Catalog of topological phonon materials.

Phonons play a crucial role in many properties of solid-state systems, and it is expected that topological phonons may lead to rich and unconventional physics. On the basis of the existing phonon materials databases, we have compiled a catalog of topological phonon bands for more than 10,000 three-dimensional crystalline materials. Using topological quantum chemistry, we calculated the band representations, compatibility relations, and band topologies of each isolated set of phonon bands for the materials in the phonon databases. Additionally, we calculated the real-space invariants for all the topologically trivial bands and classified them as atomic or obstructed atomic bands. We have selected more than 1000 "ideal" nontrivial phonon materials to motivate future experiments. The datasets were used to build the Topological Phonon Database.

Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries.

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the nonperturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this Letter, we present a real space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected real space invariants, which reveal the bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology.

All topological bands of all nonmagnetic stoichiometric materials.

Topological quantum chemistry and symmetry-based indicators have facilitated large-scale searches for materials with topological properties at the Fermi energy (EF). We report the implementation of a publicly accessible catalog of stable and fragile topology in all of the bands both at and away from EF in the 96,196 processable entries in the Inorganic Crystal Structure Database. Our calculations, which represent the completion of the symmetry-indicated band topology of known nonmagnetic materials, have enabled the discovery of repeat-topological and supertopological materials, including rhombohedral bismuth and Bi2Mg3. We find that 52.65% of all materials are topological at EF, roughly two-thirds of bands across all materials exhibit symmetry-indicated stable topology, and 87.99% of all materials contain at least one stable or fragile topological band.

Catalogue of flat-band stoichiometric materials.

Topological electronic flattened bands near or at the Fermi level are a promising route towards unconventional superconductivity and correlated insulating states. However, the related experiments are mostly limited to engineered materials, such as moiré systems1-3. Here we present a catalogue of the naturally occuring three-dimensional stoichiometric materials with flat bands around the Fermi level. We consider 55,206 materials from the Inorganic Crystal Structure Database catalogued using the Topological Quantum Chemistry website4,5, which provides their structural parameters, space group, band structure, density of states and topological characterization. We combine several direct signatures and properties of band flatness with a high-throughput analysis of all crystal structures. In particular, we identify materials hosting line-graph or bipartite sublattices-in either two or three dimensions-that probably 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 the database to extract a curated list of 2,379 high-quality flat-band materials, from which we identify 345 promising candidates that potentially host flat bands with charge centres that are not strongly localized on the atomic sites. We showcase five representative materials and provide a theoretical explanation for the origin of their flat bands close to the Fermi energy using the S-matrix method introduced in a parallel work6.

Magnetic topological quantum chemistry.

For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete 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 corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend 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). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states.

High-throughput calculations of magnetic topological materials.

The discoveries of intrinsically magnetic topological materials, including semimetals with a large anomalous Hall effect and axion insulators1-3, have directed fundamental research in solid-state materials. Topological quantum chemistry4 has enabled the understanding of and the search for paramagnetic topological materials5,6. Using magnetic topological indices obtained from magnetic topological quantum chemistry (MTQC)7, 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.

Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallographic Server.

Two new programs, MTENSOR and TENSOR, hosted on the open-access website known as the Bilbao Crystallographic Server, are presented. The programs provide automatically the symmetry-adapted form of tensor properties for any magnetic or non-magnetic point group or space group. The tensor is chosen from a list of 144 known tensor properties gathered from the scientific literature or, alternatively, the user can also build a tensor that possesses an arbitrary intrinsic symmetry. Four different tensor types are considered: equilibrium, transport, optical and nonlinear optical susceptibility tensors. For magnetically ordered structures, special attention is devoted to a detailed discussion of the transformation rules of the tensors under the time-reversal operation 1'. It is emphasized that for non-equilibrium properties it is the Onsager theorem, and not the constitutive relationships, that indicates how these tensors transform under 1'. In this way it is not necessary to restrict the validity of Neumann's principle. New Jahn symbols describing the intrinsic symmetry of the tensors are introduced for several transport and optical properties. In the case of some nonlinear optical susceptibilities of practical interest, an intuitive method is proposed based on simple diagrams, which allows easy deduction of the action of 1' on the susceptibilities. This topic has not received sufficient attention in the literature and, in fact, it is usual to find published results where the symmetry restrictions for such tensors are incomplete.

Monophosphate tungsten bronzes with pentagonal tunnels: reinvestigation through the peephole of the superspace.

The large family of monophosphate tungsten bronzes with pentagonal tunnels (MPTBp), (PO2)4(WO3)2m with m ranging from 4 to 14, can be considered as modular structures via a description with (PO2)2(WO3)m modules related together by a symmetry operation and alternating along the z axis. Following the success of the application of the superspace for the description of the lillianites homologous series, a (3+1)-dimensional superspace model is efficiently defined to unify the structural analysis of the MPTBp. The (3+1)-dimensional model reveals hidden common characteristics such as the symmetry. An evaluation of the model for six well known members of the series was carried out from experimental data collected to this aim.

Compositional uniformity, domain patterning and the mechanism underlying nano-chessboard arrays.

We propose that systems exhibiting compositional patterning at the nanoscale, so far assumed to be due to some kind of ordered phase segregation, can be understood instead in terms of coherent, single phase ordering of minority motifs, caused by some constrained drive for uniformity. The essential features of this type of arrangement can be reproduced using a superspace construction typical of uniformity-driven orderings, which only requires the knowledge of the modulation vectors observed in the diffraction patterns. The idea is discussed in terms of a simple two-dimensional lattice-gas model that simulates a binary system in which the dilution of the minority component is favoured. This simple model already exhibits a hierarchy of arrangements similar to the experimentally observed nano-chessboard and nano-diamond patterns, which are described as occupational modulated structures with two independent modulation wavevectors and simple step-like occupation modulation functions.

Unified (3 + 1)-dimensional superspace description of the 2212-type stair-like [Bi2Sr3Fe2O9]m[Bi4Sr6Fe2O16] family of compounds.

The (3 + 1)-dimensional superspace approach is applied to describe and refine a series of sheared compounds related to layered high T(c) superconducting oxides. Two commensurate members (m = 4, 5) of the 2212 stair-like [Bi(2)Sr(3)Fe(2)O(9)](m)[Bi(4)Sr(6)Fe(2)O(16)] family of compounds, previously studied using single-crystal diffraction data, are analyzed. A common average unit cell has been identified and a composition-dependent modulation wavevector is proposed. The model is built using only three independent atomic domains, one for the metal atoms and two for the O atoms. The three Sr, Bi and Fe species are described using close-connected crenel-like functions forming a continuous atomic domain along the internal space. The two oxygen domains are represented by crenel functions and the displacive modulation functions are built up by Legendre polynomials recently implemented in the program JANA2006. Surprisingly, the results of the refinements show a striking similarity of the displacive modulations for the two compounds analyzed, indicating that a unique model can be used to describe the correlations between the atomic displacements of the 2212 stair-like series. This final model is then applied to predict the structure of new members of the family.

Structure determination of two modulated gamma-brass structures in the Zn-Pd System through a (3 + 1)-dimensional space description.

The structure determination of two composite compounds in the Zn-Pd system with close relationships to the cubic gamma-brass structure Zn(11-delta)Pd(2+delta) is reported. Their structures have been solved from single crystal X-ray diffraction data within a (3 + 1)-dimensional [(3 + 1)D] formalism. Zn(75.7(7))Pd(24.3) and Zn(78.8(7))Pd(21.2) crystallize with orthorhombic symmetry, superspace group Xmmm(00gamma)0s0 (X = [(1/2,1/2,0,0); (0,1/2,1/2,1/2); (1/2,0,1/2,1/2)]), with the following lattice parameters, respectively: a(s) = 12.929(3) A, b(s) = 9.112(4) A, c(s) = 2.5631(7) A, q = 8/13 c* and V(s) = 302.1(3) A(3) and a(s) = 12.909(3) A, b(s) = 9.115(3) A, c(s) = 2.6052(6) A, q = 11/18 c* and V(s) = 306.4(2) A(3). Their structures may be considered as commensurate because they can be refined in the conventional 3D space groups (Cmce and Cmcm, respectively) using supercells, but they also refined within the (3 + 1)D formalism to residual factors R = 3.14% for 139 parameters and 1184 independent reflections for Zn(75.7(7))Pd(24.3) and R = 3.16% for 175 parameters and 1804 independent reflections for Zn(78.8(7))Pd(21.2). The use of the (3 + 1)D formalism improves the results of the refinement and leads to a better understanding of the complexity of the atomic arrangement through the various modulations (occupation waves and displacive waves). Our refinements emphasize a unique Pd/Zn occupancy modulation at the center of distorted icosahedra, a modulation which correlates with the distortion of these polyhedra.

Modular crystals as modulated structures: the case of the lillianite homologous series.

The use of the superspace formalism is extended to the description and refinement of the homologous series of modular structures with two symmetry-related modules with different orientations. The lillianite homologous series has been taken as a study case. Starting from a commensurate modulated composite description with two basic subsystems corresponding to the two different modules, it is shown how a more efficient description can be achieved using so-called zigzag modulation functions. These linear zigzag modulations, newly implemented in the program JANA2006, have very large fixed amplitudes and introduce in the starting model the two orientations of the underlying module sublattices. We show that a composite approach with this type of function, which treats the cations and anions as two separate subsystems forming a misfit compound, is the most appropriate and robust method for the refinements.

Revision of pyrrhotite structures within a common superspace model.

The structure of pyrrhotite (Fe(1 - x)S with 0.05 < or = x < or = 0.125) has been reinvestigated in the framework of the superspace formalism. A common model with a centrosymmetric superspace group is proposed for the whole family. The atomic domains in the internal space representing the Fe atoms are parametrized as crenel functions that fulfil the closeness condition. The proposed model explains the x-dependent space groups observed and the basic features of the structures reported up to now. Our model yields for any x value a well defined ordered distribution of Fe vacancies in contrast to some of the structural models proposed in the literature. A new (3 + 1)-dimensional refinement of Fe(0.91)S using the deposited dataset [Yamamoto & Nakazawa (1982). Acta Cryst. A38, 79-86] has been performed as a benchmark of the model. The consistency of the proposed superspace symmetry and its validity for other compositions has been further checked by means of ab initio calculations of both atomic forces and equilibrium atomic positions in non-relaxed and relaxed structures, respectively.

Composite behavior and multidegeneracy in high-pressure phases of Cs and Rb.

The pressure-driven phases Cs III and Rb III having large unit cells are shown to be peculiar examples of commensurate modulated composites with two monatomic subsystems of striking simplicity. The two subsystems are obverse-reverse layers, symmetry related but misfitted. Modulations are smooth and describable by a few parameters within a well-defined superspace symmetry. Ab initio density-functional theory calculations show that the composite character is reflected in their physical behavior. Cs III has a low-energy mode with phason character corresponding to the relative sliding of the neighboring misfitted layers, the energy barrier being lower than 0.01 meV/atom, which is most favorable for transforming to other configurations. These phases possess a quasidegenerate energy landscape, close to the signature of incommensurate systems and quasicrystals.

Apparently complex high-pressure phase of gallium as a simple modulated structure.

The phase of gallium GaII, with symmetry C222(1) and 104 atoms per unit cell, has been recently reported as an example of structural complexity under high pressure. It is shown here that this phase is a simple modulated distortion of an average structure of Fddd symmetry with all atoms structurally equivalent. The modulation can be described with only 4 parameters and satisfies symmetry properties described by a centrosymmetric superspace group. The structural distortion is dominated by a frozen transversal mode associated with a single irreducible representation of Fddd, with a wave vector on the line Q, at an edge of the Brillouin zone. The average structure can be related with an hcp configuration through simple sliding of hcp layers, reminiscent of the hcp-bcc Bürgers mechanism.

Superspace description of NaCa4Nb5O17 (a perovskite-related compound of the type AnBnO3n+2) as a modulated layered structure.

The recently determined structure of NaCa(4)Nb(5)O(17) is reanalyzed within the superspace framework. The material follows the general features of the superspace model proposed for the perovskite-related compounds of the general formula A(n)B(n)O(3n+2) [Elcoro et al. (2001). Acta Cryst. B57, 471-484]. It can be described as a commensurate modulated structure with discontinuous atomic domains given by occupational crenel functions. These atomic domains automatically introduce the layered configuration of the actual structure in real space. However, the displacive modulations follow a quite different pattern from that observed in the Sr(n)(Nb,Ti)(n)O(3n+2) series, discussed in the above-mentioned reference. The superspace group relevant in this new case has been identified through a systematic search of all the possible groups and a comparison of the resulting superspace embeddings of the experimental three-dimensional structure. Being a commensurate structure, the fundamental ambiguity of the superspace description was broken by choosing the highest possible symmetry that yields smooth displacive atomic modulations and hence minimizes the number of parameters. The efficiency of the proposed superspace model is demonstrated with a new refinement of the structure. Assuming the model's general relevance, symmetry properties for the whole [Ca,Na](n)Nb(n)O(3n+2) series are predicted.

superspace description of trigonal and orthorhombic A(1)+(x)A(x)'B(1)-(x)O(3) compounds as modulated layered structures; application to the refinement of trigonal Sr(6)Rh(5)O(15).

The structures of the trigonal compounds A(1+x)A(x')B(1-x)O(3) are described, to a first approximation, as a hexagonal close-packed stacking of A(3)O(9) and A(3)A'O(6) layers. However, quantitative analyses are usually performed in superspace, with the structures considered as modulated composites made of two subsystems: chains of A cations, and columns of trigonal prisms, A'O(6), and octahedra, BO(6). It is demonstrated that an alternative superspace description as a single modulated structure can be found in terms of the aforementioned layers, with a composition-dependent modulation parameter and discontinuous atomic domains. In this approach, these compounds fulfill layer-stacking rules analogous to those observed in other layered compounds. These rules translate into a so-called closeness condition for the discontinuous atomic domains in superspace; this condition is analogous to that postulated in quasicrystals. Both superspace models, the composite and the layered model, when considered without displacive modulations, can be taken as two limiting idealized paradigms and can be used as the starting point of a structure refinement. As an example, the structure of the trigonal phase Sr(6)Rh(5)O(15), which was previously refined as a modulated composite [Stitzer, El Abed et al. (2001), J. Am. Chem. Soc. 123, 8790-8796], has been refined anew, with equivalent results, as a single modulated structure taking as reference the ideal layered structure. A similar superspace layer description is applied to the recently reported orthorhombic family A(4m+4n)A(n')B(4m+2n)O(12m+9n). This description allows the a priori derivation of a refineable superspace model that includes the superspace symmetry and crenel functions and is valid for the whole family. This model has been successfully applied to the refinement of the compound Ba(12)Co(11)O(33) [Darriet et al. (2002), Chem. Mater. 14, 3349-3363].