We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of β-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that β-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.

Dirac Semimetal in Three Dimensions

Topological states of quantum matter have attracted great attention in condensed matter physics and materials science. The study of time-reversal-invariant topological states in quantum materials has made tremendous progress. However, the study of magnetic topological states falls much behind due to the complex magnetic structures. Here, we predict the tetradymite-type compound ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ and its related materials host topologically nontrivial magnetic states. The magnetic ground state of ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ is an antiferromagetic topological insulator state with a large topologically nontrivial energy gap ($\ensuremath{\sim}0.2\text{ }\text{ }\mathrm{eV}$). It presents the axion state, which has gapped bulk and surface states, and the quantized topological magnetoelectric effect. The ferromagnetic phase of ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ might lead to a minimal ideal Weyl semimetal.

Topological Axion States in the Magnetic Insulator MnBi 2 Te 4 with the Quantized Magnetoelectric Effect

Recent advances in experimental and computational methods are increasing the quantity and complexity of generated data. This massive amount of raw data needs to be stored and interpreted in order to advance the materials science field. Identifying correlations and patterns from large amounts of complex data is being performed by machine learning (ML) algorithms for decades. Recently, the materials science community started to invest in these methodologies to extract knowledge and insights from the accumulated data. This review follows a logical sequence starting from density functional theory (DFT) as the representative instance of electronic structure methods, to the subsequent high-throughput (HT) approach, used to generate large amounts of data. Ultimately, data-driven strategies which include data mining, screening, and machine learning techniques, employ the data generated. We show how these approaches to modern computational materials science are being used to uncover complexities and design novel materials with enhanced properties. Finally, we point to the present research problems, challenges, and potential future perspectives of this new exciting field.

https://iopscience.iop.org/article/10.1088/2515-7639/ab084b/pdf

From DFT to machine learning: recent approaches to materials science–a review

Non-Hermiticity enriches topological phases beyond the existing Hermitian framework. Whereas their unusual features with no Hermitian counterparts were extensively explored, a full understanding about the role of symmetry in non-Hermitian physics has still been elusive, and there remains an urgent need to establish their topological classification in view of rapid theoretical and experimental progress. Here, we develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 instead of 10, which describe intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Furthermore, due to the complex nature of energy spectra, non-Hermitian systems feature two different types of complex-energy gaps, pointlike and linelike vacant regions. On the basis of these concepts and K-theory, we complete classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. Remarkably, non-Hermitian topology depends on the type of complex-energy gaps, and multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Moreover, the bulk-boundary correspondence in non-Hermitian systems is elucidated within our framework, and symmetries preventing the non-Hermitian skin effect are identified. Our classification not only categorizes recently observed lasing and transport topological phenomena, but also predicts a new type of symmetry-protected topological lasers with lasing helical edge states and dissipative topological superconductors with nonorthogonal Majorana edge states. Furthermore, our theory provides topological classification of Hermitian and non-Hermitian free bosons. Our work establishes a theoretical framework for the fundamental and comprehensive understanding of non-Hermitian topological phases and paves the way toward uncovering unique phenomena and functionalities that emerge from the interplay of non-Hermiticity and topology.

Symmetry and Topology in Non-Hermitian Physics

A topological superconductor A promising path toward topological quantum computing involves exotic quasiparticles called the Majorana bound states (MBSs). MBSs have been observed in heterostructures that require careful nanofabrication, but the complexity of such systems makes further progress tricky. Zhang et al. identified a topological superconductor in which MBSs may be observed in a simpler way by looking into the cores of vortices induced by an external magnetic field. Using angle-resolved photoemission, the researchers found that the surface of the iron superconductor FeTe0.55Se0.45 satisfies the required conditions for topological superconductivity. Science, this issue p. 182 Angle-resolved photoemission spectroscopy indicates that FeTe0.55Se0.45 harbors Dirac-cone–type spin-helical surface states. Topological superconductors are predicted to host exotic Majorana states that obey non-Abelian statistics and can be used to implement a topological quantum computer. Most of the proposed topological superconductors are realized in difficult-to-fabricate heterostructures at very low temperatures. By using high-resolution spin-resolved and angle-resolved photoelectron spectroscopy, we find that the iron-based superconductor FeTe1–xSex (x = 0.45; superconducting transition temperature Tc = 14.5 kelvin) hosts Dirac-cone–type spin-helical surface states at the Fermi level; the surface states exhibit an s-wave superconducting gap below Tc. Our study shows that the surface states of FeTe0.55Se0.45 are topologically superconducting, providing a simple and possibly high-temperature platform for realizing Majorana states.

Observation of topological superconductivity on the surface of iron-based superconductor

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.

Catalogue of topological electronic materials

Topological electronic materials are new quantum states of matter hosting novel linear responses in the bulk and anomalous gapless states at the boundary, and are for scientific and applied reasons under intensive research in physics and in materials sciences. The detection for such materials has so far been hindered by the level of complication involved in the calculation of the so-called topological invariants, and is hence considered a specialized task that requires both experience with materials and expertise with advanced theoretical tools. Here we introduce an effective, efficient and fully automated algorithm in obtaining the topological invariants for all non-magnetic materials that are known to human, based on recently developed principles that allow for exhaustive mappings between the symmetry representation of occupied bands and the topological invariants. Our algorithm requires as input only the occupied-band information (energy and wavefunction) at a handful (up to eight) of high-symmetry points in the Brillouin zone, which is readily calculable with any first-principles software. In return, it is capable of providing a detailed topological classification of all non-magnetic materials. Equipped with this method we have scanned through a total of 39519 materials available in structural databases, and found that as many as 8056 of them are actually topological (8889 if spin-orbital coupling is neglected). These are further catalogued into classes of 5005 topological semimetals,1814 topological insulators and 1237 topological crystalline insulators, most of which are new to human knowledge. All the results are available and searchable at this http URL .

Catalogue of Topological Electronic Materials

https://iopscience.iop.org/article/10.1088/1674-1056/28/8/087102/pdf

SymTopo:An automatic tool for calculating topological properties of nonmagnetic crystalline materials

Both the intrinsic anisotropic optical materials and fullerene-assembled 2D materials have attracted much interest in fundamental science and potential applications. The synthesis of a monolayer (ML) fullerene makes the combination of these two features plausible. In this work, using first-principles calculations, we systematically study the electronic structure and optical properties of quasi-hexagonal phase (qHP) ML and quasi-tetragonal phase (qTP) ML fullerenes. The calculations of qHP ML show that it is a semiconductor with small anisotropic optical absorption, which agrees with the recent experimental measurements. However, the results for qTP ML reveal that it is a semimetal with highly in-plane anisotropic absorption. The dichroic ratio, namely the absorption ratio of x- and y-polarized light αxx/αyy, is around 12 at photon energy of 0.29 eV. This anisotropy is much more pronounced when the photon energy is between 0.7 and 1.4 eV, where αxx becomes nearly zero while αyy is more than two orders of magnitude larger. This indicates qTP ML as a candidate for long-pursuit lossless metal and a potential material for atomically thin polarizer. We hope this will stimulate further experimental efforts in the study of qTP ML and other fullerene-assembled 2D materials. 

/pdf/highly-in-plane-anisotropic-optical-properties-of-fullerene-34qni11p.pdf

Highly in-plane anisotropic optical properties of fullerene monolayers

Symmetries of three-dimensional periodic scalar fields are described by 230 space groups (SGs). Symmetries of three-dimensional periodic (pseudo-) vector fields, however, are described by the spin-space groups (SSGs), which were initially used to describe the symmetries of magnetic orders. In SSGs, the real-space and spin degrees of freedom are unlocked in the sense that an operation could have different spacial and spin rotations. SSGs gives a complete symmetry description of magnetic structures, and have natural applications in the band theory of itinerary electrons in magnetically ordered systems with weak spin-orbit coupling.\textit{Altermagnetism}, a concept raised recently that belongs to the symmetry-compensated collinear magnetic orders but has non-relativistic spin splitting, is well described by SSGs. Due to the vast number and complicated group structures, SSGs have not yet been systematically enumerated. In this work, we exhaust SSGs based on the invariant subgroups of SGs, with spin operations constructed from three-dimensional (3D) real representations of the quotient groups for the invariant subgroups. For collinear and coplanar magnetic orders, the spin operations can be reduced into lower dimensional real representations. As the number of SSGs is infinite, we only consider SSGs that describe magnetic unit cells up to 12 times crystal unit cells. We obtain 157,289 non-coplanar, 24,788 coplanar-non-collinear, and 1,421 collinear SSGs. The enumerated SSGs are stored in an online database at \url{https://cmpdc.iphy.ac.cn/ssg} with a user-friendly interface. We also develop an algorithm to identify SSG for realistic materials and find SSGs for 1,626 magnetic materials. Our results serve as a solid starting point for further studies of symmetry and topology in magnetically ordered materials.

Yi Jiang

Papers

Catalogue of topological electronic materials

Catalogue of Topological Electronic Materials

SymTopo:An automatic tool for calculating topological properties of nonmagnetic crystalline materials

Highly in-plane anisotropic optical properties of fullerene monolayers

Enumeration of spin-space groups: Towards a complete description of symmetries of magnetic orders