There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

J. Appl. Cryst.の発刊に際して

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.

Topological quantum chemistry

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.

Higher-Order Topological Insulators

Theorists have discovered topological insulators that are insulating in their interior and on their surfaces but have conducting channels at corners or along edges.

(d-2)-Dimensional Edge States of Rotation Symmetry Protected Topological States.

Commensurate and incommensurate structures, which can be described as composites but do not fit into the usual picture of a composite are briefly reviewed. The modulation is an essential part of these systems and cannot be considered as a perturbation. In many cases, the superspace phason sliding mode does not have a physical meaning.

Composite structures beyond a perturbative modulated picture

Topological electronic flatten bands near or at the Fermi level are a promising avenue towards unconventional superconductivity and correlated insulating states. In this work, we present a catalogue of all the three-dimensional stoichiometric materials with flat bands around the Fermi level that exist in nature. We consider 55,206 materials from the Inorganic Crystal Structure Database (ICSD) catalogued using the Topological Quantum Chemistry website which provides their structural parameters, space group (SG), band structure, density of states and topological characterization. We identify several signatures and properties of band flatness: bandwidth, peaks in the density of states, band topology, range of momenta over which the band is flat and the energy window around the Fermi level where the flat bands is situated. Moreover, we perform a high-throughput analysis of all crystal structures to identify those hosting line-graph or bipartite sublattices -- either in two or three dimensions -- that likely lead to flat bands. From this trove of information, we create the Materials Flatband Database website, a powerful search engine for future theoretical and experimental studies. We use it to manually extract a curated list of 2,379 materials potentially hosting flat bands whose charge centers are not strongly localized on the atomic sites, out of which we present in minute details the 345 most promising ones. In addition, we showcase five representative materials (KAg[CN]2 in SG 163 $(P\bar{3}1c)$, Pb2Sb2O7 in SG 227 $(Fd\bar{3}m)$, Rb2CaH4 in SG 139 $(I4/mmm)$, Ca2NCl in SG 166 $(R\bar{3}m)$ and WO3 in SG 221 $(Pm\bar{3}m)$) and provide a theoretical explanation for the origin of their flat bands that close to the Fermi energy using the S-matrix method introduced in a parallel work [Calugaru et al.].

Catalogue of Flat Band Stoichiometric Materials

A general procedure for the refinement of quasicrystalline structures using diffraction data is proposed. The hypersurfaces in superspace describing the atomic positions are parametrized in terms of linear combinations of symmetry adapted functions associated to their point group symmetry. No assumption on the form of these surfaces is necessary except for the symmetry restrictions resulting from the postulated quasicrystal superspace symmetry. Using some examples, it is shown that a few harmonics in the linear combination are sufficient for describing surfaces varying from the spherical shape to forms close to polyhedra. In a typical case, the adjustable parameters and their number can be small enough for allowing a successful least-squares refinement process. Any physical restriction on the surfaces, like those coming from realistic interatomic distances, composition, etc., can be introduced in the refinement by means of penalty functions.

Systematic structure refinement of quasicrystals using symmetry-adapted parameters

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Maximal Magnetic Groups for a given propagation vector in the Bilbao Crystallographic Server: Applications.

/pdf/pyrrhotites-revisited-in-superspace-with-ab-initio-3lvof36bz5.pdf

Luis Elcoro

Papers

Composite structures beyond a perturbative modulated picture

Catalogue of Flat Band Stoichiometric Materials

Systematic structure refinement of quasicrystals using symmetry-adapted parameters

Maximal Magnetic Groups for a given propagation vector in the Bilbao Crystallographic Server: Applications.

Pyrrhotites Revisited in Superspace with ab-initio Calculations Insights