Pursuing topological phases in natural and artificial materials is one of the central topics in modern physical science and engineering. In classical magnetic systems, spin waves (or magnons) and magnetic solitons (such as domain wall, vortex, skyrmion, etc) represent two important excitations. Recently, the topological insulator and semimetal states in magnon- and soliton-based crystals (or metamaterials) have attracted growing attention owing to their interesting dynamics and promising applications for designing robust spintronic devices. Here, we give an overview of current progress of topological phases in structured classical magnetism. We first provide a brief introduction to spin wave, and discuss its topological properties including magnon Hall effects, topological magnon insulators, and Dirac (Weyl) magnon semimetals. Appealing proposal of topological magnonic devices is also highlighted. We then review the collective-coordinate approach for describing the dynamics of magnetic soliton lattice. Pedagogical topological models such as the Su-Schrieffer-Heeger model and the Haldane model and their manifestation in magnetic soliton crystals are elaborated. Then we focus on the topological properties of magnetic solitons, by theoretically analyzing the first-order topological insulating phases in low dimensional systems and higher-order topological states in breathing crystals. Finally, we discuss the experimental realization and detection of the edge states in both the magnonic and solitonic crystals. We remark the challenges and future prospects before concluding this article.
The recently discovered topological Dirac semimetal represents a new exotic quantum state of matter. Topological Dirac semimetals can be viewed as three dimensional analogues of graphene, in which the Dirac nodes are protected by crystalline symmetry. It has been found that quantum confinement effect can gap out Dirac nodes and convert Dirac semimetal to a band insulator. The band insulator is either normal insulator or quantum spin Hall insulator depending on the thin film thickness. We present the study of disorder effects in thin film of Dirac semimetals. It is found that moderate Anderson …
We propose a different route to time-reversal invariant Weyl semimetals employing multilayer heterostructures comprising ordinary "trivial" insulators and nontrivial insulators with \textit{pairs} of protected Dirac cones on the surface. We consider both the case of weak topological insualtors, where surface Dirac cones are pinned to time-reversal invariant momenta, and of topological crystalline insulators with unpinned surface Dirac cones. For both realizations we explain phenomenologically how the proposed construction leads to the emergence of a Weyl semimetal phase. We further formulate e…
In this Book Chapter (invited) we briefly review the basic concepts defining topological insulators and focus on elaborating on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We then present key experimental results that demonstrate magnetism, Kondo insulation, mirror chirality or topological crystalline order and superconductivity in spin-orbit topological insulator settings and how these new (bulk insulating) phases of matter arise through topological quantum phase transitions from Bloch band insulators via topological or Dirac s…
By first-principles calculations, we find that Ba2X(X=Si, Ge) hosts a topological semimetal phase with one nodal ring in the kx=0 plane, which is protected by the glide mirror symmetry when spin-orbit coupling (SOC) is ignored. The corresponding drumheadlike surface flat band appears on the (100) surface in surface Green function calculation. Furthermore, a topological-semimetal-to-insulator transition (TSMIT) is found. The nodal line semimetal would evolve into topological insulator as SOC is turned on. The topologically protected metallic surface states emerge around the Gamma=0 point, which…
We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically-doped 3D topological insulator (TI), separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its bandstructure. This particular type of Weyl semimetal has a finite anomalous Hall conductivity, chiral edge states, and occurs as an intermediate phase b…
The electronic bands are classified according to their topology. We compute the connection and curvature for the electronic bands and show that the physical properties are determined by topological invariants which are equivalent to the existence of the zero modes. We apply this method to the Topological Insulators and Topogical Superconductors.
In this work, we propose a ferromagnetic Bi$_2$Se$_3$ as a candidate to hold the coexistence of Weyl- and nodal-line semimetal phases, which breaks the time reversal symmetry. We demonstrate that the type-I Weyl semimetal phase, type-I-, type-II- and their hybrid nodal-line semimetal phases can arise by tuning the Zeeman exchange field strength and the Fermi velocity. Their topological responses under U(1) gauge field are also discussed. Our results raise a new way for realizing Weyl and nodal-line semimetals and will be helpful in understanding the topological transport phenomena in three-dim…
We report a gate-controlled transition of a semimetallic InAs/GaSb heterostructure to a topological insulator. The transition is induced by decreasing the degree of band inversion with front and back gate voltages. Temperature dependence of the longitudinal resistance peak shows the energy gap opening in the bulk region with increasing gate electric field. The suppression of bulk conduction and the transition to a topological insulator are confirmed by nonlocal resistance measurements using a dual lock-in technique, which allows us to rigorously compare the voltage distribution in the sample f…
Symmetries play an essential role in identifying and characterizing topological states of matter. Here, we classify topologically two-dimensional (2D) insulators and semimetals with vanishing spin-orbit coupling using time-reversal ($\mathcal{T}$) and inversion ($\mathcal{I}$) symmetry. This allows us to link the presence of edge states in $\mathcal{I}$ and $\mathcal{T}$ symmetric 2D insulators, which are topologically trivial according to the Altland-Zirnbauer table, to a $\mathbb{Z}_2$ topological invariant. This invariant is directly related to the quantization of the Zak phase. It also pre…
We consider a natural generalization of the lattice model for a periodic array of two layers, A and B, of spinless electrons proposed by Fu [Phys. Rev. Lett. 106, 106802 (2011)] as a prototype for a crystalline insulator. This model has time-reversal symmetry and broken inversion symmetry. We show that when the intralayer next-nearest-neighbor hoppings ta2, a = A, B vanish, this model supports a Weyl semimetal phase for a wide range of the remaining model parameters. When the effect of ta2 is considered, topological crystalline insulating phases take place within the Weyl semimetal one. By map…
We study the problem of phase transitions from 3D topological to normal insulators without inversion symmetry. In contrast with the conclusions of some previous work, we show that a Weyl semimetal always exists as an intermediate phase regardless of any constriant from lattice symmetries, although the interval of the critical region is sensitive to the choice of path in the parameter space and can be very narrow. We demonstrate this behavior by carrying out first-principles calculations on the noncentrosymmetric topological insulators LaBiTe$_3$ and LuBiTe$_3$ and the trivial insulator BiTeI. …
We propose magnetic second-order topological insulators (SOTIs). First, we study a three-dimensional model. It is pointed out that the previously proposed topological hinge insulator has actually surface states along the [001] direction in addition to hinge states. We gap out these surface states by introducing magnetization, obtaining a SOTI only with hinge states. The bulk topological number is the $Z_2$ index protected by the combined symmetry of the four-fold rotation and the inversion symmetry. We next study two dimensional magnetic SOTIs, where the corner states are robust also in the pr…
The strong Coulomb interaction between massless Dirac fermions can drive a semimetal-insulator transition in single-layer graphene by dynamically generating an excitonic fermion gap. There is a critical interaction strength $λ_c$ that separates the semimetal phase from the insulator phase. We calculate the specific heat and susceptibility of the system and show that they exhibit distinct behaviors in the semimetal and insulator phases.
We investigate the effects of periodic fields and disorder on topological properties of a Weyl-semimetal thin film. The two periodic fields, i.e., a periodic magnetic field and elliptically polarized light, are discussed respectively. By use of the Floquet theory, we find that both the two periodic drives can resonantly induce the topological transitions from normal insulator (NI) phases to Floquet topological insulator (FTI) phases. The Floquet topological transitions are characterized by variation of Chern number. Moreover, we show that the Floquet topological transitions can be explained by…
In this work we provide a new direct and non-numerical technique to obtain the surface Green's functions for three-dimensional systems. This technique is based on the ideas presented in Phys. Rev. B 100, 081106(R), in which we start with an infinite system and model the boundary using a plane-like infinite-amplitude potential. Such a configuration can be solved exactly using the T-matrix formalism. We apply our method to calculate the surface Green's function and the corresponding Fermi-arc states for Weyl semimetals. We also apply the technique to systems of lower dimensions, such as Kane-Mel…
Regarding three-dimensional (3D) topological insulators and semimetals as a stack of constituent 2D topological (or sometimes non-topological) layers is a useful viewpoint. Primarily, concrete theoretical models of the paradigmatic 3D topological phases such as Weyl semimetal (WSM), strong and weak topological insulators (STI/WTI), and Chern insulator (CI), are often constructed in that way. Secondarily, fabrication of the corresponding 3D topological material is also done in the same spirit; epitaxial growth technique is employed, making the resulting sample in the form of a thin film. Here, …
The notion of higher-order topological insulators has endowed materials with topological states beyond the first order. Particularly, a three-dimensional (3D) higher-order topological insulator can host topologically protected 1D hinge states, referred to as the second-order topological insulator, or 0D corner states, referred to as the third-order topological insulator. Similarly, a 3D higher-order topological semimetal can be envisaged if it hosts states on the 1D hinges. Here we report the realization of a second-order topological Weyl semimetal in a 3D-printed acoustic crystal, which posse…
We use first principles calculations to study the electronic properties of rock salt rare earth monopnictides La$X$ ($X=$N, P, As, Sb, Bi). A new type of topological band crossing termed `linked nodal rings' is found in LaN when the small spin-orbital coupling (SOC) on nitrogen orbitals is neglected. Turning on SOC gaps the nodal rings at all but two points, which remain gapless due to $C_4$-symmetry and leads to a 3D Dirac semimetal. Interestingly, unlike LaN, compounds with other elements in the pnictogen group are found to be topological insulators (TIs), as a result of band reordering due …
Lattices with a basis can host crystallographic defects which share the same topological charge (e.g.~the Burgers vector $\vec b$ of a dislocation) but differ in their microscopic structure of the core. We demonstrate that in insulators with particle-hole symmetry and an odd number of orbitals per site, the microscopic details drastically affect the electronic structure: modifications can create or annihilate non-trivial bound states with an associated fractional charge. We show that this observation is related to the behavior of end modes of a dimerized chain and discuss how the end or defect…
We discuss the possibility of realizing Weyl semimetal phase in the magnetically doped topological band insulators. When the magnetic moments are ferromagnetically polarized, we show that there are three phases in the system upon the competition between topological mass and magnetic mass: topological band insulator phase, Weyl semimetal phase, and trivial phase. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian of topological insulators near the Dirac point, e.g. ${\bf k} \cdot {\bf p}$ theory near $Γ$ point for Bi_{2}Se_{3}. Furthermore, we intro…
A heterostructure consisting of a magnetic Weyl semimetal and a conventional superconductor exhibits an equilibrium current parallel to the superconductor interface and perpendicular to the magnetization. Analyzing a minimal model, which as a function of parameters may be in a trivial magnetic insulator phase, a Weyl semimetal phase, or a three-dimensional weak Chern insulator phase, we find that the equilibrium current is sensitive to the presence of surface states, such as the topological Fermi-arc states of the Weyl semimetal or the chiral surface states of the weak Chern insulator. While t…
We investigate multilayer systems of a normal insulator and a Weyl semimetal. We calculate the bulk band structure and determine phase diagrams by changing the thickness of the normal insulator and that of the Weyl semimetal layer using two models; one is from the effective model for a Weyl semimetal, and the other is the lattice model. We compare the results between the two models, and found that they agree well. As a result, we find that the multilayer system shows a qualitatively different behavior depending on the stacking direction relative to the displacement vector connecting between tw…
Topological semimetals, such as Dirac, Weyl, or line-node semimetals, are gapless states of matter characterized by their nodal band structures and surface states. In this work, we consider layered (topologically trivial) insulating systems in $D$ dimensions that are composed of coupled multi-layers of $d$-dimensional topological semimetals. Despite being nominal bulk insulators, we show that crystal defects having co-dimension $(D-d)$ can harbor robust lower dimensional topological semimetals embedded in a trivial insulating background. As an example we show that defect-bound topological semi…
Floquet engineering, the concept of tailoring a system by a periodic drive, is increasingly exploited to design and manipulate topological phases of matter. In this work, we study periodically driven higher-order topological Dirac semimetals associated with a $k$-dependent quantized quadrupole moment by applying circularly polarized light. The undriven Dirac semimetals feature gapless higher-order hinge Fermi arc states which are the consequence of the higher-order topology of the Dirac nodes. Floquet Weyl semimetal phases with hybrid-order topology, characterized by both a $k$-dependent quant…
We propose topological semimetals generated by the square-root operation for tight-binding models in two and three dimensions, which we call square-root topological semimetals. The square-root topological semimetals host topological band touching at finite energies, whose topological protection is inherited from the squared Hamiltonian. Such a topological character is also reflected in emergence of boundary modes with finite energies. Specifically, focusing on topological properties of squared Hamiltonian in class AIII, we reveal that a decorated honeycomb (decorated diamond) model hosts finit…
For conventional topological phases, the boundary gapless modes are determined by bulk topological invariants. Based on developing an analytic method to solve higher-order boundary modes, we present $PT$-invariant $2$D topological insulators and $3$D topological semimetals that go beyond this bulk-boundary correspondence framework. With unchanged bulk topological invariant, their first-order boundaries undergo transitions separating different phases with second-order-boundary zero-modes. For the $2$D topological insulator, the helical edge modes appear at the transition point for two second-or…
We theoretically study the electric polarization in magnetic topological nodal semimetal thin films. In magnetically doped topological insulators, topological nodal semimetal phases emerge once the exchange coupling overcomes the band gap. Changing the magnetization direction, nodal structure is modulated and the system becomes topological nodal point or line semimetals. We find that nodal line semimetals are characterized by non-linear electric polarization, which is not observed in nodal point semimetals. The non-linear response originates from the existence of the surface states. Screening …
A three-dimensional (3D) nodal-loop semimetal phase is exploited to engineer a number of intriguing phases featuring different peculiar topological surface states. In particular, by introducing various two-dimensional gap terms to a 3D tight-binding model of a nodal-loop semimetal, we obtain a rich variety of topological phases of great interest to ongoing theoretical and experimental studies, including chiral insulator, degenerate-surface-loop insulator, second-order topological insulator, as well as Weyl semimetal with tunable Fermi arc profiles. The unique concept underlying our approach is…
Topological insulators (TIs) and Weyl semimetals (WSMs) are two realizations of topological matter usually appearing separately in nature. However, they are directly related to each other via a topological phase transition. In this paper, we investigate the question whether these two topological phases can exist together at the same time, with a combined, hybrid surface state at the joint boundaries. We analyze effective models of a 3D TI and an inversion symmetric WSM and couple them in a way that certain symmetries, like inversion, are preserved. A tunnel coupling approach enables us to obta…
We solve the two-band model for the transport across a junction between a semimetal and an excitonic insulator. We analyze the current in terms of two competing terms associated with neutral excitons and charged carriers, respectively. We find a high value for the interface resistance, extremely sensitive to the junction transparency. We explore favorable systems for experimental confirmation.
A band gap for electronic states in crystals governs various properties of solids, such as transport, optical and magnetic properties. Its estimation and control have been an important issue in solid state physics. The band gap can be controlled externally by various parameters, such as pressure, atomic compositions and external field. Sometimes, the gap even collapses by tuning some parameter. In the field of topological insulators, such closing of the gap at a time-reversal invariant momentum indicates a band inversion, i.e. it leads to a topological phase transition from a normal insulator …
Chiral topological semimetals host multifold degenerate band crossing points under the protection of crystalline symmetries. In this paper, we suggest that the recently discovered chiral topological semimetals in space group 198, parts of which are superconducting upon doping, can be new candidates of time reversal invariant topological superconductors. By investigating the Fermi surfaces around the band crossing points that carry nonzero Chern numbers, we clarify how the nontrivial topology of chiral topological semimetals affects their superconducting state and show the existence of topologi…
We propose the creation of a two-dimensional topological semimetal in a semiconductor artificial lattice with triangular symmetry. An in-plane magnetic field drives a quantum phase transition between the topological insulating and topological semimetal phases. The topological semimetal is characterized by robust band touching points which carry quantized Berry flux and edge states which terminate at the band touching points. The topological phase transition is predicted to occur at magnetic fields $\sim 4\text{T}$ in high mobility GaAs artificial lattices, and can be detected via the anomalous…
The Weyl semimetal (WSM) is a newly proposed quantum state of matter. It has Weyl nodes in bulk excitations and Fermi arcs surface states. We study the effects of disorder and localization in WSMs and find three exotic phase transitions. (I) Two Weyl nodes near the Brillouin zone boundary can be annihilated pairwise by disorder scattering, resulting in the opening of a topologically nontrivial gap and a transition from a WSM to a three-dimensional (3D) quantum anomalous Hall state. (II) When the two Weyl nodes are well separated in momentum space, the emergent bulk extended states can give ris…
Topological semimetals and metals have emerged as a new frontier in the field of quantum materials. Novel macroscopic quantum phenomena they exhibit are not only of fundamental interest, but may hold some potential for technological applications.
We study what kind of topological crystalline insulator phase emerges from nodal line semimetal phases in the face-centered cubic system which is not indicated by topological indices when spin-orbit coupling (SOC) is introduced. We construct an effective model which hosts two different nodal lines phases, and calculated mirror Chern numbers in it by introducing SOC. As a result, we find that the two nodal line phases with different nodal line configurations are mapped to different topological crystalline insulator phases. This result shows that turning off SOC and checking the nodal line confi…
We investigate the effect of a vertical electric field on a Dirac semimetal thin film. We show that through the interplay between the quantum confinement effect and the field-induced coupling between sub-bands, the sub-band gap can be tuned and inverted, during which the system undergoes a topological phase transition between a trivial band insulator and a quantum spin Hall insulator. Consequently, one can electrically switch the topological edge channels on and off, making the system a promising platform for constructing a topological field effect transistor.
A pair of Dirac points (analogous to a vortex-antivortex pair) associated with opposite topological numbers (with $\pmπ$ Berry phases) can be merged together through parameter tuning and annihilated to gap the Dirac spectrum, offering a canonical example of a topological phase transition. Here, we report transport studies on thin films of BiSbTeSe$_2$ (BSTS), which is a 3D TI that hosts spin-helical gapless (semi-metallic) Dirac fermion surface states (SS) for sufficiently thick samples, with an observed resistivity close to $h/4e^2$ at the charge neutral point. When the sample thickness is re…
In this work we predict several new types of topological semimetals that exhibit a bulk quadrupole moment. These semimetals are modeled with a 3D extension of the 2D quadrupole topological insulator. One type of semimetal has bulk nodes and gapped, topological surfaces. A second type, which we may call a higher order topological semimetal, has a gapped bulk, but harbors a Dirac semimetal with an even number of nodes on one or more surfaces. The final type has a gapped bulk, but harbors half of a Dirac semimetal on multiple surfaces. Each of these semimetals gives rise to mid-gap hinge states a…
We investigate electromagnetic properties of a topological crystalline insulator (TCI) thin film under external electromagnetic fields. The TCI thin film is a topological insulator indexed by the mirror-Chern number. It is demonstrated that the gap closes together with the emergence of a pair of gapless cones carrying opposite chirarities by applying in-plane magnetic field. A pair of gapless points have opposite vortex numbers. This is a reminiscence of a pair of Weyl cones in 3D Weyl semimetal. We thus present an a magnetic-field induced semimetal-semiconductor transition in 2D material. Thi…
The three dimensional (3D) topological insulators are predicted to exhibit a 3D Dirac semimetal state in critical regime of topological to trivial phase transition. Here we demonstrate the first experimental evidence of 3D Dirac semimetal state in topological insulator Bi2Se3 with bulk carrier concentration of ~ 10^19 cm^{-3}, using magneto-transport measurements. At low temperatures, the resistivity of our Bi2Se3 crystal exhibits clear Shubnikov-de Haas (SdH) oscillations above 6T. The analysis of these oscillations through Lifshitz-Onsanger and Lifshitz-Kosevich theory reveals a non-trivial …
We study a general phase transition between spinless topological nodal-line semimetal and Weyl semimetal phases. We classify topological nodal lines into two types based on their positions and shapes, and their phase transitions depends on their types. We show that the topological nodal-line semimetal becomes the Weyl semimetal by breaking time-reversal symmetry when the nodal lines enclose time-reversal invariant momenta (type-A nodal lines). We also discuss an effect of crystallographic symmetries determining the band structure of the topological nodal-line semimetals. Thanks to protection b…
Recently, we have introduced in [A. M. Marques et al., Phys. Rev. B 103, 235425 (2021)] the concept of $2^n$-root topology and applied it to one-dimensional systems. These models require $n$ squaring operations to their Hamiltonians, intercalated with different constant energy downshifts at each level, in order to arrive at a decoupled block corresponding to a known topological insulator (TI) that acts as the source of the topological features of the starting $2^n$-root TI ($\sqrt[2^n]{\text{TI}}$). In the process, $n$ non-topological residual models with degenerate spectra and in-gap impurity…
We review the recent, mainly theoretical, progress in the study of topological nodal line semimetals in three dimensions. In these semimetals, the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone, and any perturbation that preserves a certain symmetry group (generated by either spatial symmetries or time-reversal symmetry) cannot remove this crossing line and open a full direct gap between the two bands. The nodal line(s) is hence topologically protected by the symmetry group, and can be associated with a topological invari…
$α$-Sn is well known as a typical Luttinger semimetal with a quadratic band touching at the $Γ$ point. Based on the effective $k\cdot p$ analysis as well as first-principles calculations, we demonstrate that multiple topological phases with a rich diagram, including topological insulator, Dirac semimetal, and Weyl semimetal phases, can be induced and engineered in $α$-Sn by external strains, magnetic fields, and circularly polarized light (CPL). Intriguingly, not only the conventional type-I Weyl nodes, but also type-II Weyl nodes and double-Weyl nodes can be generated directly from the quadra…
When high quality bismuth or graphite crystals are placed in a magnetic field directed along the c-axis (trigonal axis for bismuth) and the temperature is lowered, the resistance increases as it does in an insulator but then saturates. We show that the combination of unusual features specific to semimetals, i.e., low carrier density, small effective mass, high purity, and an equal number of electrons and holes (compensation), gives rise to a unique ordering and spacing of three characteristic energy scales, which not only is specific to semimetals but which concomitantly provides a wide window…
We theoretically show that IV-VI semiconducting compounds with low-temperature rhombohedral crystal structure represent a new potential platform for topological semimetals. By means of minimal $\mathbf{k}\cdot\mathbf{p}$ models we find that the two-step structural symmetry reduction of the high-temperature rocksalt crystal structure, comprising a rhombohedral distortion along the [111] direction followed by a relative shift of the cation and anion sublattices, gives rise to topologically protected Weyl semimetal and nodal line semimetal phases. We derive general expressions for the nodal featu…
We review our recent works on the quantum transport, mainly in topological semimetals and also in topological insulators, organized according to the strength of the magnetic field. At weak magnetic fields, we explain the negative magnetoresistance in topological semimetals and topological insulators by using the semiclassical equations of motion with the nontrivial Berry curvature. We show that the negative magnetoresistance can exist without the chiral anomaly. At strong magnetic fields, we establish theories for the quantum oscillations in topological Weyl, Dirac, and nodal-line semimetals. …
We propose an unconventional type of Hall effect in a topological Dirac semimetal with ferromagnetic electrodes. The topological Dirac semimetal itself has time-reversal symmetry, whereas attached ferromagnetic electrodes break it, causing the large Hall response. This induced Hall effect is a characteristic of the helical surface/edge states that arise in topological materials, such as topological Dirac semimetals or quantum spin Hall insulators. We compute the Hall conductance/resistance and the Hall angle by using a lattice model with four-terminal geometry. For topological Dirac semimetals…
We study two Weyl semimetal generalizations in five dimensions (5d) which have Yang monopoles and linked Weyl surfaces in the Brillouin zone, respectively, and carry the second Chern number as a topological number. In particular, we show a Yang monopole naturally reduces to a Hopf link of two Weyl surfaces when the $\mathbf{TP}$ (time-reversal combined with space-inversion) symmetry is broken. We then examine the phase transition between insulators with different topological numbers in 5d. In analogy to the 3d case, 5d Weyl semimetals emerge as intermediate phases during the topological phase …
Starting from a minimal model for a 2D nodal loop semimetal, we study the effect of chiral mass gap terms. The resulting Dirac loop anomalous Hall insulator's Chern number is the phase winding number of the mass gap terms on the loop. We provide simple lattice models, analyze the topological phases and generalize a previous index characterizing topological transitions. The responses of the Dirac loop anomalous Hall and quantum spin Hall insulators to a magnetic field's vector potential are also studied both in weak and strong field regimes, as well as the edge states in a ribbon geometry.
In ultra-thin film of topological insulator, the hybridization between the top and bottom surfaces opens an energy gap and forms two degenerate quantum anomalous Hall states, which give rise to a quantum spin Hall state. In this paper, we demonstrate that a Weyl semimetal can be realized in an ultra-thin film of topological insulator heterostructure in a similar way to that of the surface state of a strong three-dimensional (3D) topological insulator studied by Burkov and Balents . We find that the system realizes both 3D quantum anomalous Hall phase and 3D quantum spin Hall phase, and the Wey…
We report on a fully self-consistent Hartree-Fock calculation of interaction effects on the Moiré flat bands of twisted bilayer graphene, assuming that valley U(1) symmetry is respected. We use realistic band structures and interactions and focus on the charge neutrality point, where experiments have variously reported either insulating or semimetallic behavior. Restricting the search to orders for which the valley U(1) symmetry remains unbroken, we find three types of self-consistent solutions with competitive ground state energy (i) insulators that break $C_2 {\mathcal T}$ symmetry, includin…
