Topological band theory has achieved great success in the high-throughput search for topological band structures both in paramagnetic and magnetic crystal materials. However, a significant proportion of materials are topologically trivial insulators
at the Fermi level. In this paper, we show that, remarkably, for a subset of the topologically trivial insulators, knowing only their electron number and the Wyckoff positions of the atoms we can separate them into two groups: the obstructed atomic insulator (OAI) and the atomic insulator (AI). The interesting group, the OAI, have a center of charge not localized on the atoms. Using the theory of topological quantum chemistry, in this work we first derive the necessary and sufficient conditions for a topologically trivial insulator to be a filling enforced obstructed atomic insulator (feOAI) in the 1651 Shubnikov space groups. Remarkably, the filling enforced criteria enable the identification of obstructed atomic bands without knowing the representations of the band structures. Hence, no ab-initio calculations are needed for the filling enforced criteria, although they are needed to obtain the band gaps. With the help of the Topological Quantum Chemistry website, we have performed a high-throughput search for feOAIs and have found that 957 ICSD entries (638 unique materials) are paramagnetic feOAIs, among which 738 (475) materials have an indirect gap. The metallic obstructed surface states of feOAIs are also showcased by several material examples.
Exotic phases of matter emerge from the interplay between strong electron interactions and non-trivial topology. Owing to their lack of dispersion at the single-particle level, systems harboring flat bands are excellent testbeds for strongly interact
ing physics, with twisted bilayer graphene serving as a prime example. On the other hand, existing theoretical models for obtaining flat bands in crystalline materials, such as the line-graph formalism, are often too restrictive for real-life material realizations. Here we present a generic technique for constructing perfectly flat bands from bipartite crystalline lattices. Our prescription encapsulates and generalizes the various flat band models in the literature, being applicable to systems with any orbital content, with or without spin-orbit coupling. Using Topological Quantum Chemistry, we build a complete topological classification in terms of symmetry eigenvalues of all the gapped and gapless flat bands, for all 1651 Magnetic Space Groups. In addition, we derive criteria for the existence of symmetry-protected band touching points between the flat and dispersive bands, and we identify the gapped flat bands as prime candidates for fragile topological phases. Finally, we show that the set of all (gapped and gapless) perfectly flat bands is finitely generated and construct the corresponding bases for all 1651 Shubnikov Space Groups.
The recently introduced theories of Topological Quantum Chemistry and Symmetry-Based Indicators (SIs) have facilitated the discovery of novel topological phases of matter and large-scale searches for materials with experimentally accessible topologic
al properties at the Fermi energy ($E_F$). In this work, we have completed the first catalog of stable and fragile topology in all of the bands both at and away from $E_F$ in the Inorganic Crystal Structure Database (ICSD), which we have made accessible through a substantial upgrade of the Topological Materials Database. We have computed the electronic structure, topological class, and stable and fragile SIs of all bands in the 96,196 processable ICSD entries with stoichiometric chemical formulas in the presence and absence of SOC. Our calculations represent the completion of the symmetry-indicated band topology of known nonmagnetic materials, and a doubling of the number of materials accessible in previous topological material catalogs. Through our calculations, we discover the existence of repeat-topological (RTopo) materials with stable topological insulating (TI) gaps at and just below $E_F$, and supertopological (STopo) materials in which every isolated set of bands above the core shell is stable topological. Our findings recontextualize several previous experimental investigations of topological materials. We find that Ta$_2$NiSe$_5$ and Ta$_2$NiSe$_7$, respectively previously highlighted for hosting exciton-insulator and CDW phases, are 3D TIs in their normal states, and that rhombohedral bismuth and Bi$_2$Mg$_3$ are both RTopo and STopo materials. We present detailed statistics for our computations revealing that 52.65% of all materials are topological at $E_F$, roughly 2/3 of bands across all materials exhibit symmetry-indicated stable topology, and that shockingly, 87.99% of all materials contain at least one topological band.
Over the last 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. There exist two classes of crystalline solids: nonmagnetic crystals left invaria
nt by space groups (SGs), and solids with commensurate magnetic order that respect the symmetries of magnetic space groups (MSGs). Whereas many of the properties of the SGs, such as their momentum-space corepresentations (coreps) and elementary band coreps (EBRs) were tabulated with relative ease, progress on deriving the analogous properties of the MSGs has largely stalled for the past 70 years due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small coreps, momentum stars, compatibility relations, and magnetic EBRs (MEBRs) of the single (spinless) and double (spinful) MSGs. We have implemented freely-accessible tools on the Bilbao Crystallographic Server for accessing the coreps of the MSGs, whose wide-ranging applications include neutron diffraction investigations of magnetic structure, the interplay of lattice regularization and (symmetry-enhanced) fermion doubling, and magnetic topological phases, such as axion insulators and spin liquids. Using the MEBRs, we extend the earlier theory of Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids - Magnetic Topological Quantum Chemistry (MTQC). We then use MTQC to derive the complete set of symmetry-based indicators (SIs) of band topology in all spinful (fermionic) crystals, for which we identify symmetry-respecting bulk and anomalous surface and hinge states. Lastly, using the SIs, we discover several novel non-axionic magnetic higher-order topological insulators.
The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators, have directed fundamental research in solid-state materials. Topological quantum chemistry has enabled the
understanding of and the search for paramagnetic topological materials. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC), here we perform a high-throughput search for magnetic topological materials based on first-principles calculations. We use as our starting point the Magnetic Materials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic compounds with magnetic structures deduced from neutron-scattering experiments, and identify 130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and topological insulators. For each compound, we perform complete electronic structure calculations, which include complete topological phase diagrams using different values of the Hubbard potential. Using a custom code to find the magnetic co-representations of all bands in all magnetic space groups, we generate data to be fed into the algorithm of MTQC to determine the topology of each magnetic material. Several of these materials display previously unknown topological phases, including symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators and higher-order magnetic semimetals. We analyse topological trends in the materials under varying interactions: 60 per cent of the 130 topological materials have topologies sensitive to interactions, and the others have stable topologies under varying interactions. We provide a materials database for future experimental studies and open-source code for diagnosing topologies of magnetic materials.
The electronic properties in a solid depend on the specific form of the wave-functions that represent the electronic states in the Brillouin zone. Since the discovery of topological insulators, much attention has been paid to the restrictions that th
e symmetry imposes on the electronic band structures. In this work we apply two different approaches to characterize all types of bands in a solid by the analysis of the symmetry eigenvalues: the induction procedure and the Smith Decomposition method. The symmetry eigenvalues or irreps of any electronic band in a given space group can be expressed as the superposition of the eigenvalues of a relatively small number of building units (the emph{basic} bands). These basic bands in all the space groups are obtained following a group-subgroup chain starting from P1. Once the whole set of basic bands are known in a space group, all other types of bands (trivial, strong topological or fragile topological) can be easily derived. In particular, we confirm previous calculations of the fragile root bands in all the space groups. Furthermore, we define an automorphism group of equivalences of the electronic bands which allows to define minimum subsets of, for instance, independent basic or fragile root bands.
