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The interplay between strong electron-electron interaction and symmetry breaking can have profound influence on the topological properties of materials. In magic angle twisted bilayer graphene (MATBG), the flat band with a single SU(4) flavor associated with the spin and valley degrees of freedom gains non-zero Chern number when C2z symmetry or C2zT symmetry is broken. Electron-electron interaction can further lift the SU(4) degeneracy, leading to the Chern insulator states. Here we report a complete sequence of zero-field Chern insulators at all odd integer fillings (v = +-1, +-3) with different chirality (C = 1 or -1) in hBN aligned MATBG which structurally breaks C2z symmetry. The Chern states at hole fillings (v = -1, -3), which are firstly observed in this work, host an opposite chirality compared with the electron filling scenario. By slightly doping the v = +-3 states, we have observed new correlated insulating states at incommensurate moir\'e fillings which is highly suggested to be intrinsic Wigner crystals according to our theoretical calculations. Remarkably, we have observed prominent Streda-formula violation around v = -3 state. By doping the Chern gap at v = -3 with notable number of electrons at finite magnetic field, the Hall resistance Ryx robustly quantizes to ~ h/e2 whereas longitudinal resistance Rxx vanishes, indicating that the chemical potential is pinned within a Chern gap, forming an incommensurate Chern insulator. By providing the first experimental observation of zero-field Chern insulators in the flat valence band, our work fills up the overall topological framework of MATBG with broken C2z symmetry. Our findings also demonstrate that doped topological flat band is an ideal platform to investigate exotic incommensurate correlated topological states., Comment: 28 pages. 5 figures

Ingap states are commonly observed in semiconductors and are often well characterized by a hydrogenic model within the effective mass approximation. However, when impurities are strong, they significantly perturb all momentum eigenstates, leading to deep-level bound states that reveal the global properties of the unperturbed band structure. In this work, we discover that the topology of band wavefunctions can impose zeros in the impurity-projected Green's function within topological gaps. These zeros can be interpreted as spectral attractors, defining the energy at which ingap states are pinned in the presence of infinitely strong local impurities. Their pinning energy is found by minimizing the level repulsion of band eigenstates onto the ingap state. We refer to these states as ring states, marked by a mixed band character and a node at the impurity site, guaranteeing their orthogonality to the bare impurity eigenstates and a weak energy dependence on the impurity strength. We show that the inability to construct symmetric and exponentially localized Wannier functions ensures topological protection of ring states. Linking ring states together, the edge or surface modes can be recovered for any topologically protected phase. Therefore, ring states can also be viewed as building blocks of boundary modes, offering a framework to understand bulk-boundary correspondence., Comment: 12 pages, 5 figures + 20 pages supplementary material

Fractional quantum anomalous hall effect (FQAHE), a transport effect with fractionally quantized Hall plateau emerging under zero magnetic field, provides a radically new opportunity to engineer topological quantum electronics. By construction of topological flat band with moire engineering, intrinsic FQAHE has been observed in twisted MoTe2 system and rhombohedral pentalayer graphene/hBN moire superlattices with anomalous Hall resistivity quantization number C <= 2/3 including the gapless composite Fermi-liquid state with C = 1/2. Here we experimentally demonstrate a new system of rhombohedral hexalayer graphene (RHG)/hBN moire superlattices showing both fractional and integer quantum anomalous Hall effects when the lowest flat Chern band is fractionally and fully filled at zero magnetic field. The zero-field Hall resistance Rho_xy = h/Ce2 is quantized to values corresponding to C = 3/5, 2/3, 5/7, 3/4, 7/9 and 1 at moire filling factors v = 3/5, 2/3, 5/7, 3/4, 7/9 and 1, respectively. Particularly, the C = 3/4 FQAHE state at v = 3/4 moire filling featuring a minimum of longitudinal resistance Rho_xx and fractionally quantized Hall resistance Rho_xy = 4h/3e2, is observed for the first time under zero magnetic field. Such a state may be similar to the C = 3/4 fractional quantum hall (FQHE) state recently observed at high magnetic fields9,10 and possibly host fractional charge excitations obeying non-Abelian statistics. By tuning the electrical and magnetic fields at 0 < v < 1, we have observed a sign reversal of the Hall resistivity for v = 2/3 state, indicating a transition from quasi-electron-like excitations to quasi-hole ones. Our experiment has established RHG/hBN moire superlattices a promising platform to explore quasi-particles with fractional charge excitations and non-Abelian anyons at zero magnetic field.

Electrons residing in flat-band system can play a vital role in triggering spectacular phenomenology due to relatively large interactions and spontaneous breaking of different degeneracies. In this work we demonstrate chirally twisted triple bilayer graphene, a new moir\'e structure formed by three pieces of helically stacked Bernal bilayer graphene, as a highly tunable flat-band system. In addition to the correlated insulators showing at integer moir\'e fillings, commonly attributed to interaction induced symmetry broken isospin flavors in graphene, we observe abundant insulating states at half-integer moir\'e fillings, suggesting a longer-range interaction and the formation of charge density wave insulators which spontaneously break the moir\'e translation symmetry. With weak out-of-plane magnetic field applied, as observed half-integer filling states are enhanced and more quarter-integer filling states appear, pointing towards further quadrupling moir\'e unit cells. The insulating states at fractional fillings combined with Hartree-Fock calculations demonstrate the observation of a new type of correlated charge density wave insulators in graphene and points to a new accessible twist manner engineering correlated moir\'e electronics.

Fragile topology, akin to twisted bilayer graphene and the exotic phases therein, is a notable topological class with intriguing properties. However, due to its unique nature and the lack of bulk-edge correspondence, the experimental signature of fragile topology has been under debated since its birth. Here, we demonstrate experimentally that fragile topological phases with filling anomaly can be probed via screw dislocations, despite that they do not support gapless edge states. Using a designer hexagonal phononic crystal with a fragile topological band gap, we find that 1D gapless bound modes can emerge at a screw dislocation due to the bulk fragile topology. We then establish a connection between our system and the twisted boundary condition via the gauge invariance principle and illustrate that such an emergent phenomenon is an intrinsic property of fragile topological phases with filling anomaly. We observe experimentally the 1D topological bound states using the pump-probe measurements of their dispersion and wavefunctions, which unveils a novel bulk-defect correspondence of fragile topology and a powerful tool for probing fragile topological phases and materials., Comment: Submitted to Science Bulletin

Magic-angle twisted bilayer graphene (MATBG) exhibits correlated phenomena such as superconductivity and Mott insulating state related to the weakly dispersing flat band near the Fermi energy. Beyond its moir\'e period, such flat band is expected to be sensitive to lattice relaxations. Thus, clarifying the evolution of the electronic structure with twist angle is critical for understanding the physics of MATBG. Here, we combine nanospot angle-resolved photoemission spectroscopy and atomic force microscopy to resolve the fine electronic structure of the flat band and remote bands, and their evolution with twist angles from 1.07$^\circ$ to 2.60$^\circ$. Near the magic angle, dispersion is characterized by a flat band near the Fermi energy with a strongly reduced bandwidth. Moreover, near 1.07$^\circ$, we observe a spectral weight transfer between remote bands at higher binding energy and extract the modulated interlayer spacing near the magic angle. Our work provides direct spectroscopic information on flat band physics and highlights the role of lattice relaxations., Comment: 22 pages, 5 figures, Nature Materials, in press

The discovery of new catalysts that are efficient, sustainable, and low-cost is a major research endeavor for many industrial chemical processes. This requires an understanding and determination of the catalytic origins for the given catalysts, which still remains a challenge. Here we describe a novel method to identify new catalysts based on searching for crystalline symmetry-protected obstructed atomic insulators (OAIs) that have metallic surface states on otherwise semiconducting or insulating compounds. The Wannier charge centers in OAIs are pinned by symmetries at some empty Wyckoff positions so that surfaces that accommodate these sites are guaranteed to have metallic obstructed surface states (OSSs). Beyond the well-studied 2H-MoS2, we further verified our theory on the catalysts, 2H-MoTe2, and 1T'-MoTe2, whose catalytic active sites are consistent with our calculations of obstructed Wannier charge centers (OWCCs) and OSSs. In addition, we have predicted the location of catalytic active sites and confirmed these predictions by exploring the hydrogen evolution reaction on NiPS3 bulk single crystals, which we find to be one of the most promising new catalysts with high activity and, moreover, of low cost. Most importantly, we successfully identified several high-efficient catalysts just by considering the number of OWCCs and the crystal symmetry of the OAIs. Using the real space invariant theory and high-throughput computational methods applied to a database of 34013 topologically trivial insulators, we have identified 1788 unique OAIs (3383 ICSDs), of which 465 are potential high-quality catalysts for heterogeneous reactions. The Miller indices of the active surfaces are also obtained. Our new methodology will facilitate and accelerate the discovery of new catalysts for a wide range of heterogeneous redox reactions, where sustainability, toxicity, and cost must be considered., Comment: 17+26 pages, 4+14 figures. See also "Three-Dimensional Real Space Invariants, Obstructed Atomic Insulators and A New Principle for Active Catalytic Sites", Yuanfeng Xu et al

We theoretically study potential unconventional superconductivity in doped AB-type IV-VI semi-conductors, based on a minimal effective model with interaction up to the next-nearest neighbors. According to the experimental implications, we focus on the spin-triplet channels and obtain the superconducting phase diagram with respect to the anisotropy of the Fermi surfaces and the interaction strength. Abundant nodal and nodeless states with different symmetry breaking appear in the phase diagram, and all the states are time reversal invariant and topologically nontrivial. Specifically, the various nodal superconducting ground states, dubbed as the topological Dirac superconductors, are featured by Dirac nodes in the bulk and Majorana arcs on the surface; among the full-gap states, there exist a mirror-symmetry-protected second-order topological superconductor state favoring helical Majorana hinge cones, and different first-order topological superconductor states supporting 4 surface Majorana cones. The experimental verification of the different kinds of superconducting ground states is also discussed., Comment: 31 pages, 7 figures, Accepted version

By means of the first-principles calculations and magnetic topological quantum chemistry, we demonstrate that the low energy physics in the checkerboard antiferromagnetic (AFM) monolayer FeSe, very close to an AFM topological insulator that hosts robust edge states, can be well captured by a double-degenerate fragile topologically flat band just below the Fermi level. The Wilson loop calculations identify that such fragile topology is protected by the $S_{4z}$ symmetry, which gives rise to an AFM higher-order topological insulator that support the bound state with fractional charge $e/2$ at the sample corner. This is the first reported $S_{4z}$-protected fragile topological material, which provides a new platform to study the intriguing properties of magnetic fragile topological electronic states. Previous observations of the edge states and bound states in checkerboard AFM monolayer FeSe can also be well understood in our work.

The interplay between topology and correlations can generate a variety of quantum phases, many of which remain to be explored. Recent advances have identified monolayer WTe2 as a promising material for doing so in a highly tunable fashion. The ground state of this two-dimensional (2D) crystal can be electrostatically tuned from a quantum spin Hall insulator (QSHI) to a superconductor. However, much remains unknown about the gap-opening mechanism of the insulating state. Here we report evidence that the QSHI is also an excitonic insulator (EI), arising from the spontaneous formation of electron-hole bound states (excitons). We reveal the presence of an intrinsic insulating state at the charge neutrality point (CNP) in clean samples and confirm the correlated nature of this charge-neutral insulator by tunneling spectroscopy. We provide evidence against alternative scenarios of a band insulator or a localized insulator and support the existence of an EI phase in the clean limit. These observations lay the foundation for understanding a new class of correlated insulators with nontrivial topology and identify monolayer WTe2 as a promising candidate for exploring quantum phases of ground-state excitons., Comment: 39 pages; Nat. Phys. (2021)

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., Comment: Final version, 10 pg main text + 184 pg appendix, 5 + 25 figures

Topological flat bands, such as the band in twisted bilayer graphene, are becoming a promising platform to study topics such as correlation physics, superconductivity, and transport. In this work, we introduce a generic approach to construct two-dimensional (2D) topological quasi-flat bands from line graphs and split graphs of bipartite lattices. A line graph or split graph of a bipartite lattice exhibits a set of flat bands and a set of dispersive bands. The flat band connects to the dispersive bands through a degenerate state at some momentum. We find that, with spin-orbit coupling (SOC), the flat band becomes quasi-flat and gapped from the dispersive bands. By studying a series of specific line graphs and split graphs of bipartite lattices, we find that (i) if the flat band (without SOC) has inversion or $C_2$ symmetry and is non-degenerate, then the resulting quasi-flat band must be topologically nontrivial, and (ii) if the flat band (without SOC) is degenerate, then there exists an SOC potential such that the resulting quasi-flat band is topologically nontrivial. This generic mechanism serves as a paradigm for finding topological quasi-flat bands in 2D crystalline materials and meta-materials.

In flat bands, superconductivity can lead to surprising transport effects. The superfluid "mobility", in the form of the superfluid weight $D_s$, does not draw from the curvature of the band but has a purely band-geometric origin. In a mean-field description, a non-zero Chern number or fragile topology sets a lower bound for $D_s$, which, via the Berezinskii-Kosterlitz-Thouless mechanism, might explain the relatively high superconducting transition temperature measured in magic-angle twisted bilayer graphene (MATBG). For fragile topology, relevant for the bilayer system, the fate of this bound for finite temperature and beyond the mean-field approximation remained, however, unclear. Here, we use numerically exact Monte Carlo simulations to study an attractive Hubbard model in flat bands with topological properties akin to those of MATBG. We find a superconducting phase transition with a critical temperature that scales linearly with the interaction strength. We then investigate the robustness of the superconducting state to the addition of trivial bands that may or may not trivialize the fragile topology. Our results substantiate the validity of the topological bound beyond the mean-field regime and further stress the importance of fragile topology for flat-band superconductivity., Comment: 5 pages, 3 figures + supplemental material (14 pages, 4 figures)

The quantum level interplay between geometry, topology, and correlation is at the forefront of fundamental physics. Owing to the unusual lattice geometry and breaking of time-reversal symmetry, kagome magnets are predicted to support intrinsic Chern quantum phases. However, quantum materials hosting ideal spin-orbit coupled kagome lattices with strong out-of-plane magnetization have been lacking. Here we use scanning tunneling microscopy to discover a new topological kagome magnet TbMn6Sn6, which is close to satisfying the above criteria. We visualize its effectively defect-free purely Mn-based ferromagnetic kagome lattice with atomic resolution. Remarkably, its electronic state exhibits distinct Landau quantization upon the application of a magnetic field, and the quantized Landau fan structure features spin-polarized Dirac dispersion with a large Chern gap. We further demonstrate the bulk-boundary correspondence between the Chern gap and topological edge state, as well as the Berry curvature field correspondence of Chern gapped Dirac fermions. Our results point to the realization of a quantum-limit Chern phase in TbMn6Sn6, opening up an avenue for discovering topological quantum phenomena in the RMn6Sn6 (R = rare earth element) family with a variety of magnetic structures. Our visualization of the magnetic bulk-boundary-Berry correspondence covering real and momentum space demonstrates a proof-of-principle method revealing topological magnets., Comment: To appear in Nature (2020)

We propose a universal practical approach to realize magnetic second-order topological insulator (SOTI) materials, based on properly breaking the time reversal symmetry in conventional (first-order) topological insulators. The approach works for both three dimensions (3D) and two dimensions (2D), and is particularly suitable for 2D, where it can be achieved by coupling a quantum spin Hall insulator with a magnetic substrate. Using first-principles calculations, we predict bismuthene on EuO(111) surface as the first realistic system for a 2D magnetic SOTI. We explicitly demonstrate the existence of the protected corner states. Benefited from the large spin-orbit coupling and sizable magnetic proximity effect, these corner states are located in a boundary gap $\sim 83$ meV, hence can be readily probed in experiment. By controlling the magnetic phase transition, a topological phase transition between a first-order TI and a SOTI can be simultaneously achieved in the system. The effect of symmetry breaking, the connection with filling anomaly, and the experimental detection are discussed., Comment: 6 pages, 4 figures

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., Comment: 228 pages; Full topologies and band structures are provided on the Topological Magnetic Materials Database https://www.topologicalquantumchemistry.fr/magnetic/

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 the 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., Comment: 17 pages, 0 figures 129 pages of Supplementary Material

In the time-reversal-breaking centrosymmetric systems, the appearance of Weyl points can be guaranteed by an odd number of all the even/odd parity occupied bands at eight inversion-symmetry-invariant momenta. Here, based on symmetry analysis and first-principles calculations, we demonstrate that for the time-reversal-invariant systems with $S_4$ symmetry, the Weyl semimetal phase can be characterized by the inequality between a well-defined invariant $\eta$ and an $S_4$ indicator $z_2$. By applying this criterion, we find that some candidates, previously predicted to be topological insulators, are actually Weyl semimetals in the noncentrosymmetric space group with $S_4$ symmetry. Our first-principles calculations show that four pairs of Weyl points are located in the $k_{x,y}$ = 0 planes, with each plane containing four same-chirality Weyl points. An effective model has been built and captures the nontrivial topology in these materials. Our strategy to find the Weyl points by using symmetry indicators and invariants opens a new route to search for Weyl semimetals in the time-reversal-invariant systems., Comment: 11 pages,8 figures

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries (LMSs), in which the time-reversal symmetry $\mathcal{T}$ is broken but its combinations with certain crystalline symmetry such as $M_x \mathcal{T}$, $C_{2z} \mathcal{T}$, $C_{4z}\mathcal{T}$, and $C_{6z}\mathcal{T}$ are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the $M_x \mathcal{T}$ or $C_{2z} \mathcal{T}$ superconducting wire, which is characterized by a chiral $Z^c$ invariant. More interestingly, we also find two types of totally new TSC phases in the $C_{4z}\mathcal{T}$, and $C_{6z}\mathcal{T}$ superconducing wires, which are beyond the known AZ class, and are characterized by a helical $Z^h$ invariant and $Z^h\oplus Z^c$ invariants, respectively. In the $Z^h$ TSC phase, $Z$-pairs of MZMs are protected at each end. In the $C_{6z}\mathcal{T}$ case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving $C_{4z}\mathcal{T}$ or $C_{6z}\mathcal{T}$ symmetry are presented to illustrate their novel TSC properties and MZMs., Comment: 5 pages, 3 figures

In the present paper, we propose a new way to classify centrosymmetric metals by studying the Zeeman effect caused by an external magnetic field described by the momentum dependent g-factor tensor on the Fermi surfaces. Nontrivial U(1) Berry's phase and curvature can be generated once the otherwise degenerate Fermi surfaces are splitted by the Zeeman effect, which will be determined by both the intrinsic band structure and the structure of g-factor tensor on the manifold of the Fermi surfaces. Such Zeeman effect generated Berry's phase and curvature can lead to three important experimental effects, modification of spin-zero effect, Zeeman effect induced Fermi surface Chern number and the in-plane anomalous Hall effect. By first principle calculations, we study all these effects on two typical material, ZrTe$_5$ and TaAs$_2$ and the results are in good agreement with the existing experiments., Comment: 10 pages, 6 figures

While regular flat bands are good for enhancing the density of states and hence the gap, they are detrimental to the superfluid weight. We show that the predicted nontrivial topology of the two lowest flat bands of twisted bilayer graphene plays an important role in the enhancement of the superfluid weight and hence of superconductivity. We derive the superfluid weight (phase stiffness) of the TBLG superconducting flat bands with a uniform pairing, and show that it can be expressed as an integral of the Fubini-Study metric of the flat bands. This mirrors results already obtained for nonzero Chern number bands even though the TBLG flat bands have zero Chern number. We further show the metric integral is lower bounded by the topological $C_{2z}T$ Wilson loop winding number of the TBLG flat bands, which renders the superfluid weight has a topological lower bound proportional to the pairing gap. In contrast, trivial flat bands have a zero superfluid weight. The superfluid weight is crucial in determining the BKT transition temperature of the superconductor. Based on the transition temperature measured in TBLG experiments, we estimate the topological contribution of the superfluid weight in TBLG., Comment: 7+22 pages, 4 figures

Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an Elementary Band Representation (Atomic Limit Band) splits into a particular set of bands. We obtain, for the first time, a complete classification of the fragile topological phases which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure which far surpasses that of stable/strong topological states. We find and enumerate all hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues (available at http://www.cryst.ehu.es/html/doc/FragileRoots.pdf), and link the mathematical structure of these phases to that of Affine Monoids in mathematics. Furthermore, we predict and calculate, for the first time, (hundred of realistic) materials where fragile topological bands appear, and show-case the very best ones., Comment: Accepted in PRX. Complete supplementary tables (979 pages) containing all the fragile criteria and fragile roots are available at http://www.cryst.ehu.es/html/doc/FragileRoots.pdf

Based on first-principles calculations and symmetry analysis, we propose that EuIn$_2$As$_2$ is a long awaited axion insulator with antiferromagnetic (AFM) long range order. Characterized by the parity-based invariant $\mathbb Z_4=2$, the topological magneto-electric effect is quantized with $\theta=\pi$ in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moment of the AFM phase is along $a/b$ axis, it's also a TCI phase. Gapless surface states emerge on (100), (010) and (001) surfaces, protected by mirror symmetries (nonzero mirror Chern numbers). When the magnetic moment is along $c$ axis, the (100) and (001) surfaces are gapped. As a consequence of a high-order topological insulator with $\mathbb Z_4=2$, the one-dimensional (1D) chiral state can exist on the hinge between those gapped surfaces. We have calculated both the topological surface states and hinge state in different phases of the system, respectively, which can be detected by ARPES or STM experiments.

Quasiparticles and collective modes are two fundamental aspects that characterize a quantum matter in addition to its ground state features. For example, the low energy physics for Fermi liquid phase in He-III was featured not only by Fermionic quasiparticles near the chemical potential but also by fruitful collective modes in the long-wave limit, including several different sound waves that can propagate through it under different circumstances. On the other hand, it is very difficult for sound waves to be carried by the electron liquid in the ordinary metals, due to the fact that long-range Coulomb interaction among electrons will generate plasmon gap for ordinary electron density fluctuation and thus prohibits the propagation of sound waves through it. In the present paper, we propose a unique type of acoustic collective modes in Weyl semimetals under the magnetic field called chiral zero sound. The chiral zero sound can be stabilized under so-called "chiral limit", where the intra-valley scattering time is much shorter than the inter-valley one, and only propagates along an external magnetic field for Weyl semimetals with multiple-pairs of Weyl points. The sound velocity of the chiral zero sound is proportional to the field strength in the weak field limit, whereas it oscillates dramatically in the strong field limit, generating an entirely new mechanism for quantum oscillations through the dynamics of neutral bosonic excitation, which may manifest itself in the thermal conductivity measurements under magnetic field., Comment: 9+16 pages, 2+0 figures, a new appendix added, accepted in PRX

Topological crystalline states are short-range entangled states jointly protected by onsite and crystalline symmetries. While the non-interacting limit of these states, e.g., the topological crystalline insulators, have been intensively studied in band theory and have been experimentally discovered, the classification and diagnosis of their strongly interacting counterparts are relatively less well understood. Here we present a unified scheme for constructing all topological crystalline states, bosonic and fermionic, free and interacting, from real-space "building blocks" and "connectors". Building blocks are finite-size pieces of lower dimensional topological states protected by onsite symmetries alone, and connectors are "glue" that complete the open edges shared by two or multiple pieces of building blocks. The resulted assemblies are selected against two physical criteria we call the "no-open-edge condition" and the "bubble equivalence", which, respectively, ensure that each selected assembly is gapped in the bulk and cannot be deformed to a product state. The scheme is then applied to obtaining the full classification of bosonic topological crystalline states protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our real-space recipes give the complete set of topological crystalline states for bosons and fermions, and prove the boson case analytically using a spectral sequence expansion of group cohomology., Comment: 17+44 pages, 7+1 figures, 0+2 tables. The content is the same as the published version, but arranged differently

We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry protected topological states can be adiabatically deformed (allowing for stacking of trivial states) into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for non-interacting time-reversal symmetric electronic insulators with significant spin-orbit coupling and any one of the 230 space groups in three dimensions. Because every topological crystalline insulator can be deformed into a topological crystal, the enumeration of the latter gives topological crystalline insulators a full classification and for each class an explicit real-space construction. We also extend these results to give a unified classification including both strong topological insulators and topological crystalline insulators., Comment: 5 + 12 pages. Improvements to figures, presentation and referencing

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semi-metallic 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 has what we conjecture to be a stable topological index stabilized by a magnetic symmetry and reflected in the odd winding of the Wilson loop in the Moir\'e BZ. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: when strongly broken, non-topological phases can appear. Our paper underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a $4$-band short range tight-binding model whose $2$ lower bands have the same topology, symmetry, and flatness as those of the twisted 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 $\bf k$-point computing time) ab-initio calculations of a series of small angles, from $3^\circ$ to $1^\circ$, which show a more complex and somewhat qualitatively different evolution of the symmetry of the low-energy bands than that of the theoretical Moir\'e model, but which confirms the topological nature of the system. At certain angles, we find no insulating filling in graphene at $-4$ electrons per Moir\'e unit cell. The ab-initio evolution of gaps tends to differ from that of the continuum Moir\'e model., Comment: 7+23 pages, 3+12 figures, 2+3 tables; v2: references added, note added

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 http://materiae.iphy.ac.cn/ ., Comment: 122 pages, 3 figures, 1 short table and 5 long tables

A universal mechanism to generate chiral hinge states in the ferromagnetic axion insulator phase is proposed, which leads to an exotic transport phenomena, the quantum anomalous Hall effect (QAHE) on some particular surfaces determined by both the crystalline symmetry and the magnetization direction. A realistic material system Sm doped $\text{Bi}_2\text{Se}_3$ is then proposed to realize such exotic hinge states by combing the first principle calculations and the Green's function techniques. A physically accessible way to manipulate the surface QAHE is also proposed, which makes it very different from the QAHE in ordinary 2D systems., Comment: 8 pages, 5 figures

We present a prediction of the Dirac semimetal (DSM) phase in MgTa2N3 based on first-principles calculations and symmetry analysis. In this material, the Fermi level is located exactly at the Dirac point without additional Fermi surface pockets. The band inversion associated with the Dirac cone involves the d orbitals of two structurally inequivalent Ta atoms with octahedral and trigonal prismatic coordination spheres. We further show that the lattice symmetry breaking can realize topological phase transitions from the DSM phase to a triple nodal point semimetal, Weyl semimetal or topological insulator. The topologically protected surface states and the non-protected Fermi arc surface states are also studied., Comment: 4 figures, 11 pages

Charge transport of topological semimetals has been in the focus of intensive investigations because of their non-trivial band topology. Heat transport of these materials, on the other hand, is largely unexplored and remains elusive. Here we report on an observation of unprecedented, giant magnetic quantum oscillations of thermal conductivity in the prototypical Weyl semimetal TaAs. The oscillations are antiphase with the quantum oscillating electronic density of states of a Weyl pocket, and their amplitudes amount to two orders of magnitude of the estimation based on the Wiedemann-Franz law. Our analyses show that all the conventional heat-transport mechanisms through diffusions of propagating electrons, phonons and electron-hole bipolar excitations, are far inadequate to account for these phenomena. Taking further experimental facts that the parallel field configuration favors much higher magneto-thermal conductivity, we propose that the newly proposed chiral zero sound provides a reasonable explanation to these exotic phenomena. More work focusing on other topological semimetals along the same line is badly called for., Comment: 15 pages, 5 figures

Analogues of the elementary particles, Dirac fermions in condensed matter have received extensive attention for both scientific interest and device applications. In this work, we generalize the concept of Dirac semimetal (DSM) to the magnetic space groups (MSGs), and define a new category of DSM in type IV MSGs, which is protected by the antiunitarity of the product of the inversion symmetry and the nonsymmophic time-reversal symmetry. Moreover, we propose the interlayer antiferromagnetic (AFM) EuCd2As2 as a promising candidate with only one pair of Dirac points at the Fermi level. Many exotic topological states, such as the triple point semimetal and the AFM topological insulator holding of the half-quantum Hall effect, can be derived from such AFM DSMs by breaking certain symmetry, providing an ideal platform to study topological phase transitions. Our results extend the range of DSM, and open a new way to study the interplay between the DSM and other exotic topological states., Comment: 16 pages, 2 figures

The study of spatial symmetries was accomplished during the last century, and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here, we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided., Comment: 8+108 pages, 3+9 figures, 0+8 tables;v4: accepted in Nat. Commun, title changed, summary of surface states added

Topological semimetals are under intensive theoretical and experimental studies. The first step of these studies is always the theoretical (numerical) predication of one of several candidate materials, starting from first principles. In these calculations, it is crucial that all topological band crossings, including their types and positions in the Brillouin zone, are found. While band crossings along high-symmetry lines, which are routinely scanned in numerics, are simple to locate, the ones at generic momenta are notoriously time-consuming to find, and may be easily missed. In this paper, we establish a theoretical scheme of diagnosis for topological semimetals where all band crossings are at generic momenta in systems with time-reversal symmetry and negligible spin-orbital coupling. The scheme only uses the symmetry (inversion and rotation) eigenvalues of the valence bands at high-symmetry points in the BZ as input, and provides the types, numbers and configurations of all topological band crossings, if any, at generic momenta. The nature of new diagnosis scheme allows for full automation and parallelizations, and paves way to high throughput numerical predictions of topological materials., Comment: 21 pages, 5 figures, 1 table; v4: accepted in PRX, a "PRELIMINARIES" section added

We study fourfold rotation invariant gapped topological systems with time-reversal symmetry in two and three dimensions ($d=2,3$). We show that in both cases nontrivial topology is manifested by the presence of the $(d-2)$-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands, and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through explicit construction of microscopic models having robust $(d-2)$-dimensional edge states., Comment: v5, accepted in prl, the explicit Z8 Fu-Kane-like formulas provided

Topological states of electrons and photons have attracted significant interest recently. Topological mechanical states also being actively explored, have been limited to macroscopic systems of kHz frequency. The discovery of topological phonons of atomic vibrations at THz frequency can provide a new venue for studying heat transfer, phonon scattering and electron-phonon interaction. Here, we employed ab initio calculations to identify a class of noncentrosymmetric materials of $M$Si ($M$=Fe,Co,Mn,Re,Ru) having double Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed "spin-1 Weyl point" at the Brillouin zone~(BZ) center and "charge-2 Dirac point" at the zone corner. The corresponding gapless surface phonon dispersions are double helicoidal sheets whose isofrequency contours form a single non-contractible loop in the surface BZ. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions. Our prediction of topological bulk and surface phonons can be experimentally verified by neutron scattering and electron energy loss spectroscopy, opening brand new directions for topological phononics., Comment: 5 pages, 4 figures

The quantum limit can be easily reached in the Dirac semimetals under the magnetic field, which will lead to some exotic many-body physics due to the high degeneracy of the topological zeroth Landau bands (LBs). By solving the effective Hamiltonian, which is derived by tracing out the high energy degrees of freedom, at the self-consistent mean field level, we have systematically studied the instability of Dirac semimetal under a strong magnetic field. A charge density wave (CDW) phase and a polarized nematic phase formed by "exciton condensation" are predicted as the ground state for the tilted and untilted bands, respectively. Furthermore, we propose that, distinguished from the CDW phase, the polarized nematic phase can be identified in experiments by anisotropic transport and Raman scattering.

Based on first-principles calculations and effective model analysis, a Dirac nodal-net semimetal state is recognized in AlB$_2$-type TiB$_2$ and ZrB$_2$ when spin-orbit coupling (SOC) is ignored. Taking TiB$_2$ as an example, there are several topological excitations in this nodal-net structure including triple point, nexus, and nodal link, which are protected by coexistence of spatial-inversion symmetry and time reversal symmetry. This nodal-net state is remarkably different from that of IrF$_4$, which requires sublattice chiral symmetry. In addition, linearly and quadratically dispersed two-dimensional surface Dirac points are identified as having emerged on the B-terminated and Ti-terminated (001) surfaces of TiB$_2$ respectively, which are analogous to those of monolayer and bilayer graphene., Comment: 12 pages, 13 figures, Corrected some citations

In the present paper, we propose that the chiral magnetic effect, the direct consequence of the presence of Weyl points in the band structure, can be detected by its coupling to certain phonon modes, which behave like pseudo scalars under point group transformations. Such coupling can generate resonance between intrinsic plasmon oscillation and the corresponding phonon modes, leading to dramatic modification of the optical response by the external magnetic field, which provides a new way to study chiral magnetic effect by optical measurements.

By using first-principles calculations, we propose that ZrSiO can be looked as a three-dimensional (3D) oxide weak topological insulator (TI) and its single layer is a long-sought-after 2D oxide TI with a band gap up to 30 meV. Calculated phonon spectrum of the single layer ZrSiO indicates it is dynamically stable and the experimental achievements in growing oxides with atomic precision ensure that it can be readily synthesized. This will lead to novel devices based on TIs, the so called "topotronic" devices, operating under room-temperature and stable when exposed in the air. Thus, a new field of "topotronics" will arise. Another intriguing thing is this oxide 2D TI has the similar crystal structure as the well-known iron-pnictide superconductor LiFeAs. This brings great promise in realizing the combination of superconductor and TI, paving the way to various extraordinary quantum phenomena, such as topological superconductor and Majorana modes. We further find that there are many other isostructural compounds hosting the similar electronic structure and forming a $WHM$-family with $W$ being Zr, Hf or La, $H$ being group IV or group V element, and $M$ being group VI one., Comment: adding results of phonon spectrum, supported substrate and binding energy. Related with our recent preprints for 2D TIs: arXiv:1507.01172, arXiv:1503.09040 and arXiv:1508.05220

We find that quantum spin Hall (QSH) state can be obtained on a square-like or rectangular lattice, which is generalized from two-dimensional (2D) transition metal dichalcogenide (TMD) haeckelites. Band inversion is shown to be controled by hopping parameters and results in Dirac cones with opposite or same vorticity when spin-orbit coupling (SOC) is not considered. Effective k$\cdot$p model has been constructed to show the merging or annihilation of these Dirac cones, supplemented with the intuitive pseudospin texture. Similar to graphene based honeycomb lattice system, the QSH insulator is driven by SOC, which opens band gap at the Dirac cones. We employ the center evolution of hybrid Wannier function from Wilson-loop method, as well as the direct integral of Berry curvature, to identify the $Z_2$ number. We hope our detailed analysis will stimulate further efforts in searching for QSH insulators in square or rectangular lattice, in addition to the graphene based honeycomb lattice., Comment: The detailed description and discussion on the effective model Hamiltonian for MX2-4-8 haeckelite in our previous paper arXiv:1503.09040