How to characterize topological quantum phases is a fundamental issue in the broad field of topological matter. From a dimension reduction approach, we propose the concept of high-order band inversion surfaces (BISs) which enable the optimal schemes to characterize equilibrium topological phases by far-from-equilibrium quantum dynamics, and further report the experimental simulation. We show that characterization of a d-dimensional (dD) topological phase can be reduced to lower-dimensional topological invariants in the high-order BISs, of which the nth-order BIS is a (d-n)D interface in momentum space. In quenching the system from trivial phase to topological regime, we unveil a high-order dynamical bulk-surface correspondence that the quantum dynamics exhibits nontrivial topological pattern in arbitrary nth-order BISs, which universally corresponds to and so characterizes the equilibrium topological phase of the post-quench Hamiltonian. This high-order dynamical bulk-surface correspondence provides new and optimal dynamical schemes with fundamental advantages to simulate and detect topological states, in which through the highest-order BISs that are of zero dimension, the detection of topological phase relies on only minimal measurements. We experimentally build up a quantum simulator with spin qubits to investigate a 3D chiral topological insulator through emulating each momentum one by one and measure the high-order dynamical bulk-surface correspondence, with the advantages of topological characterization via highest-order BISs being demonstrated.
We have experimentally realized novel space-time inversion (P-T) invariant Z2-type topological semimetal-bands, via an analogy between the momentum space and a controllable parameter space in superconducting quantum circuits. By measuring the whole energy spectrum of system, we imaged clearly an exotic tunable gapless band structure of topological semimetals. Two topological quantum phase transitions from a topological semimetal to two kinds of insulators can be manipulated by continuously tuning the different parameters in the experimental setup, one of which captures the Z2 topology of the P…
Higher-order topological phases (HOTPs) hold gapped bulk bands and topological boundary states localized in boundaries with codimension higher than one. In this paper, we provide a unified construction and topological characterization of HOTPs for the full Altland-Zirnbauer tenfold symmetry classes, based on a method known as nested band inversion surfaces (BISs). Specifically, HOTPs built on this method are decomposed into a series of subsystems, and higher-order topological boundary states emerges from the interplay of their first-order topology. Our analysis begins with a general discussion…
In this paper, we discuss the characteristic features of one-dimensional topological insulators with inversion symmetry but noncentered inversion axis in the unit cell, for any choice of the unit cell. In these systems, the global inversion operation generates a k-dependent inversion operator within the unit cell and this implies a nonquantized Zak's phase both for the trivial and nontrivial topological phases. By relating the Zak's phase with the eigenvalues of modified parity operators at the inversion invariant momenta, a corrected quantized form of the Zak's phase is derived. We show that …
Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge sta…
We investigate the effect of a high frequency electromagnetic field with both of circularly and linearly polarization, on the emergence of quantum phases on thin topological insulators. Simultaneously, the influence of the system parameters (such as magnetic impurity, thickness engineering and structural inversion asymmetry of the potential) on emergence of topological phases is studied. We take our attention to the high frequency regime in which it is possible to consider an expansion for the Floquet Hamiltonian in terms of orders of 1/Ω. The topological invariants are determined and it is de…
Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We a…
We study positions of chiral hinge states in higher-order topological insulators (HOTIs) with inversion symmetry. First, we exhaust all possible configurations of the hinge states in the HOTIs in all type-I magnetic space groups with inversion symmetry by studying dependence of the sign of the surface Dirac mass on surface orientations. In particular, in the presence of glide symmetry, for particular surface orientations, the surface Dirac mass changes sign by changing the surface terminations. By applying this result to a layered antiferromagnet (AFM), we find a difference in the hinge states…
Recent realizations of exotic topological states in condensed matter and cold atoms have advanced the exploration for topological characteristics, such as invariant topological orders and band inversion. Here we construct a 1D optical diatomic system made of split ring resonators with electric-magnetic couplings to implement the topological features. We experimentally measure the dispersion relationship and sub-lattice pseudo-spin vectors by detecting field strength distributions, which determines the accompanying winding number of bulk band as the topological order. The sub-lattice pseudo-spi…
We determine the optical response of ultrathin film topological insulators in the presence of a quantizing external magnetic field taking into account both hybridization between surface states, broken inversion symmetry and explicit time reversal symmetry breaking by the magnetic field. We find that breaking of inversion symmetry in the system, which can be due to interaction with a substrate or electrical gating, results in Landau level crossings which lead to additional optical transition channels that were previously forbidden. We show that by tuning the hybridization and symmetry breaking …
We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of "higher-order" topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We show that the surface states of higher-order topological insulators and superconductors can be thought of as globally irremovable topological defects and provide a complete classification of these inversion-protected phases in any spatial dimension for…
We prove the existence of higher-order topological insulators in: {\it i}) fourfold rotoinversion invariant bulk crystals, and {\it ii}) inversion-symmetric systems with or without an additional three-fold rotation symmetry. These states of matter are characterized by a non-trivial $\mathbb{Z}_2$ topological invariant, which we define in terms of symmetric hybrid Wannier functions of the filled bands, and can be readily calculated from the knowledge of the crystalline symmetry labels of the bulk band structure. The topological invariant determines the generic presence or absence of protected c…
Based on first principles calculations, we predicate that Bi on a graphene derivate, $g$-C$_{14}$N$_3$, which involves a $3\times 3$ unit cell of graphene with four C atoms substituted by three N atoms, is a topological insulator with a gap of 50~meV. With the help of maximally localized Wannier functions, we find that the band inversion gap can be determined by examining a pair of interaction parameters between the two involved bands. Accordingly, a phase diagram for band inversion of topological materials as a function of the interactions is obtained. The conclusion also holds for Sb, Ir and…
In commonly employed models for 2D topological insulators, bulk gapless states are well known to form at the band inversion points where the degeneracy of the states is protected by symmetries. It is thus sometimes quite tempting to consider this feature, the occurrence of gapless states, a result of the band inversion process under protection of the symmetries. Similarly, the band inversion process might even be perceived as necessary to induce 2D topological phase transitions. To clarify these misleading perspectives, we propose a simple model with a flexible Chern number to demonstrate that…
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…
The band inversion has led to rich physical effects in both topological insulators and topological semimetals. It has been found that the inverted band structure with the Mexican-hat dispersion could enhance the interband correlation leading to a strong intrinsic plasmon excitation. Its frequency ranges from several $\mathrm{meV}$ to tens of $\mathrm{meV}$ and can be effectively tuned by the external fields. The electron-hole asymmetric term splits the peak of the plasmon excitation into double peaks. The fate and properties of this plasmon excitation can also act as a probe to characterize th…
We present a theory of the electron structure and the Zeeman effect for the helical edge states emerging in two-dimensional topological insulators based on HgTe/HgCdTe quantum wells with strong natural interface inversion asymmetry. The interface inversion asymmetry, reflecting the real atomistic structure of the quantum well, drastically modifies both bulk and edge states. For the in-plane magnetic field, this asymmetry leads to a strong anisotropy of the edge-state effective $g$-factor which becomes dependent on the edge orientation. The interface inversion asymmetry also couples the counter…
We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the ass…
We study translationally-invariant insulators with inversion symmetry that fall outside the established classification of topological insulators. These insulators are not required to have gapless boundary modes in the energy spectrum. However, they do exhibit protected modes in the entanglement spectrum localized on the cut between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the inversion eigenvalues of the band structure and the mid-gap states in the entanglem…
The effect of magnetic field on an ultrathin magnetic topological insulator film with structural inversion asymmetry is investigated. We introduce the phase diagram, calculate the Landau-level spectrum analytically and simulate the transport behavior in Landauer-Büttiker formalism. The quantum anomalous Hall phase will survive increasing magnetic field. Due to the two spin polarized zero modes of Landau levels a nontrivial phase similar with quantum spin Hall effect can be induced by magnetic field which is protected by structural inversion symmetry. Some exotic longitudinal and Hall resistanc…
The influence of bulk inversion asymmetry in [001] and [013] grown HgTe quantum wells is investigated theoretically. The bulk inversion asymmetry leads to an anti-crossing gap between two zero-mode Landau levels in a HgTe topological insulator, i.e., the quantum well with inverted band structure. It is found that this is the main contribution to the anti-crossing splitting observed in recent experimental magneto spectroscopic measurements. The relevant optical transitions involve different subbands, but the electron-electron interaction induced depolarization shift is found to be negligibly sm…
A topological boundary can be formed at the interface between a trivial and a topological insulator. The difference in the topological index across the junction leads to robust gapless surface states. Optical studies of these states are scarce in the literature, the reason being the difficulty to isolate their response from that of the bulk. In this work, we propose to deposit a $δ$ layer of donor impurities in close proximity to a topological boundary to help detecting gapless surface states. As we will show, gapless surface states are robust against this perturbation and they enhance intraba…
Floquet engineering provides a toolbox for the realization of novel quantum phases without static counterparts, while conventionally the realization may rely on the manipulation of complex temporal evolution. Here we propose a systematic and high-precision scheme to realize unconventional Floquet topological phases by engineering local band structures in particular momentum subspace called band inversion surfaces (BISs). This scheme is based on a new bulk-boundary correspondence that for a class of generic $d$-dimensional periodically driven systems, the local topological structure formed in e…
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 …
We examine the nature of topological currents in black phosphorus when its inversion symmetry is deliberately broken. Here, the conduction and valence band edges are located at the $Γ$ point of the rectangular Brillouin zone, and they exhibit strong anisotropy along its two crystal axes. We will show below that these salient features lead to a linear transverse neutral topological currents, accompanied also by a non-linear transverse charge current at the Fermi surface. These topological currents are maximal when the in-plane electric field is applied along the zigzag crystal axes, but zero al…
Higher-order topological insulators (HOTIs) are systems with topologically protected in-gap boundary states localized at their $(d-n)$-dimensional boundaries, with $d$ the system dimension and $n$ the order of the topology. This work proposes a rather universal dynamics-based characterization of one large class of $Z$-type HOTIs without specifically relying on any symmetry considerations. The key element of our innovative approach is to connect quantum quench dynamics with nested configurations of the so-called band inversion surfaces (BISs) of momentum-space Hamiltonians as a sum of operators…
Three-dimensional (3D) topological insulators (TIs) are new forms of quantum matter that are characterized by their insulating bulk state and exotic metallic surface state, which hosts helical Dirac fermions1-2. Very recently, BiTeCl, one of the polar semiconductors, has been discovered by angle-resolved photoemission spectroscopy to be the first strong inversion asymmetric topological insulator (SIATI). In contrast to the previously discovered 3D TIs with inversion symmetry, the SIATI are expected to exhibit novel topological phenomena, including crystalline-surface-dependent topological surf…
Recent ab initio calculations and experiments reported insulating-semimetallic phase transitions in multilayer phosphorene under a perpendicular dc field, pressure or doping, as a possible route to realize topological phases. In this work, we show that even a monolayer phosphorene may undergo Lifshitz transitions toward semimetallic and topological insulating phases, provided it is rapidly driven by in-plane time-periodic laser fields. Based on a four-orbital tight-binding description, we give an inversion-symmetry-based prescription in order to apprehend the topology of the photon-renormalize…
One of the hallmarks of topological insulators is the correspondence between the value of its bulk topological invariant and the number of topologically protected edge modes observed in a finite-sized sample. This bulk-boundary correspondence has been well-tested for strong topological invariants, and forms the basis for all proposed technological applications of topology. Here, we report that a group of weak topological invariants, which depend only on the symmetries of the atomic lattice, also induces a particular type of bulk-boundary correspondence. It predicts the presence or absence of s…
Depositing Au on a graphene derivate, which involves substituting four C atoms with three N atoms in a $3\times 3$ cell graphene, we realized a topological insulator of the Kane-Mele model with a gap of 50~meV surrounding the Dirac point of graphene. In this material, we observed an anomalous band inversion (BI) protected by the symmetry with character $e$ of group C$_{\rm 3V}$. The symmetry constrains two $e$ bands with mirror-symmetry combination (MSC) and mirror-antisymmetry combination (MAC) of Au and N orbitals degenerate at $Γ$, whereas the interaction of $π^*$ of graphene on the $e$-MAC…
Three-dimensional topological gapless phases have attracted significant attention due to their unique electronic properties. A flagship example are Weyl semimetals, which require breaking time-reversal or inversion symmetry. In two dimensions, the dimensionality reduction requires imposing an additional symmetry, thereby weakening the phase. Like its three-dimensional counterpart, these two-dimensional Weyl semimetals present edge states directly related to Weyl nodes. The direct comparison with the edge states in zigzag-like terminated graphene ribbons is unavoidable, offering the question of…
The precise determination of critical point is the basis to extract various critical properties of phase transitions. We identify that for two-dimensional inversion asymmetric insulators, with and without time-reversal symmetry, when topological phase transitions take place, all nonvanishing components of band-edge shift current tensor will reverse their signs in a singular way, regardless of what realistic value the temperature takes. This remarkable sign-reversal behavior of band-edge shift current tensor thus can be applied to determine the critical points of various topological phase trans…
We prove the existence of higher-order topological insulators with protected chiral hinge modes in quasi-two-dimensional systems made out of coupled layers stacked in an inversion-symmetric manner. In particular, we show that an external magnetic field drives a stack of alternating p- and n-doped buckled honeycomb layers into a higher-order topological phase, characterized by a non-trivial three-dimensional ${\mathbb Z}_2$ invariant. We identify silicene multilayers as a potential material platform for the experimental detection of this novel topological insulating phase.
Symmetry indicators have proven to be extremely helpful in identifying topologically non-trivial crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a $\mathbb{Z}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors of $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the …
We construct a general theory of $Z_2$ topological phase transitions in two-dimensional systems with time-reversal symmetry. We investigate the possibilities of $Z_2$ topological phase transitions at band inversions at all high-symmetry points in $k$-space in all the 80 layer groups. We exclude the layer groups with inversion symmetry because the $Z_2$ topological phase transition is known to be associated with band inversions with an exchange of parities. Among the other layer groups, we find 21 layer groups with insulator-to-insulator transitions with band inversion, and this problem is fina…
Topological insulators are materials with a bulk excitation gap generated by the spin orbit interaction, and which are different from conventional insulators. This distinction is characterized by Z_2 topological invariants, which characterize the groundstate. In two dimensions there is a single Z_2 invariant which distinguishes the ordinary insulator from the quantum spin Hall phase. In three dimensions there are four Z_2 invariants, which distinguish the ordinary insulator from "weak" and "strong" topological insulators. These phases are characterized by the presence of gapless surface (or ed…
We consider a disordered topological insulator thin film placed on the top of a ferromagnetic insulator with a perpendicular exchange field $M$ and subjected to a perpendicular electric field. The presence of ferromagnetic insulator causes that bottom surface states of the topological insulator thin film become spin polarized and the electric field provides a potential difference $V$ between the two surface states, resulting in breaking of time-reversal and inversion symmetry in the system. Using Kubo formalism and employing the first Born approximation as well as the self-consistent Born appr…
Nodal superconductors without inversion symmetry exhibit nontrivial topological properties, manifested by topologically protected flat-band edge states. Here we study the effects of breaking translational symmetry, crucial to the definition of the topological invariant. We show using large-scale numerical simulations that bulk-edge correspondence remains valid in the presence of edge roughness and strong edge disorder. Moderately strong nonmagnetic disorder shifts some of the edge states away from zero energy, but does not change their total number. Strong spin-independent edge disorder, on th…
It is known that three-dimensional magnetic systems with glide symmetry can be characterized by a $Z_2$ topological invariant together with the Chern number associated with the normal vector of the glide plane, and they are expressed in terms of integrals of the Berry curvature. In the present paper, we study the fate of this topological invariant when inversion symmetry is added while time-reversal symmetry is not enforced. There are two ways to add inversion symmetry, leading to space groups No.~13 and No.~14. In space group No.~13, we find that the glide-$Z_2$ invariant is expressed solely …
Macroscopic two-dimensional sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be controlled simultaneously. The bulk band gap forms an acoustic analog of topological crystalline insulators with edge states which are gapped due to symmetry reduction on the edges. In the presence of mirror symmetry, the band topology of the edge states can be characterized by the Zak phase, illustrating the band topology in a hierarchy of dimensions, which …
A Z2 topological insulator protected by time-reversal symmetry is realized via spin-orbit interaction driven band inversion. For example, the topological phase in the Bi-Sb system is due to an odd number of band inversions. A related spin-orbit system, the (Pb/Sn)Te class, has been known to contain an even number of inversions based on band theory. Here we experimentally investigate the possibility of a mirror symmetry protected topological crystalline insulator phase in the (Pb/Sn)Te class of materials which has been theoretically predicted to exist in its non-alloyed version. Our experimenta…
We study topological states of honeycomb photonic crystals in absence of inversion symmetry using plane wave expansion and finite element methods. The breaking of inversion symmetry in honeycomb lattice leads to contrasting topological valley indices, i.e., the valley-dependent Chern numbers in momentum space. We find that the topological corner states appear for 60$^\circ$ degree corners, but absent for other corners, which can be understood as the sign flip of valley Chern number at the corner. Our results provide an experimentally feasible platform for exploring valley-dependent higher-orde…
We propose a two-dimensional non-Hermitian Chern insulator with inversion symmetry, which is anisotropic and has staggered gain and loss in both x and y directions. In this system, conventional bulk-boundary correspondence holds. The Chern number is a topological invariant that accurately predicts the topological phase transition and the existence of helical edge states in the topologically nontrivial gapped phase. In the gapless phase, the band touching points are isolated and protected by the symmetry. The degenerate points alter the system topology, and the exceptional points can destroy th…
The interface between two materials described by spectrally gapped Hamiltonians is expected to host an in-gap interface mode, whenever a certain topological invariant changes across the interface. We provide a precise statement of this bulk-interface correspondence, and its rigorous justification. The correspondence applies to continuum and lattice models of interfaces between one-dimensional materials with inversion symmetry, with dislocation models being of particular interest. For continuum models, the analysis of the parity of the "edge" Bloch modes is the key component in our argument, wh…
We demonstrate via three examples that topological insulators (TI) offer a new platform for plasmonics. First, we show that the collective excitations of a thin slab of a TI display spin-charge separation. This gives rise to purely charge-like optical and purely spin-like acoustic plasmons, respectively. Second, we argue that the depletion layer mixes Dirac and Schrödinger electrons which can lead to novel features such as high modulation depths and interband plasmons. The analysis is based on an extension of the usual formula for optical plasmons that depends on the slab width and on the diel…
We study a topological phase transition between a normal insulator and a quantum spin Hall insulator in two-dimensional (2D) systems with time-reversal and two-fold rotation symmetries. Contrary to the case of ordinary time-reversal invariant systems where a direct transition between two insulators is generally predicted, we find that the topological phase transition in systems with an additional two-fold rotation symmetry is mediated by an emergent stable two-dimensional Weyl semimetal phase between two insulators. Here the central role is played by the so-called space-time inversion symmetry…
Valley pseudospin, the quantum degree of freedom characterizing the degenerate valleys in energy bands, is a distinct feature of two-dimensional Dirac materials. Similar to spin, the valley pseudospin is spanned by a time reversal pair of states, though the two valley pseudospin states transform to each other under spatial inversion. The breaking of inversion symmetry induces various valley-contrasted physical properties; for instance, valley-dependent topological transport is of both scientific and technological interests. Bilayer graphene (BLG) is a unique system whose intrinsic inversion sy…
The recently discovered three dimensional or bulk topological insulators are expected to exhibit exotic quantum phenomena. It is believed that a trivial insulator can be twisted into a topological state by modulating the spin-orbit interaction or the crystal lattice via odd number of band inversions, driving the system through a topological quantum phase transition. By directly measuring the topological invariants (for the method to directly measure Fu-Kane {$ν_0$}, see Hsieh \textit{et.al.,} Science 323, 919 (2009) at http://www.sciencemag.org/content/323/5916/919.abstract) we report the obse…
We examine different topological phases in three-dimensional non-centrosymmetric superconductors with time-reversal symmetry by using three different types of topological invariants. Due to the bulk boundary correspondence, a non-zero value of any of these topological numbers indicates the appearance of zero-energy Andreev surface states. We find that some of these boundary modes in nodal superconducting phases are dispersionless, i.e., they form a topologically protected flat band. The region where the zero-energy flat band appears in the surface Brillouin zone is determined by the projection…
As reflection symmetry or space-time inversion symmetry is preserved, with a non-contractible integral loop respecting the symmetry in the Brilliouin zone, Berry phase is quantized in proper basis. Topological nodal lines can be enclosed in the integral loop and $π$-Berry phase topologically protects the nodal lines. In this work, we show that to have quantized Berry phase restricted by the symmetry in any crystal structure, we choose to use the cell-periodic convention and define the origin point in the real space at one of the reflection (inversion) centers. In addition, $π$-Berry phase is n…
In a topological crystalline insulator (TCI) the inversion of the bulk valence and conduction bands is a necessary condition to observe surface metallic states. Solid solutions of Pb$_{1-x}$Sn$_x$Te have been shown to be TCI, where band inversion occurs as a result of the band gap evolution upon alloying with Sn. The origins of this band inversion remain unclear. Herein the role of Sn insertion into the PbTe matrix is investigated for the $p$-type Pb$_{1-x}$Sn$_x$Te series with $x$ = 0, 0.35, 0.60, and 1.00 via nuclear magnetic resonance (NMR) and transport measurements. $^{207}$Pb, $^{119}$Sn…
A two-dimensional topological crystalline insulator (TCI) with a single unit cell (u.c.) thickness is demonstrated here. To that end, one first shows that tetragonal ($C_4$ in-plane) symmetry is not a necessary condition for the creation of zero-energy metallic surface states on TCI slabs of finite-thicknesses, because zero-energy states persist even as all the in-plane rotational symmetries--furnishing topological protection--are completely removed. In other words, zero-energy levels on the model are not due to (nor are they protected by) topology. Furthermore, effective twofold energy degene…
We study a special class of topological phase transitions in two dimensions described by the inversion of bands with relative angular momentum higher than 1. A band inversion of this kind, which is protected by rotation symmetry, separates the trivial insulator from a Chern insulating phase with higher Chern number, and thus generalizes the quantum Hall transition described by a Dirac fermion. Higher angular momentum band inversions are of special interest, as the non-vanishing density of states at the transition can give rise to interesting many-body effects. Here we introduce a series of min…
Topological flat bands emerging at the inversion of stacking order in rhombohedral graphite
Comisión Nacional de Energía Atómica (CNEA)（国家原子能委员会）；Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)（国家科学研究委员会）；Universidad Nacional de San Martín（圣马丁国立大学）；Instituto de Nanociencia y Nanotecnología, CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro（纳米科学与纳米技术研究所，CNEA-CONICET，GAIDI，巴里洛切原子中心及巴尔塞罗研究所）；Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，格勒诺布尔理工学院，内耳研究所）
