Quantum transport and two-parameter scaling at the surface of a weak topological insulator

Emergent p-Wave Superconductivity in a Dual Topological Insulator BiSe via Superconducting Proximity Effect

The quest for anisotropic superconductors has been a long-standing pursuit due to their potential applications in quantum computing. In this regard, experimentally, d-wave and anisotropic s-wave superconducting order parameters are predominantly observed, while p-wave superconductors remain largely elusive. Achieving p-wave superconductivity in topological phases is highly desirable, as it is considered suitable for creating topologically protected qubits. To achieve topological superconductivity in the dual topological insulator BiSe, we place an s-wave superconductor NbSe2 in its close proximity employing the van der Waals epitaxy technique. Low-temperature differential conductance measurements performed at the heterojunction exhibit a dual-dip feature with a V-shaped inner dip, a characteristic of p-wave superconductivity. This observation is corroborated by the multiband 2D Blonder-Tinkham-Klapwijk (BTK) fitting, where the inner and outer gaps exhibit p-wave and s-wave character, respectively. Furthermore, the BTK analysis reveals that the two superconducting gaps experience distinct effective critical fields and transition temperatures.

Robust Weak Topological Insulator in the Bismuth Halide Bi_{4}Br_{2}I_{2}

We apply a topological material design concept for selecting a bulk topology of 3D crystals by different van der Waals stackings of 2D topological insulator layers, and find a bismuth halide Bi_{4}Br_{2}I_{2} to be an ideal weak topological insulator (WTI) with the largest band gap (∼300  meV) among all the WTI candidates, by means of angle-resolved photoemission spectroscopy (ARPES), density functional theory (DFT) calculations, and resistivity measurements. Furthermore, we reveal that the topological surface state of a WTI is not "weak" but rather robust against external perturbations against the initial theoretical prediction by performing potassium deposition experiments. Our results vastly expand future opportunities for fundamental research and device applications with a robust WTI.

Anderson critical metal phase in trivial states protected by average magnetic crystalline symmetry

Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale-invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average C2zT, m, and C4zT and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups P m , P 2 ' , P 4 ' , and P 6 ' with and without spin-orbit coupling, most of which can support CMP.

Topological zero-dimensional defect and flux states in three-dimensional insulators

In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We demonstrate the presence of first-order 0D defect states in PbTe monolayers and HEND states in 3D SnTe crystals. We relate our analysis to magnetic flux insertion in insulating crystals. We find that π-flux tubes in inversion- and time-reversal-symmetric (helical) higher-order topological insulators bind Kramers pairs of spin-charge-separated HEND states, which represent observable signatures of anomalous surface half quantum spin Hall states.

Delocalization Transition of a Disordered Axion Insulator

The axion insulator is a higher-order topological insulator protected by inversion symmetry. We show that, under quenched disorder respecting inversion symmetry on average, the topology of the axion insulator stays robust, and an intermediate metallic phase in which states are delocalized is unavoidable at the transition from an axion insulator to a trivial insulator. We derive this conclusion from general arguments, from classical percolation theory, and from the numerical study of a 3D quantum network model simulating a disordered axion insulator through a layer construction. We find the localization length critical exponent near the delocalization transition to be ν=1.42±0.12. We further show that this delocalization transition is stable even to weak breaking of the average inversion symmetry, up to a critical strength. We also quantitatively map our quantum network model to an effective Hamiltonian and we find its low-energy k·p expansion.

Moiré Band Topology in Twisted Bilayer Graphene

Recently twisted bilayer graphene (t-BLG) has emerged as a strongly correlated physical platform. Besides the apparent significance of band flatness, band topology may be another critical element in t-BLG and yet receives much less attention. Here we report the compelling evidence for nontrivial noninteracting Moiré band topology in t-BLG through a systematic nonlocal transport study and a K-theory examination. The nontrivial topology manifests itself as two pronounced nonlocal responses in the electron and hole superlattice gaps. We show that the nonlocal responses are robust to the twist angle and edge termination, exhibiting a universal scaling law. We elucidate that, although Berry curvature is symmetry-trivialized, two nontrivial Z₂ invariants characterize the Moiré Dirac bands, validating the topological origin of the observed nonlocal responses. Our findings not only provide a new perspective for understanding the strongly correlated t-BLG but also suggest a potential strategy to achieve topological metamaterials from trivial vdW materials.

Artificial oxide heterostructures with non-trivial topology

In the quest for topological insulators with large band gaps, heterostructures with Rashba spin-orbit interactions come into play. Transition metal oxides with heavy ions are especially interesting in this respect. We discuss the design principles for stacking oxide Rashba layers. Assuming a single layer with a two-dimensional electron gas (2DEG) on both interfaces as a building block, a two-dimensional topological insulating phase is present when negative coupling between the 2DEGs exists. When stacking multiple building blocks, a two-dimensional or three-dimensional topological insulator is artificially created, depending on the intra- and interlayer coupling strengths and the number of building blocks. We show that the three-dimensional topological insulator is protected by reflection symmetry, and can therefore be classified as a topological crystalline insulator. In order to isolate the topological states from bulk states, the intralayer coupling term needs to be quadratic in momentum. It is described how such a quadratic coupling could potentially be realized by taking buckling within the layers into account. The buckling, thereby, brings the idea of stacked Rashba system very close to the alternative approach of realizing the buckled honeycomb lattice in [111]-oriented perovskite oxides.

Z_{4} Topological Superconductivity in UCoGe

Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCSs are classified into Z_{4} topological phases, which have unique surface states referred to as a Möbius strip or an hourglass, and they have not been achieved in symmorphic superconductors. However, material realization of Z_{4} TNCS has never been known, to the best of our knowledge. Here, we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of Z_{4}-nontrivial TNCS enriched by glide symmetry. We evaluate Z_{4} invariants of UCoGe by deriving the formulas relating Z_{4} invariants to the topology of Fermi surfaces. Applying the formulas and previous ab initio calculations, we clarify that three odd-parity representations out of four are Z_{4}-nontrivial TNCS, whereas the other is also Z_{2}-nontrivial TNCS. We also discuss possible Z_{4} TNCS in CrAs and related materials.

Realization of a three-dimensional photonic topological insulator

Confining photons in a finite volume is highly desirable in modern photonic devices, such as waveguides, lasers and cavities. Decades ago, this motivated the study and application of photonic crystals, which have a photonic bandgap that forbids light propagation in all directions^1-3. Recently, inspired by the discoveries of topological insulators^4,5, the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic topological insulators^6-8, with promising applications in topological lasers^9,10 and robust optical delay lines¹¹. However, a fully three-dimensional (3D) topological photonic bandgap has not been achieved. Here we experimentally demonstrate a 3D photonic topological insulator with an extremely wide (more than 25 per cent bandwidth) 3D topological bandgap. The composite material (metallic patterns on printed circuit boards) consists of split-ring resonators (classical electromagnetic artificial atoms) with strong magneto-electric coupling and behaves like a 'weak' topological insulator (that is, with an even number of surface Dirac cones), or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk band structure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D topological insulators from fermions to bosons and paves the way for applications in topological photonic cavities, circuits and lasers in 3D geometries.

Interacting topological insulators: a review

The discovery of the quantum spin Hall effect and topological insulators more than a decade ago has revolutionized modern condensed matter physics. Today, the field of topological states of matter is one of the most active and fruitful research areas for both experimentalists and theorists. The physics of topological insulators is typically well described by band theory and systems of non-interacting fermions. In contrast, several of the most fascinating effects in condensed matter physics merely exist due to electron-electron interactions, examples include unconventional superconductivity, the Kondo effect, and the Mott-Hubbard transition. The aim of this review article is to give an overview of the manifold directions which emerge when topological bandstructures and correlation physics interfere and compete. These include the study of the stability of topological bandstructures and correlated topological insulators. Interaction-induced topological phases such as the topological Kondo insulator provide another exciting topic. More exotic states of matter such as topological Mott insulator and fractional Chern insulators only exist due to the interplay of topology and strong interactions and do not have any bandstructure analogue. Eventually the relation between topological bandstructures and frustrated quantum magnetism in certain transition metal oxides is emphasized.

Selective state spectroscopy and multifractality in disordered Bose-Einstein condensates: a numerical study

We propose to apply a modified version of the excitation scheme introduced by Volchkov et al. on bosons experiencing hyperfine state dependent disorder to address the critical state at the mobility edge of the Anderson localization transition, and to observe its intriguing multifractal structure. An optimally designed, spatially focused external radio frequency pulse can be applied to generate transitions to eigenstates in a narrow energy window close to the mobility edge, where critical scaling and multifractality emerge. Alternatively, two-photon laser scanning microscopy is proposed to address individual localized states even close to the transition. The projected image of the cloud is shown to inherit multifractality and to display universal density correlations. Interactions - unavoidably present - are taken into account by solving the Gross-Pitaevskii equations, and their destructive effect on the spectral resolution and the multifractal spectrum is analyzed. Time of flight images of the excited states are predicted to show interference fringes in the localized phase, while they allow one to map equal energy surfaces deep in the metallic phase.

Topological superconductors: a review

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

Aperiodic Weak Topological Superconductors

Weak topological phases are usually described in terms of protection by the lattice translation symmetry. Their characterization explicitly relies on periodicity since weak invariants are expressed in terms of the momentum-space torus. We prove the compatibility of weak topological superconductors with aperiodic systems, such as quasicrystals. We go beyond usual descriptions of weak topological phases and introduce a novel, real-space formulation of the weak invariant, based on the Clifford pseudospectrum. A nontrivial value of this index implies a nontrivial bulk phase, which is robust against disorder and hosts localized zero-energy modes at the edge. Our recipe for determining the weak invariant is directly applicable to any finite-sized system, including disordered lattice models. This direct method enables a quantitative analysis of the level of disorder the topological protection can withstand.

Electronic Structure of the Dark Surface of the Weak Topological Insulator Bi14Rh3I9

Compound Bi14Rh3I9 consists of ionic stacks of intermetallic [(Bi4Rh)3I](2+) and insulating [Bi2I8](2-) layers and has been identified to be a weak topological insulator. Scanning tunneling microscopy revealed the robust edge states at all step edges of the cationic layer as a topological fingerprint. However, these edge states are found 0.25 eV below the Fermi level, which is an obstacle for transport experiments. Here, we address this obstacle by comparing results of density functional slab calculations with scanning tunneling spectroscopy and angle-resolved photoemission spectroscopy. We show that the n-type doping of the intermetallic layer is intrinsically caused by the polar surface and is well-screened toward the bulk. In contrast, the anionic "spacer" layer shows a gap at the Fermi level, both on the surface and in the bulk; that is, it is not surface-doped due to iodine desorption. The well-screened surface dipole implies that a buried edge state, probably already below a single spacer layer, is located at the Fermi level. Consequently, a multilayer step covered by a spacer layer could provide access to the transport properties of the topological edge states. In addition, we find a lateral electronic modulation of the topologically nontrivial surface layer, which is traced back to the coupling with the underlying zigzag chain structure of the spacer layer.

Correlation between topological band character and chemical bonding in a Bi14Rh3I9-based family of insulators

Recently the presence of topologically protected edge-states in Bi₁₄Rh₃I₉ was confirmed by scanning tunnelling microscopy consolidating this compound as a weak 3D topological insulator (TI). Here, we present a density-functional-theory-based study on a family of TIs derived from the Bi₁₄Rh₃I₉ parent structure via substitution of Ru, Pd, Os, Ir and Pt for Rh. Comparative analysis of the band-structures throughout the entire series is done by means of a unified minimalistic tight-binding model that evinces strong similarity between the quantum-spin-Hall (QSH) layer in Bi₁₄Rh₃I₉ and graphene in terms of -molecular orbitals. Topologically non-trivial energy gaps are found for the Ir-, Rh-, Pt- and Pd-based systems, whereas the Os- and Ru-systems remain trivial. Furthermore, the energy position of the metal -band centre is identified as the parameter which governs the evolution of the topological character of the band structure through the whole family of TIs. The -band position is shown to correlate with the chemical bonding within the QSH layers, thus revealing how the chemical nature of the constituents affects the topological band character.

Weak Topological Insulators and Composite Weyl Semimetals: β-Bi4X4 (X=Br, I)

While strong topological insulators (STIs) were experimentally realized soon after they were theoretically predicted, a weak topological insulator (WTI) has yet to be unambiguously confirmed. A major obstacle is the lack of distinct natural cleavage surfaces to test the surface selective hallmark of a WTI. With a new scheme, we discover that β-Bi4X4 (X=Br, I), dynamically stable or synthesized before, can be a prototype WTI with two natural cleavage surfaces, where two anisotropic Dirac cones stabilize and annihilate, respectively. We further find four surface-state Lifshitz transitions under charge doping and two bulk topological phase transitions under uniaxial strain. Near the WTI-STI transition, there emerges a novel Weyl semimetal phase, in which the Fermi arcs generically appear at both cleavage surfaces whereas the Fermi circle only appears at one selected surface.

Anomalous Quasiparticle Symmetries and Non-Abelian Defects on Symmetrically Gapped Surfaces of Weak Topological Insulators

We show that boundaries of 3D weak topological insulators can become gapped by strong interactions while preserving all symmetries, leading to Abelian surface topological order. The anomalous nature of weak topological insulator surfaces manifests itself in a nontrivial action of symmetries on the quasiparticles; most strikingly, translations change the anyon types in a manner impossible in strictly 2D systems with the same symmetry. As a further consequence, screw dislocations form non-Abelian defects that trap Z_{4} parafermion zero modes.

Dirac Semimetals in Two Dimensions

Graphene is famous for being a host of 2D Dirac fermions. However, spin-orbit coupling introduces a small gap, so that graphene is formally a quantum spin Hall insulator. Here we present symmetry-protected 2D Dirac semimetals, which feature Dirac cones at high-symmetry points that are not gapped by spin-orbit interactions and exhibit behavior distinct from both graphene and 3D Dirac semimetals. Using a two-site tight-binding model, we construct representatives of three possible distinct Dirac semimetal phases and show that single symmetry-protected Dirac points are impossible in two dimensions. An essential role is played by the presence of nonsymmorphic space group symmetries. We argue that these symmetries tune the system to the boundary between a 2D topological and trivial insulator. By breaking the symmetries we are able to access trivial and topological insulators as well as Weyl semimetal phases.

Layered Topological Crystalline Insulators

Topological crystalline insulators (TCIs) are insulating materials whose topological property relies on generic crystalline symmetries. Based on first-principles calculations, we study a three-dimensional (3D) crystal constructed by stacking two-dimensional TCI layers. Depending on the interlayer interaction, the layered crystal can realize diverse 3D topological phases characterized by two mirror Chern numbers (MCNs) (μ1,μ2) defined on inequivalent mirror-invariant planes in the Brillouin zone. As an example, we demonstrate that new TCI phases can be realized in layered materials such as a PbSe (001) monolayer/h-BN heterostructure and can be tuned by mechanical strain. Our results shed light on the role of the MCNs on inequivalent mirror-symmetric planes in reciprocal space and open new possibilities for finding new topological materials.

Engineering Dirac electrons emergent on the surface of a topological insulator

The concept of the topological insulator (TI) has introduced a new point of view to condensed-matter physics, relating a priori unrelated subfields such as quantum (spin, anomalous) Hall effects, spin-orbit coupled materials, some classes of nodal superconductors, superfluid 3He, etc. From a technological point of view, TIs are expected to serve as platforms for realizing dissipationless transport in a non-superconducting context. The TI exhibits a gapless surface state with a characteristic conic dispersion (a surface Dirac cone). Here, we review peculiar finite-size effects applicable to such surface states in TI nanostructures. We highlight the specific electronic properties of TI nanowires and nanoparticles, and in this context we contrast the cases of weak and strong TIs. We study the robustness of the surface and the bulk of TIs against disorder, addressing the physics of Dirac and Weyl semimetals as a new research perspective in the field.

Transport discovery of emerging robust helical surface States in Z2 = 0 systems

We study the possibility of realizing robust helical surface states in Z(2) = 0 systems. We find that the combination of anisotropy and finite-size confinement leads to the emergence of robust helical edge states in both two-dimensional and three-dimensional Z(2) = 0 systems. By investigating an anisotropic Bernevig-Hughes-Zhang model in a finite sample, we demonstrate that the transport manifestation of the surface states is robust against nonmagnetic disorder, resembling that of a Z(2) = 1 phase. Notably, the effective energy gap of the robust helical states can be efficiently engineered, allowing for potential applications as valley filters and valley valves. The realization of emerging robust helical surface states in realistic materials is also discussed.

Symmetry-protected Majorana fermions in topological crystalline superconductors: theory and application to Sr2RuO4

Crystal point group symmetry is shown to protect Majorana fermions (MFs) in spinfull superconductors (SCs). We elucidate the condition necessary to obtain MFs protected by the point group symmetry. We argue that superconductivity in Sr2RuO4 hosts a topological phase transition to a topological crystalline SC, which accompanies a d-vector rotation under a magnetic field along the c axis. Taking all three bands and spin-orbit interactions into account, symmetry-protected MFs in the topological crystalline SC are identified. Detection of such MFs provides evidence of the d-vector rotation in Sr2RuO4 expected from Knight shift measurements but not yet verified.

Existence of Majorana-fermion bound states on disclinations and the classification of topological crystalline superconductors in two dimensions

We prove a topological criterion for the existence of a zero-energy Majorana bound state on a disclination, a rotation symmetry breaking point defect, in fourfold symmetric topological crystalline superconductors (TCS) in two dimensions. We first establish a complete topological classification of TCS using the Chern invariant and three integral rotation invariants. By analytically and numerically studying disclinations, we algebraically deduce a Z2 index that identifies the parity of the number of Majorana zero modes at a disclination. Surprisingly, we also find weakly protected Majorana fermions bound at the corners of superconductors with trivial Chern and weak invariants.

Disordered weak and strong topological insulators

A global phase diagram of disordered weak and strong topological insulators is established numerically. As expected, the location of the phase boundaries is renormalized by disorder, a feature recognized in the study of the so-called topological Anderson insulator. Here, we report unexpected quantization, i.e., robustness against disorder of the conductance peaks on these phase boundaries. Another highlight of the work is on the emergence of two subregions in the weak topological insulator phase under disorder. According to the size dependence of the conductance, the surface states are either robust or "defeated" in the two subregions. The nature of the two distinct types of behavior is further revealed by studying the Lyapunov exponents.

Quantum interference and Aharonov-Bohm oscillations in topological insulators

Topological insulators (TIs) have an insulating bulk but a metallic surface. In the simplest case, the surface electronic structure of a three-dimensional (3D) TI is described by a single two-dimensional (2D) Dirac cone. A single 2D Dirac fermion cannot be realized in an isolated 2D system with time-reversal symmetry, but rather owes its existence to the topological properties of the 3D bulk wavefunctions. The transport properties of such a surface state are of considerable current interest; they have some similarities with graphene, which also realizes Dirac fermions, but have several unique features in their response to magnetic fields. In this review we give an overview of some of the main quantum transport properties of TI surfaces. We focus on the efforts to use quantum interference phenomena, such as weak anti-localization and the Aharonov-Bohm effect, to verify in a transport experiment the Dirac nature of the surface state and its defining properties. In addition to explaining the basic ideas and predictions of the theory, we provide a survey of recent experimental work.

Stacked topological insulator built from bismuth-based graphene sheet analogues

Commonly, materials are classified as either electrical conductors or insulators. The theoretical discovery of topological insulators has fundamentally challenged this dichotomy. In a topological insulator, the spin-orbit interaction generates a non-trivial topology of the electronic band structure dictating that its bulk is perfectly insulating, whereas its surface is fully conducting. The first topological insulator candidate material put forward--graphene--is of limited practical use because its weak spin-orbit interactions produce a bandgap of ~0.01 K. Recent reexaminations of Bi2Se3 and Bi2Te3, however, have firmly categorized these materials as strong three-dimensional topological insulators. We have synthesized the first bulk material belonging to an entirely different, weak, topological class, built from stacks of two-dimensional topological insulators: Bi14Rh3I9. Its Bi-Rh sheets are graphene analogues, but with a honeycomb net composed of RhBi8 cubes rather than carbon atoms. The strong bismuth-related spin-orbit interaction renders each graphene-like layer a topological insulator with a 2,400 K bandgap.

Kosterlitz-Thouless transition in disordered two-dimensional topological insulators

The disorder-driven metal-insulator transition in the quantum spin Hall systems is studied by scaling analysis of the Thouless conductance g. Below a critical disorder strength, the conductance is independent of the sample size M, an indication of critically delocalized electron states. The calculated beta function β = d ln g/d ln M indicates that the metal-insulator transition is of Kosterlitz-Thouless (KT) type, which is characterized by binding and unbinding of vortex-antivortex pairs of the local currents. The KT-like metal-insulator transition is a basic characteristic of the quantum spin Hall state, being independent of the time-reversal symmetry.

Topology, delocalization via average symmetry and the symplectic Anderson transition

A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z(2) topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N=1-[symbol: see text] expansion in the number of replicas, where for small [symbol: see text] the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Prediction of weak topological insulators in layered semiconductors

We report the discovery of weak topological insulators by ab initio calculations in a honeycomb lattice. We propose a structure with an odd number of layers in the primitive unit cell as a prerequisite for forming weak topological insulators. Here, the single-layered KHgSb is the most suitable candidate for its large bulk energy gap of 0.24 eV. Its side surface hosts metallic surface states, forming two anisotropic Dirac cones. Although the stacking of even-layered structures leads to trivial insulators, the structures can host a quantum spin Hall layer with a large bulk gap, if an additional single layer exists as a stacking fault in the crystal. The reported honeycomb compounds can serve as prototypes to aid in the finding of new weak topological insulators in layered small-gap semiconductors.