space group. This analysis can be performed with the new version of the computer program PSEUDO, located at the Bilbao Crystallographic Server. After defining the procedures for the detection and quantification of pseudosymmetry, we present the new program, illustrating its use with worked cases of polar structures from the literature which are either known or reported as possible ferroelectrics.

A new computer tool at the Bilbao Crystallographic Server to detect and characterize pseudosymmetry

The Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

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Emre Tasci

Papers

Symmetry-Based Computational Tools for Magnetic Crystallography

Crystallography online: Bilbao Crystallographic Server

Crystallography online by the Bilbao Crystallographic Server

A new computer tool at the Bilbao Crystallographic Server to detect and characterize pseudosymmetry

Brillouin-zone database on the Bilbao Crystallographic Server