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Topological Insulators with Inversion Symmetry
Abstract
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 edge) states. In the 2D quantum spin Hall phase and the 3D strong topological insulator these states are robust and are insensitive to weak disorder and interactions. In this paper we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the Z_2 invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wavefunctions at the time reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials are strong topological insulators, including the semiconducting alloy Bi_{1-x} Sb_x as well as \alpha-Sn and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments. Comment: 16 pages, 7 figures; published version
... Magnetic doping in a topological insulator (TI) with Diractype linear topological surface state (TSS) dispersions breaks time-reversal symmetry and can lead to ferromagnetic (FM) ordering [1][2][3][4][5]. In a three-dimensional (3D) TI, the onset of FM order leads to the formation of a T -dependent magnetic gap at the Dirac point (DP) [4]. ...
... Thus, in the present case, the lowest unoccupied states are confirmed to be Te 5p states. Another important piece of evidence of Te 5p character in the unoccupied states is the presence of Dirac-type linear bands associated with band inversion in TIs [1][2][3][7][8][9]. Hence, we measured ARPES with different photon energies to investigate bulk-and surface-sensitive band dispersions. ...
... The calculated Sb (green) and Te (purple) band dispersions of (Cr 0.33 Sb 0.67 ) 2 Te 3 shown in Fig. 4(c) indicate that the band inversion, which is a necessary condition to obtain TSS [1][2][3], survives after Cr doping at and near the point, although with small relative changes in the Sb 5p and Te 5p bands compared to LDA results of Sb 2 Te 3 shown in Fig. 4(b). Upon Cr substitution, the calculations show that the valence-band states have moved up towards E F compared to Sb 2 Te 3 and they cross the E F , consistent with the effective hole doping due to Cr substitution and a metallic ground state, as seen in electrical resistivity [29]. ...
We study the high Curie temperature ferromagnet (Cr0.35Sb0.65)2Te3 (TC=192K), using T-dependent x-ray absorption spectroscopy (XAS), x-ray magnetic circular dichroism (XMCD), and angle-resolved photoemission spectroscopy (ARPES). The T-dependent (25–220 K) XAS-XMCD evolution of Cr3d and Te5p unoccupied site- and orbital-projected states shows a systematic modification, which we interpret as due to spin-splitting below TC. The T-dependent XMCD intensity and leading-edge spin-sensitive shifts γexpt(T) follow bulk magnetization. ARPES measurements with hν=78eV show a metallic state with Sb 5p band dispersions at and near Fermi level (EF), consistent with bulk band-structure calculations for (Cr0.33Sb0.67)2Te3. However, surface-sensitive ARPES with hν=8.4eV above and below TC show linear band dispersions just below EF, suggesting a remnant of Dirac-type dispersions. Assuming the linear dispersion survives above EF, it implies a topologically trivial ferromagnet as the estimated Dirac point energy lies above the largest γexpt(T=25K). The Cr3d XAS-XMCD spectra can be simulated by charge transfer multiplet cluster model calculations with an exchange field Hex which quantitatively reproduces the experimental XMCD. At T=25K, the required exchange field Hex of ∼48T corresponds to a Zeeman energy ζ=2.8meV<TC=192K (= 16.5 meV) ≪γexpt∼140meV. The results indicate the role of Cr3d exchange interactions in causing spin-sensitive shifts in Cr3d states, and inducing comparable spin-sensitive shifts via hybridization in Te5p states of (Cr0.35Sb0.65)2Te3.
... These authors contributed equally. ‡ rjs269@cam.ac.uk gap phases in principle go beyond conventional single gap topological phases [12][13][14], which can be classified by comparing how irreducible band representations glue together over the Brillouin zone (BZ) [15][16][17][18] and comparing their real space Wannier description [19,20], as they are in general not symmetry indicated [11]. Notably, multi-gap invariants, such as the Euler class χ, and the corresponding braiding of band degenaracies in two-dimensional systems have been related to a variety of physical systems and phenomena, including outof-equilibrium quenches and Floquet systems [21][22][23], phonon modes [24,25], magnetic systems [26,27], and implementations in metamaterials [28][29][30][31]. ...
... The topological invariants of this phase may again be calculated using Eqs. (12), (15), (16), and (17). However, it is important to recall Eq. (7), which indicates that the presence of non-trivial weak Euler invariants leads to a reduction in the range of values which the Hopf invariant H can take. ...
We discuss a class of three-band non-Abelian topological insulators in three dimensions which carry a single bulk Hopf index protected by spatiotemporal ($\mathcal{PT}$) inversion symmetry. These phases may also host subdimensional topological invariants given by the Euler characteristic class, resulting in real Hopf-Euler insulators. Such systems naturally realize helical nodal structures in the 3D Brillouin zone, providing a physical manifestation of the linking number described by the Hopf invariant. We show that, by opening a gap between the valence bands of these systems, one finds a fully-gapped `flag' phase, which displays a three-band multi-gap Pontryagin invariant. Unlike the previously reported $\mathcal{PT}$-symmetric four-band real Hopf insulator, which hosts a $\mathbb{Z} \oplus \mathbb{Z}$ invariant, these phases are not unitarily equivalent to two copies of a complex two-band Hopf insulator. We show that these uncharted phases can be obtained through dimensional extension of two-dimensional Euler insulators, and that they support (1) an optical bulk integrated circular shift effect quantized by the Hopf invariant, (2) quantum-geometric breathing in the real space Wannier functions, and (3) surface Euler topology on boundaries. Consequently, our findings pave a way for novel experimental realizations of real-space quantum-geometry, as these systems may be directly simulated by utilizing synthethic dimensions in metamaterials or ultracold atoms.
... Since the theoretical proposal and experimental observation of 2D [1][2][3][4][5][6] and 3D [7][8][9][10][11][12] topological insulators (TIs), these materials have been the subject of substantial research in condensed matter physics and related areas of science [11,[13][14][15][16][17]. Further investigations soon resulted in the discovery of another important class of topological materials named topological crystalline insulators (TCIs) [18][19][20][21][22]. ...
... Both TIs and TCIs host mid-gap boundary states related to a bulk topological invariant via the bulk-boundary correspondence [23,24]. The TI phase can be characterized by a non-trivial spin Chern number (in 2D) [25][26][27][28] or Z 2 invariant [1,3,8], whereas the characterization of a TCI phase depends on the particular material and involves at least one crystal symmetry, e.g., inversion, rotation, or the mirror operation (it may or may not involve TRS). Well-known examples are TCIs characterized by the mirror Chern number [20,[29][30][31] and the Z 2 invariant defined in terms of the product of time reversal with a C 4 rotation [18]. ...
Dual topological insulators (DTIs) are simultaneously protected by time-reversal and crystal symmetries, representing advantageous alternatives to conventional topological insulators. By combining ab initio calculations and the $\mathbf{k}\cdot\mathbf{p}$ approach, here we investigate the electronic band structure of a Na$_2$CdSn tri-atomic layer and derive a low-energy $4\times 4$ effective model consistent with all the symmetries of this material class. We obtain the effective Hamiltonian using the L\"owdin perturbation theory, the folding down technique, and the theory of invariants, and determine its parameters by fitting our analytical dispersion relations to those of ab initio calculations. We then calculate the bulk topological invariants of the system and show that the Na$_2$CdSn tri-atomic layer is a giant-gap (hundreds of meV) quasi-2D DTI characterized by both spin and mirror Chern numbers $-2$. In agreement with the bulk-boundary correspondence theorem, we find that a finite-width strip of Na$_2$CdSn possesses two pairs of counter-propagating helical edge states per interface. We obtain analytical expressions for the edge states energy dispersions and wave functions, which are shown to agree with our numerical calculations. Our work opens a new avenue for further studies of Na$_2$CdSn as a potential DTI candidate with room-temperature applications in areas of technological interest, such as nanoelectronics and spintronics.
... Top & Bottom Anti-Symmetric Along with the special 3 + 1 space-time dimension, the Maxwell electrodynamics is allowed to be decorated with an extra θ term, which generates axion electrodynamics [115,116] to the space-time dependent θ axion field that couples with the ordinary electromagnetic field. On a practical level, based on the picture of surface Hall effect [64,117] and analogical mathematical structure between Hall current and magnetization current, people generalize and propose the topological field theory [50], where a θ term is introduced to describe the magnetoelectric effect [110][111][112][118][119][120][121][122][123] in a topological insulator medium, where the axion field is forced to gain a magnitude of π [124] by symmetry and topological requirement. ...
... The solid blue line of m1(0) represents results from solving the self-consistent equations Eq.(20), while the circles are obtained from diagonalizing the TI film Hamiltonian at k = 0 directly. The Z2 index is calculated from inversion symmetry indicator[64] method, and the solid red line represents index of n = 1 block Dirac fermions with solved m1(k), while circles are indexes calculated from TI film Hamiltonian directly. ...
We develop a Dirac fermion theory for topological phases in magnetic topological insulator films. The theory is based on exact solutions of the energies and the wave functions for an effective model of the three-dimensional topological insulator (TI) film. It is found that the TI film consists of a pair of massless or massive Dirac fermions for the surface states, and a series of massive Dirac fermions for the bulk states. The massive Dirac fermion always carries zero or integer quantum Hall conductance when the valence band is fully occupied while the massless Dirac fermion carries a one-half quantum Hall conductance when the chemical potential is located around the Dirac point for a finite range. The magnetic exchange interaction in the magnetic layers in the film can be used to manipulate either the masses or chirality of the Dirac fermions and gives rise to distinct topological phases, which cover the known topological insulating phases, such as quantum anomalous Hall effect, quantum spin Hall effect and axion effect, and also the novel topological metallic phases, such as half quantized Hall effect, half quantum mirror Hall effect, and metallic quantum anomalous Hall effect.
... At the boundary between a topological insulator (TI) and a trivial material, conducting surface states appear [1], surrounding the insulating bulk. These surface states are of interest due to their potential for dissipation-less and spinpolarized transport [1][2][3]. Additionally, these states are topologically protected, making them robust against disorder [2]. Topological crystalline insulators (TCIs) are a class of TIs, where the topology arises from the crystal mirror symmetries [4]. ...
... These surface states are of interest due to their potential for dissipation-less and spinpolarized transport [1][2][3]. Additionally, these states are topologically protected, making them robust against disorder [2]. Topological crystalline insulators (TCIs) are a class of TIs, where the topology arises from the crystal mirror symmetries [4]. ...
Topological crystalline insulators (TCIs) are interesting for their topological surface states, which hold great promise for scattering-free transport channels and fault-tolerant quantum computing. A promising TCI is SnTe. However, Sn-vacancies form in SnTe, causing a high hole density, hindering topological transport from the surface being measured. This issue could be relieved by using nanowires with a high surface-to-volume ratio. Furthermore, SnTe can be alloyed with Pb reducing the Sn-vacancies while maintaining its topological phase. Here we present the catalyst-free growth of monocrystalline PbSnTe in molecular beam epitaxy. By the addition of a pre-deposition stage before the growth, we have control over the nucleation phase and thereby increase the nanowire yield. This facilitates tuning the nanowire aspect ratio by a factor of four by varying the growth parameters. These results allow us to grow specific morphologies for future transport experiments to probe the topological surface states in a Pb1–x Sn x Te-based platform.
... topological invariant under the ten-fold classification scheme [29]. Importantly, it was shown that a direct connection can be made between the Z 2 index [41,42] and the WCC spectra in [20]. We briefly summarize that in a time-reversal symmetric system each Kramers pair is composed of two eigenstates which admit equal and opposite Chern numbers, ...
... The results in figure 2 demonstrate the resulting fully connected WCC spectra indicating a non-trivial Z 2 index. This formalism can be easily extended to threedimensional systems as computation of the weak and strong Z 2 indices is accomplished via computation of the Z 2 index in each high-symmetry plane [41]. ...
Multiple software packages currently exist for the computation of bulk topological invariants in both idealized tight-binding models and realistic Wannier tight-binding models derived from density functional theory. Currently, only one package is capable of computing nested Wilson loops and spin-resolved Wilson loops. These state-of-the-art techniques are vital for accurate analysis of band topology. In this paper we introduce BerryEasy, a python package harnessing the speed of graphical processing units to allow for efficient topological analysis of supercells in the presence of disorder and impurities. Moreover, the BerryEasy package has built-in functionality to accommodate use of realistic many-band tight-binding models derived from first-principles.
... [7,27,44]. Since the considered system possesses time-reversal symmetry, Chern numbers of upper and lower bands vanish, and the relevant invariants are either spin Chern numbers C ↑ and C ↓ calculated separately for spin-up and spin-down states in the respective band or Z 2 invariant [45]. However, a direct evaluation of Berry curvature for eigenfunctions Eq. (3) shows that it zeroes out, in consistency with the evaluation of Z 2 invariant for the Hamiltonian Eq. (1) with the help of Z2Pack [46] which yields Z 2 = 0. Thus, the considered Hamiltonian is topologically trivial. ...
We demonstrate that a bianisotropic response associated with a broken mirror symmetry of a dielectric resonator allows opening the bandgap in simple square lattice arrays of such resonators. Realizing the proposed system as an array of high-index ceramic resonators working at GHz frequencies, we numerically and experimentally demonstrate the presence of edge states at the interface between two domains with opposite orientations of the bianisotropic resonators as well as at the boundary between a single domain and free space. For both cases, we characterize the dispersion of edge states, examine their propagation along sharp bends, their resilience towards various types of geometrical defects, and a spin-momentum locked unidirectional propagation in the case of circularly polarized excitation. The considered design opens novel possibilities in constructing optical and microwave structures simultaneously featuring edge states at the interfaces between distinct resonator domains or a resonator domain and free space.
... In this intriguing landscape, inversion symmetry emerges as a savior of symmetry-based topological protection [63][64][65]. Even in NH systems, where the preservation of Hermiticity is not guaranteed, inversion symmetry plays a crucial role [66][67][68]. ...
We theoretically investigate topological features of a one-dimensional Su-Schrieffer-Heeger lattice with modulating non-Hermitian on-site potentials containing four sublattices per unit cell. The lattice can be either commensurate or incommensurate. In the former case, the entire lattice can be mapped by supercells completely. While in the latter case, there are two extra lattice points, thereby making the last cell incomplete. We find that an anti-PT transition occurs at exceptional points of edge states at certain parameters, which does not coincide with the conventional topological phase transition characterized by the Berry phase, provided the imaginary on-site potential is large enough. Interestingly, when the potential exceeds a critical value, edge states appear even in the regime with a trivial Berry phase. To characterize these novel edge states we present topological invariants associated with the system's parity. Finally, we analyze the dynamics for initial states with different spatial distributions, which exhibit distinct dynamics for the commensurate and incommensurate cases, depending on the imaginary part of edge state energy.
... is the weak topological indices defined on the high symmetry planes at the Brillouin zone's boundary 97 and G i is the reciprocal lattice vectors. M ν acts like a time-reversal invariant momentum. ...
When a two-dimensional $d$-wave altermagnet is grown on a substrate, the interplay of momentum-dependent spin splittings arising from altermagnetism and Rashba spin-orbit coupling gives rise to a nodal band structure with band degeneracies enforced by a $C_{4z}\mathcal{T}$ symmetry. If we break the $C_{4z}\mathcal{T}$ symmetry by an exchange field, the band degeneracies are found to be immediately lifted, leading to a topological band structure characterized by nontrivial strong and weak topological indices. Remarkably, both the strong topological index and the $Z_{2}$-valued weak topological indices depend sensitively on the direction of the exchange field. As a consequence of the bulk-defect correspondence, we find that the unique dependence of weak topological indices on the exchange field in this system dictates that the presence or absence of topological bound states at lattice dislocations also depends sensitively on the direction of the exchange field. When the substrate is an $s$-wave superconductor, we find that a similar dependence of band topology on the exchange field gives rise to field-sensitive dislocation Majorana zero modes. As topological dislocation bound states are easily detectable by scanning tunneling microscopy, our findings unveil a promising experimental diagnosis of altermagnetic materials among an ever growing list of candidates.
... Introduction. -Topological Insulators (TIs) are materials that behave as gapped insulators in bulk whereas also hosting metallic (gapless) topological helical states localized at their edges in 2D TIs [1][2][3][4][5][6][7][8][9][10] or surfaces in 3D TIs [11][12][13][14][15][16]. For that reason, attention to topological materials has been mainly focused on edge-and surfacelike phenomena. ...
We investigate the Shubnikov-de Haas (SdH) magneto-oscillations in the resistivity of two-dimensional topological insulators (TIs). Within the Bernevig-Hughes-Zhang (BHZ) model for TIs in the presence of a quantizing magnetic field, we obtain analytical expressions for the SdH oscillations by combining a semiclassical approach for the resistivity and a trace formula for the density of states. We show that when the non-trivial topology is produced by inverted bands with ''Mexican-hat'' shape, SdH oscillations show an anomalous beating pattern that is {\it solely} due to the non-trivial topology of the system. These beatings are robust against, and distinct from beatings originating from spin-orbit interactions. This provides a direct way to experimentally probe the non-trivial topology of 2D TIs entirely from a bulk measurement. Furthermore, the Fourier transform of the SdH oscillations as a function of the Fermi energy and quantum capacitance models allows for extracting both the topological gap and gap at zero momentum.
... Due to the conservation of TR-symmetry, the opposite edge states of the 2D nontrivial TIs possess opposite spin chirality and are charge neutral. [128][129][130][131] BiTeI well exemplifies this phenomenon. Under normal conditions, BiTeI exhibits a substantial Rashba spin splitting of 3.85 eV Å. ...
The Rashba spin-orbit coupling effect, primarily arising from structural-inversion asymmetry in periodic crystals, has garnered considerable attention due to its tunability and potential applications in spintronics. Its capability to manipulate electron spin without an external magnetic field opens new avenues for spintronic device design, particularly in semiconductor technology. Within this framework, 2D Rashba materials hold special interest due to their inherent characteristics, which facilitate miniaturization and engineering capabilities. In this Perspective article, we provide an overview of recent advancements in the research of 2D Rashba materials, aiming to offer a comprehensive understanding of the diverse manifestations and multifaceted implications of the Rashba effect in material science. Rather than merely presenting a list of materials, our approach involves synthesizing various viewpoints, assessing current trends, and addressing challenges within the field. Our objective is to bridge the gap between fundamental research and practical applications by correlating each material with the necessary advancements required to translate theoretical concepts into tangible technologies. Furthermore, we highlight promising avenues for future research and development, drawing from insights gleaned from the current state of the field.
... Following the pioneering works in [29], crystals with FTR-symmetric Hamiltonians have been analyzed in great detail in many works; see, e.g. [10,23,26,37], with techniques allowing us to compute the class of a bulk Hamiltonian from its spectral decomposition on a Brillouin zone. The FTR-symmetry is one among a large class of possible symmetries [14,30]. ...
This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.
... Since I symmetry is conserved, the hidden polarization can be captured by the parity eigenvalues of the highsymmetry invariant points for occupied bands using Eq. (S19) [75,76]. The corresponding parity eigenvalues for the two occupied bands are given in Table S1 in the SM [70]. ...
The one-dimensional Su-Schrieffer-Heeger (SSH) model is central to band topology in condensed matter physics, which allows us to understand and design distinct topological states. In this work we find another mechanism to analogize the SSH model in a spinful system, realizing an obstructed atomic insulator by introducing intrinsic spin-orbit coupling and in-plane Zeeman field. In our model the midgap states originate from a quantized hidden polarization with invariant index Z2 (0; 01) due to the local inversion symmetry breaking. When the global inversion symmetry is broken, a charge pumping is designed by tuning the polarization. Moreover, by introducing the p+ip superconductor pairing potential, a topological phase dubbed obstructed superconductor (OSC) is identified. This new state is characterized by invariant index Z2 (0; 01) and nonchiral midgap states. More interestingly, these nonchiral edge states result in a chiral-like nonlocal conductance, which is different from the traditional chiral topological superconductor. Our findings not only find another strategy to achieve a spinful SSH model but also predict the existence of OSC, providing a promising avenue for further exploration of its transport properties.
... Kane-Mele predicted the quantum spin Hall insulator [two-dimensional (2D) TIs] in 2005, which is the spin version of quantum Hall effect [46,47], and this interesting phase was confirmed in HgTe quantum well later [48][49][50]. Subsequently, the 2D TIs were generalized to three-dimensional (3D) cases [51][52][53][54], which are divided into strong and weak TIs, hosting metallic surface states but with an odd and even number of Dirac cones, respectively. For the free-fermion systems, one can classify the topological phases by their symmetries, i.e., time-reversal, particle-hole, and chiral symmetries, called class [17,55]. ...
Recently, topolectrical circuits (TECs) boom in studying the topological states of matter. The resemblance between circuit Laplacians and tight-binding models in condensed matter physics allows for the exploration of exotic topological phases on the circuit platform. In this review, we begin by presenting the basic equations for the circuit elements and units, along with the fundamentals and experimental methods for TECs. Subsequently, we retrospect the main literature in this field, encompassing the circuit realization of (higher-order) topological insulators and semimetals. Due to the abundant electrical elements and flexible connections, many unconventional topological states like the non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states that are challenging to observe in conventional condensed matter physics, have been observed in circuits and summarized in this review. Furthermore, we show the capability of electrical circuits for exploring the physical phenomena in other systems, such as photonic and magnetic ones. Importantly, we highlight TEC systems are convenient for manufacture and miniaturization because of their compatibility with the traditional integrated circuits. Finally, we prospect the future directions in this exciting field, and connect the emerging TECs with the development of topology physics, (meta)material designs, and device applications.
... Although thousands of topological insulators have been identified to date [7][8][9], experimental studies have focused primarily on materials based on Bi, Hg, and Te. Strained HgTe, in particular, which is a strong topological insulator [10] is characterized by its high carrier mobility that allows the study of quantum effects like the quantum Hall effect in moderate magnetic fields [11][12][13][14]. TIs such as HgTe accommodate different types of carrier species, depending on the gate voltage applied. ...
High mobility two-dimensional systems with superposed 1D lateral periodic potentials exhibit characteristic commensurability (Weiss) oscillations that reflect the interplay of the cyclotron radius at the Fermi level and the superlattice period. Here, we impose a one-dimensional periodic potential on strained HgTe, which is a strong 3D topological insulator. By tuning the Fermi level with top gates, the effects of the artificial potential can be studied in the bulk gap, where only Dirac surface states exist, in the conduction band, and in the valence band, where Dirac electrons and holes coexist. On the electron side, we observe clear commensurability oscillations whose period is governed by the carrier density of the top-surface Dirac electrons. Unexpectedly, weak commensurability oscillations are also observed in the valence band with a period that depends on both electron and hole density.
Published by the American Physical Society 2024
... The discovery of topological states of matter is one of the central advances in condensed-matter physics (and beyond) in the last half century [1][2][3][4][5][6][7][8][9][10][11]. In recent years, two major efforts have been to broaden and deepen the scope of this concept. ...
The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low-temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions and a series of fragile topological CSLs connected by zero-temperature transitions.
Published by the American Physical Society 2024
We propose helical topological superconductivity away from the Fermi surface in three-dimensional time-reversal-symmetric odd-parity multiband superconductors. In these systems, pairing between electrons originating from different bands is responsible for the corresponding topological phase transition. Consequently, a pair of helical topological Dirac surface states emerges at finite excitation energies. These helical Dirac surface states are tunable in energy by chemical potential and strength of band splitting. They are protected by time-reversal symmetry combined with crystalline twofold rotation symmetry. We suggest concrete materials in which this phenomenon could be observed.
We present comprehensive investigations into the structural, superconducting, and topological properties of Bi2PdPt. Magnetization and heat-capacity measurements performed on polycrystalline Bi2PdPt demonstrate a superconducting transition at ≃ 0.8 K. Moreover, muon spin relaxation/rotation (µSR) measurements present evidence for a time-reversal symmetry preserving, isotropically gapped superconducting state in Bi2PdPt. We have also performed density-functional theory (DFT) calculations on Bi2PdPt alongside the more general isostructural systems BiPdxPt1−x, of which Bi2PdPt and γ-BiPd are special cases for x=0.5 and x=1, respectively. We have calculated the Z2 topological index from our DFT calculations for a range of substitution fractions x, between x=0 and x=1, characterizing the topology of the band structure. We find a nontrivial topological state when x>0.75 and a trivial topological state when x<0.75. Therefore our results indicate that BiPdxPt1−x could be a topological superconductor for x>0.75.
Topological semimetals possess nodal or nodal-line phases where conduction and valence bands touch at points or lines in momentum space, respectively. Such band touching is symmetry protected and gives rise to exotic and interesting electronic properties. Coupling topological order with magnetism provides a platform for exploring time-reversal (TR) symmetry breaking topological physics, such as axion electrodynamics, inverse spin-galvanic effect, and the quantized anomalous Hall effect. The Weyl semimetal (WSM) requires breaking either TR symmetry or lattice inversion symmetry (I). By doping inversion-symmetry-broken WSM with magnetic dopants, one can expect to create a WSM with both symmetries breaking simultaneously. Here, structural, electrical, and magnetic properties of FexW1−xTe2 (x=0 and 0.011) are reported. It is revealed that, with a small Fe doping concentration (x=0.011), a ferromagnetism is induced at low temperature (<10 K). Scanning tunneling microscopy and spectroscopy measurements in Fe0.011W0.989Te2 further reveal only substitution and no intercalated dopants being observed. The probabilities of the Fe substitutions at the two nonequivalent W sites are quantified with equal probability. The dI/dV point spectra indicates that the Fe substitution in WTe2 manifests itself as electron doping regardless of doping sites. The results clearly reveal the possible coexistence of magnetism and Weyl points in the lightly Fe doped WTe2 at low temperature. This provides an ideal system for further study on the interplay between the topological Weyl points and the TR symmetry breaking.
Antiferromagnetic (AFM) semiconductor MnS2 possesses both high-spin and low-spin magnetic phases that can be reversibly switched by applying pressure. With increasing pressure, the high-spin state undergoes pressure-induced metallization before transforming into a low-spin configuration, which is then closely followed by a volume collapse and structural transition. We show that the pressure-driven band inversion is in fact topological, resulting in an antiferromagnetic Z2 topological metal (Z2AFTM) phase with a small gap and a Weyl metal phase at higher pressures, both of which precede the spin-state crossover and volume collapse. In the Z2AFTM phase, the magnetic order results in a doubling of the periodic unit cell, and the resulting folding of the Brillouin zone leads to a Z2 topological invariant protected by the persisting combined time-reversal and half-translation symmetries. Such a topological phase was proposed theoretically by Mong et al. [Phys. Rev. B 81, 245209 (2010)] in 2010 for a system with AFM order on a fcc lattice, which until now has not been found in the pool of real materials. MnS2 represents a realization of this original proposal through AFM order on the Mn fcc sublattice. A rich phase diagram of topological and magnetic phases tunable by pressure establishes MnS2 as a candidate material for exploring magnetic topological phase transitions and for potential applications in AFM spintronics.
Manipulating topological states is a topic of vigorous research. However, the impact of such topological phase transitions on the quasiparticle dynamics remains elusive. In this work, we present the effects of a transition from a strong to weak topological insulator in HfTe5 as a function of Te vacancy concentration. We observed a significant transition from distinct sharp surface states and Dirac crossing to a Fermi-liquid-like quasiparticle state in which these surface-localized features are heavily suppressed. Additionally, by inducing the same effect through applied uniaxial stress, we demonstrate that changes in the lattice constants play the foremost role in determining the electronic structure, self-energy, and topological states of HfTe5. Our results demonstrate the possibility of using both defect chemistry and strain as control parameters for topological phase transitions and associated many-body physics.
We investigate the electronic structure and topological properties of iron-based superconductors LaFe2As2 using density functional theory plus dynamical mean-field theory. We find that the uncollapsed tetragonal LaFe2As2 is in a nontrivial Z2 topological phase and has topological Dirac surface states near the Fermi energy which suggests there could be Majorana zero modes in the superconducting LaFe2As2. In light of the nontrivial topological properties and superconductivity of LaFe2As2 and CaKFe4As4, we predict a new iron-based compound LaBaFe4As4 and find it possesses two sets of topological Dirac surface states near the Fermi energy despite of a trivial Z2 topological index. These topological surface states are induced by a nontrivial high-order topological index Z8, a new mechanism that is distinct from all-known iron-based superconductors. Our study not only demonstrates that both LaBaFe4As4 and uncollapsed tetragonal LaFe2As2 can be good platforms for exploring topological superconductivity but also paves a new way to realize it with a nontrivial high-order topological index.
While the possibility of topological superconductivity (TSC) in hybrid heterostructures involving topologically nontrivial band structure and superconductors has been proposed, the realization of TSC in a single stoichiometric material is most desired for fundamental experimental investigation of TSC and its device applications. Bulk measurements on YRuB2 detect a single superconducting gap of ∼1 meV. This is supported by our electronic structure calculations, which also reveal the existence of topological surface states in the system. We performed surface-sensitive Andreev reflection spectroscopy on YRuB2 and detected the bulk superconducting gap as well as another superconducting gap of ∼0.5 meV. From our analysis of electronic structure, we show that the smaller gap is formed in the topological surface states in YRuB2 due to the proximity of the bulk superconducting condensate. Thus, in agreement with the past theoretical predictions, we present YRuB2 as a unique system that hosts superconducting topological surface states.
Since the advent of time-reversal-invariant topological insulators, the generalization of topology concepts has led to the discovery of a wide range of topological states. These topological states may undergo topological phase transitions (TPTs) accompanied by discontinuous changes in topological invariants. In this Letter, by combining time- and angle-resolved photoemission spectroscopy measurements with first-principles calculations, we have demonstrated the existence of distinct TPTs in the unoccupied states above the Fermi level in the M2Te2P (M=Ti, Zr, or Hf) family. These TPTs arise from the inversion of the transition metal d and Te p orbitals due to the enhancement of spin-orbit coupling and variation in crystal fields, as the transition metal changes from 3d to 4d and then to 5d. Eventually, dual topological states are achieved in the 5d system Hf2Te2P, where a type-III Dirac semimetal state coexists with a strong topological insulator state. Finally, we reveal a rapid relaxation process in the nonequilibrium population of the Dirac surface state due to the presence of an additional interband scattering channel in the bulk.
In the context of θ electrodynamics we find transverse electromagnetic wave solutions forbidden in Maxwell electrodynamics. Our results attest to different signatures of the topological magnetoelectric effect in topological insulators, resulting from a polarization rotation of an external electromagnetic field. Unlike Faraday and Kerr rotations, the effect does not rely on a longitudinal magnetic field, the reflected field, or birefringence. The rotation occurs due to transversal discontinuities of the topological magnetoelectric parameter in cylindrical geometries. The dispersion relation is linear, and birefringence is absent. Exact transversality allows electromagnetic waves to propagate in an optical fiber without successive total internal reflections, diminishing losses, and regardless of acute bends of the fiber. These results may open other possibilities in optics and photonics by utilizing topological insulators to manipulate light.
The structural, elastic, mechanical, optoelectronic, and Debye temperatures of \({\text{MAs}}_{2}\) (M = W, Cr, Mo) were explored at ambient pressure using first-principles calculations. Lattice constants and cell volume are calculated to be in good consistent with other findings. We investigated the mechanical properties of \({\text{MAs}}_{2}\) materials using elastic moduli, the machinability index, and Vickers hardness. Poisson’s and Pugh’s ratios indicate that \({\text{WAs}}_{2}\) material is ductile, whereas \({\text{CrAs}}_{2}\) and \({\text{MoAs}}_{2}\) are brittle at ambient pressure. Analyzing electronic properties offers crucial support for assessing optical performance. In the higher energy range, the refractive index value falls and flattens. Due to their high reflectivity, these phases are excellent candidates for solar heating reduction in the infrared and ultraviolet wavelength regions. The minimum thermal conductivity values for \({WAs}_{2}\), \({CrAs}_{2}\), and \(Mo{As}_{2}\) are 0.571, 0.732, and 0.666, respectively, making them attractive thermal insulators above their Debye temperatures (\({\theta }_{D}\)). Melting temperatures show that \(Cr{As}_{2}\) melts at 1389.12 °C, \({MoAs}_{2}\) at 1587.60 °C, and \({WAs}_{2}\) at 1648.86 °C. It indicates that \({WAs}_{2}\) is the most thermally stable of the three compounds, whereas \(Cr{As}_{2}\) is the least. Lastly, we anticipate that the present findings will have profound implications for future studies of various aspects of related materials.
Dynamics of submonolayer lead adsorption structures on Ni(111) surface, overlayers and surface alloy, has been studied within the embedded atom method calculations.
Compared to time-reversal symmetry-protected Z2 topological insulators and Dirac/Weyl semimetals, there are significantly fewer candidates for topological crystalline insulators. SrAg4Sb2 is predicted to exhibit topological crystalline insulator behavior when considering spin-orbit coupling. In this study, we systematically investigate single crystals of SrAg4Sb2 using electrical transport and magnetic torque measurements, along with first-principles calculations. Our transport data reveals its compensated semimetal nature with a magnetoresistance up to around 700% at 2 K and 9 T. Analysis of de Haas–van Alphen oscillations uncovers a Fermi surface consisting of three distinct Fermi pockets with light effective masses. Comparison between the three-dimensional fermiology obtained from our oscillation data and the first-principles calculations demonstrates excellent agreement. This confirms the accuracy of the calculations, which indicate a band inversion centered at the T point and identify the existence of nontrivial tube and needle hole Fermi pockets at Γ, alongside one trivial diamond electron pocket at the F point in the Brillouin zone. Furthermore, symmetry and topology analysis results in two potential sets of topological invariants, suggesting the emergence of two-dimensional gapless Dirac surface states either on the ab planes or on both the ab planes and mirror planes, protected by crystal symmetries. Therefore, SrAg4Sb2 emerges as a promising candidate topological crystalline insulator.
We theoretically and experimentally demonstrate singular topological edge states in a locally resonant metamaterial with a configuration based on inversion-symmetric extended Su-Schrieffer-Heeger chains. Such states arise from a topological gap transition from a conventional Bragg-type gap (BG) to a local resonance-induced gap (LRG), accompanied by a topological phase transition. Remarkably, nontrivial topological states can emerge in the vicinity of the singularity in the imaginary parts of the wavenumber within the bandgap, leading to highly localized modes on a scale comparable to a single subwavelength unit cell. We experimentally demonstrate distinct differences in topologically protected modes-highlighted by wave localization-between the BG and the LRG in locally resonant granular-based metamaterials. Our findings suggest the scope of topological metamaterials may be extended via their bandgap nature.
We theoretically construct a higher-order topological insulator (HOTI) on a Brillouin real projective plane enabled by momentum-space nonsymmorphic (k−NS) symmetries from synthetic gauge fields. Two anicommutative k−NS glide reflections appear in a checkerboard Z2 flux model, impose nonsymmorphic constraints on Berry curvature, and quantize bulk and Wannier-sector polarization nonlocally across different momenta. The model’s bulk exhibits an isotropic quadrupole phase diagram, where the transition appears intrinsically from bulk gap closure. The model hosts the simultaneous presence of intrinsic and extrinsic HOTI features: in a ribbon geometry where one pair of boundaries gets open, the edge termination can induce boundary-obstructed topological phase within the symmetry-protected topological phase due to the breaking of k−NS symmetry. At last, we present a concrete design for the real projective plane quadrupole insulator and show how to measure the momentum glide reflection based on acoustic resonator arrays. Our results shed light on HOTIs on deformed Brillouin manifolds via k−NS symmetries.
We theoretically investigate topological features of a one-dimensional Su-Schrieffer-Heeger lattice with modulating non-Hermitian on-site potentials containing four sublattices per unit cell. The lattice can be either commensurate or incommensurate. In the former case, the entire lattice can be mapped by supercells completely. While in the latter case, there are two extra lattice points, thereby making the last cell incomplete. We find that an anti-PT transition occurs at exceptional points of edge states at certain parameters, which does not coincide with the conventional topological phase transition characterized by the Berry phase, provided the imaginary on-site potential is large enough. Interestingly, when the potential exceeds a critical value, edge states appear even in the regime with a trivial Berry phase. To characterize these novel edge states we present topological invariants associated with the system's parity. Finally, we analyze the dynamics for initial states with different spatial distributions, which exhibit distinct dynamics for the commensurate and incommensurate cases, depending on the imaginary part of edge state energy.
The AV3Sb5 (A=K, Rb, Cs) kagome materials host an interplay of emergent phenomena including superconductivity, charge density wave states, and nontrivial electronic structure topology. The band structures of these materials exhibit a rich variety of features like Dirac crossings, saddle points associated with van Hove singularities, and flat bands prompting significant investigations into the in-plane electronic behavior. However, recent findings including the charge density wave ordering and effects due to pressure or chemical doping point to the importance of understanding interactions between kagome layers. Probing this c-axis electronic structure via experimental methods remains challenging due to limitations of the crystals and, therefore, rigorous computational approaches are necessary to study the interlayer interactions. Here we use first-principles approaches to study the electronic structure of CsV3Sb5 with emphasis on the kz dispersion. We find that the inclusion of nonlocal and dynamical many-body correlation has a substantial impact on the interlayer band structure. We present band behavior that additionally supports the integration of symmetry in accurately plotting electronic structures and influences further analysis like the calculation of topological invariants.
Resonant tunneling diodes (RTDs), which are based on double barrier quantum well structures, are typically achieved by combining different materials with varying band gap sizes. However, this approach often poses challenges such as material mismatching and dislocations. In this study, we present a novel resonant tunneling diode scheme utilizing the unique properties of topological insulator materials. Specifically, we exploit the gap opening in the band structure of the topological insulator by employing perpendicular magnetization. In this proposed RTD platform, the barrier regions are formed from a ferromagnetic topological insulator through the proximity effect. By adjusting the thickness and spacing of the ferromagnetic barriers, a well region with confined states emerges between the barrier regions. Theoretical analysis reveals that by tuning the back gate voltage, the I-V characteristics exhibit two significant behaviors: negative differential resistance (NDR) and step-like behavior for Fermi energy values of EF = −3 and EF = 3, respectively. Furthermore, we observe an increase in the peak-to-valley ratio (PVR) with higher magnetization values. Notably, the PVR reaches a value of 7.13 for a magnetization value of m = 9. Additionally, we investigate the influence of the well width and barrier thickness on the transport properties of the device.
The defects have a remarkable influence on the electronic structures and the electric transport behaviors of the matter, providing the additional means to engineering their physical properties. In this work, a comprehensive study on the effect of Br-vacancies on the electronic structures and transport behaviors in the high-order topological insulator Bi4Br4 is performed by the combined techniques of the scanning tunneling microscopy (STM), angle-resolved photoemission spectroscopy (ARPES), and physical properties measurement system along with the first-principle calculations. The STM results show the defects on the cleaved surface of a single crystal and reveal that the defects are correlated to the Br-vacancies with the support of the simulated STM images. The role of the Br-vacancies in the modulation of the band structures has been identified by ARPES spectra and the calculated energy-momentum dispersion. The relationship between the Br-vacancies and the semiconducting-like transport behaviors at low temperature has been established, implying a Mott variable ranging hopping conduction in Bi4Br4. The work not only resolves the unclear transport behaviors in this matter, but also paves a way to modulate the electric conduction path by the defects engineering.
Bismuth-antimony alloys are among the most studied topological insulators and also have very promising thermoelectric properties. In addition, in the amorphous state they exhibit superconductivity with critical temperatures in the range 6.0–6.4 K. In this work, we have prepared and studied different polycrystalline films of Bi100–xSbx (x = 0, 5, 10, 15), and we have induced, through ion beam irradiation, significant damage in their internal structure with the aim of amorphizing the material. Specifically, we have irradiated Bi ions in the 10–30 MeV range, exploiting the capabilities of a 5 MV ion beam accelerator of tandem type. We have characterized the Bi–Sb films before and after irradiation from a morphological and structural point of view and measured their electrical resistivity from room temperature to near 2 K, to evaluate the influence of the preparation method and degree of disorder. We have found that the studied Bi–Sb system always behaves as a small energy gap semiconductor that follows the empirical Meyer–Neldel rule, which correlates the conductivity prefactor with the exponential value of the energy gap.
In the present study, we have investigated the effect of magnetic Cr doping on the transport properties of a sputtered Bi2Se3 thin film on the SrTiO3 (110) substrate. The high-resolution x-ray diffraction and Raman spectroscopy measurements revealed the growth of rhombohedral Bi2Se3 thin films. Further electronic and compositional analysis was done by x-ray photoemission spectroscopy and Rutherford backscattering spectroscopy, and the x-value was estimated to be 0.18 in the Bi2−xCrxSe3 thin film. The variation in the resistivity with temperature (2–300 K) revealed the metallic nature in undoped Bi2Se3 up to 30 K and upturn resistivity below 30 K. The Cr-doped Bi2Se3 resistivity data show a traditional semiconducting nature up to 25 K and take an abrupt upturn resistivity below 25 K. The resistivity behavior of both samples was explained by adopting a model that consists of the total resistance, a combination of bulk and surface resistance in parallel. The bulk bandgap value determined by this method is obtained to be 256 meV in an undoped Bi2Se3 thin film. Magnetoconductance data of the undoped thin film revealed a weak anti-localization (WAL) effect, while the Cr-doped thin film showed a weak localization (WL) effect at low temperatures (<50 K). At low magnetic field and low temperature, a competing nature of WAL and WL effects was prominent in the Cr-doped film. A drastic increase in the electrical resistance suggests that Cr doping can significantly modify the electrical properties of Bi2Se3 thin films, which could have potential applications in futuristic devices.
Band topology of anomalous quantum Hall insulators can be precisely addressed by computing the Chern numbers of constituent nondegenerate bands, describing the presence of quantized, Abelian Berry flux through the two-dimensional Brillouin zone. Can Berry flux be captured for the SU(2) Berry connection of two-fold degenerate bands in spinful materials preserving space-inversion (P) and time-reversal (T) symmetries without detailed knowledge of underlying basis? We address this question by investigating the correspondence between a non-Abelian generalization of Stokes' theorem and the manifestly gauge-invariant eigenvalues of Wilson loops computed along in-plane contours which preserve the underlying crystalline symmetry. The importance of this correspondence is elucidated by performing natural number resolved classification of ab initio band structures of three-dimensional, Dirac materials. Our work underscores how identification of quantized Berry flux, both Abelian and non-Abelian, offers a unified framework for addressing first-order and higher-order topology of insulators and semimetals.
A unique co-existence of extremely large magnetoresistance (XMR) and topological characteristics in non-magnetic rare-earth monopnictides has stimulated intensive research on these materials. Yttrium monobismuthide (YBi) has been reported to exhibit XMR up to 105% but its Topological properties still need clarification. Here we use the hybrid density functional theory to probe the structural, electronic, and topological properties of YBi in detail. We observe that YBi is topologically trivial semimetal at ambient pressure which is in accordance with reported experimental results. The topological phase transitions i.e., trivial to non-trivial are obtained with volumetric pressure of 6.5 GPa and 3% of epitaxial strain. This topological phase transitions are well within the structural phase transition of YBi (24.5 GPa). The topological non-trivial state is characterized by band inversions among Y-d band and Bi-p band at Γ- and X-point which is further verified with the help of surface band structure along (001) plane. The Z2 topological invariants are calculated with the help of product of parities and evolution of Wannier charge centers. The occurrence of non-trivial phase in YBi with a relatively small epitaxial strain, which a thin film geometry can naturally has, make it an ideal candidate to probe inter-relationship between XMR and non-trivial topology.
Having previously been the subject of decades of semiconductor research, cadmium arsenide (Cd3As2) has now reemerged as a topological material, realizing ideal three-dimensional Dirac points at the Fermi level. These topological Dirac points lead to a number of extraordinary transport phenomena, including strong quantum oscillations, large magnetoresistance, ultrahigh mobilities, and Fermi velocities exceeding graphene. The large mobilities persist even in thin films and nanowires of Cd3As2, suggesting the involvement of topological surface states. However, computational studies of the surface states in this material are lacking, in part due to the large 80-atom unit cell. Here we present the computed Fermi-arc surface states of a Cd3As2 thin film, based on a tight-binding model derived directly from the electronic structure. We show that despite the close proximity of the Dirac points, the Fermi arcs are very long and straight, extending through nearly the entire Brillouin zone. The shape and spin properties of the Fermi arcs suppress both back- and side scattering at the surface, which we show by explicit integrals over the phase space. The introduction of a small symmetry-breaking term, expected in a strong electric field, gaps the electronic structure, creating a weak topological insulator phase that exhibits similar transport properties. Crucially, the mechanisms suppressing scattering in this material differ from those in other topological materials such as Weyl semimetals and topological insulators, suggesting a new route for engineering high-mobility devices based on Dirac semimetal surface states.
We propose to implement tunable higher-order topological states in a heterojunction consisting of a two-dimensional (2D) topological insulator and the recently discovered altermagnets, whose unique spin-polarization in both real and reciprocal space and null magnetization are in contrast to conventional ferromagnets and antiferromagnets. Based on symmetry analysis and effective edge theory, we show that the special spin splitting in altermagnets with different symmetries, such as d wave, can introduce Dirac mass terms with opposite signs on the adjacent boundaries of the topological insulator, resulting in the higher-order topological state with mass-domain-bound corner states. Moreover, by adjusting the direction of the Néel vector, we can manipulate such topological corner states by moving their positions. By first-principles calculations, taking a 2D topological insulator bismuthene with a square lattice on an altermagnet MnF2 as an example, we demonstrate the feasibility of creating and manipulating the higher-order topological states through altermagnets. Finally, we discuss the experimental implementation and detection of the tunable topological corner states, as well as the potential non-Abelian braiding of the Dirac corner fermions.
Recently, the quantum spin-Hall edge channels of two-dimensional colloidal nanocrystals of the topological insulator Bi2Se3 were observed directly. Motivated by this development, we reconsider the four-band effective model which has been traditionally employed in the past to describe thin nanosheets of this material. Derived from a three-dimensional k·p model, it physically describes the top and bottom electronic surface states at the Γ point that become gapped due to the material's small thickness. However, we find that the four-band model for the surface states alone, as derived directly from the three-dimensional theory, is inadequate for the description of thin films of a few quintuple layers and even yields an incorrect topological invariant within a significant range of thicknesses. To address this limitation we propose an eight-band model which, in addition to the surface states, also incorporates the set of bulk states closest to the Fermi level. We find that the eight-band model not only captures most of the experimental observations, but also agrees with previous first-principles calculations of the Z2 invariant in thin films of varying thickness. The band inversion around the Γ point, which endows the surfacelike bands with topology, is shown to be enabled by the presence of the additional bulklike states without requiring any reparametrization of the resulting effective Hamiltonian.
Nontrivial bulk topological invariants of quantum materials can leave their signatures on charge, thermal, and spin transports. In two dimensions, their imprints can be experimentally measured from well-developed multiterminal Hall bar arrangements. Here, we numerically compute the low temperature (T) thermal (κxy) and zero temperature spin (σxysp) Hall conductivities, and longitudinal thermal conductance (Gxxth) of various prominent two-dimensional fully gapped topological superconductors, belonging to distinct Altland-Zirnbauer symmetry classes, namely p+ip (class D), d+id (class C), and p±ip (class DIII) paired states, in mesoscopic six-terminal Hall bar setups from the scattering matrix formalism using kwant. In both clean and weak disorder limits, the time-reversal symmetry breaking p+ip and d+id pairings show half-quantized and quantized κxy [in units of κ0=π2kB2T/(3h)], respectively, while the latter one in addition accommodates a quantized σxysp [in units of σ0sp=ℏ/(8π)]. By contrast, the time-reversal invariant p±ip pairing only displays a quantized Gxxth at low T up to a moderate strength of disorder. In the strong disorder regime, all these topological responses (κxy, σxysp, and Gxxth) vanish. Possible material platforms hosting such paired states and manifesting these robust topological thermal and spin responses are discussed.
We reconsider the phase diagram of a three-dimensional Z2 topological insulator in the presence of short-ranged potential disorder, with the insight that nonperturbative rare states destabilize the noninteracting Dirac semimetal critical point separating different topological phases. Based on our numerical data on the density of states, conductivity, and wave functions, we argue that the putative Dirac semimetal line is destabilized into a diffusive metal phase of finite extent due to nonperturbative effects of rare regions. We discuss the implications of these results for past and current experiments on doped topological insulators.
We have grown Bi1-xSbx alloy thin films on
CdTe(111)B over a wide range of Sb concentrations (0<=x<=0.183)
using molecular-beam epitaxy. Temperature-dependent electrical
resistivity (ρ) and thermoelectric power (S) were studied. We have
observed several differences over the bulk system. The 3.5 and 5.1% Sb
alloys show semiconducting behavior, and the Sb concentration with
maximum band gap shifted to a lower Sb concentration from 15% in bulk to
9%. Based on a simple interpretation of the temperature-dependent
resistivity the maximum gap would be 40 meV, which is larger than that
observed in bulk alloys. In addition, we have observed that the power
factor S2/ρ peaks at a significantly higher temperature
(250 K) than previously reported for the bulk alloy (80 K). Differences
between thin film grown on CdTe(111) and bulk alloy may arise from the
effects of strain, which is supported by theoretical electronic band
calculations. These results show that BiSb films may be useful as
band-engineered materials in thermoelectric devices.
We present a topological description of the quantum spin-Hall effect (QSHE) in a two-dimensional electron system on a honeycomb lattice with both intrinsic and Rashba spin-orbit couplings. We show that the topology of the band insulator can be characterized by a 2x2 matrix of first Chern integers. The nontrivial QSHE phase is identified by the nonzero diagonal matrix elements of the Chern number matrix (CNM). A spin Chern number is derived from the CNM, which is conserved in the presence of finite disorder scattering and spin nonconserving Rashba coupling. By using the Laughlin gedanken experiment, we numerically calculate the spin polarization and spin transfer rate of the conducting edge states and determine a phase diagram for the QSHE.
I show that a lattice theory of massive interacting fermions in 2n+1 dimensions may be used to simulate the behavior of massless chiral fermions in 2n dimensions if the fermion mass has a step function shape in the extra dimension. The massless states arise as zero modes bound to the mass defect, and all doublers can be given large gauge invariant masses. The manner in which the anomalies are realized is transparent: apparent chiral anomalies in the 2n-dimensional subspace correspond to charge flow into the extra dimension.
Electrically induced electron-spin polarization near the edges of a semiconductor channel was detected and imaged with the
use of Kerr rotation microscopy. The polarization is out-of-plane and has opposite sign for the two edges, consistent with
the predictions of the spin Hall effect. Measurements of unstrained gallium arsenide and strained indium gallium arsenide
samples reveal that strain modifies spin accumulation at zero magnetic field. A weak dependence on crystal orientation for
the strained samples suggests that the mechanism is the extrinsic spin Hall effect.
Whenever the Fermi level lies in a gap (or mobility gap) the bulk Hall conductance can be expressed in a topologically invariant form showing the quantization explicitly. The new formulation generalizes the earlier result by Thouless, Kohmoto, Nightingale, and den Nijs to the situation where many-body interaction and substrate disorder are also present. When applying to the fractional quantized Hall effect, we draw the conclusion that there must be a symmetry breaking in the many-body ground state. The possibility of writing the fractionally quantized Hall conductance as a topological invariant is also discussed. Journal Article
The surface electronic structure of 13000) was studied by angle-resolved photoemission and the full-potential linearized-augmented plane-wave film method. Experimentally, several electronic surface states were identified in the gaps of the projected-bulk band structure close to the Fermi level. Theory shows that these states belong to a spin-orbit split-surface band that extends through the whole Brillouin zone, and that some surface states penetrate very deeply into the bulk. In the experiment, the surface Fermi surface was found to consist of three features: an electron pocket at the (Gamma) over bar point, a hole pocket in the (Gamma) over bar-(M) over bar direction (i.e., in the direction of the surface-mirror line), and a small Fermi-surface element close to the (M) over bar' points.
Synchrotron radiation angle-resolved photoemission spectroscopy of Bi(111) shows that the Fermi surface consists of six elongated hole pockets along the gamma M right macro directions surrounding a ring-shaped electron pocket centered at gamma, all of which have two-dimensional character. The associated hole and electron sheet densities are p(s) = 1.1 x 10(13) cm(-2) and n(s) = 5.5 x 10(12) cm(-2), respectively. A weak emission feature associated with the bulk hole pocket in the Fermi surface was identified. The Fermi momentum of the bulk hole band near the T point is k(F) = 0.013+/-0.003 A(-1).
We show that a PbTe-type narrow-gap semiconductor with an antiphase boundary (or domain wall) has currents of abnormal parity and induced fractional charges. A model is introduced which reduces the problem to the physics of a Dirac equation with a soliton in background electric and magnetic fields. We show that this system is a physical realization of the parity anomaly.
This chapter discusses the crystal momentum representation (CMR) that is evidently analogous to the momentum representation of ordinary continuum mechanics. It describes the Wannier representation’ (WR), which is analogous to the coordinate representation. The chapter discusses the development of the Crystal momentum representation (CMR) ab initio and relates the other methods to it. These include the Kohn–Luttinger or modified crystal momentum representation (MCMR) and the Wannier Representation (WR). Wannier was interested in the problem of the large exciton that is similar to but more difficult than the usual type of problem to be attacked by the methods. Wannier introduced Wannier functions to have localized states to describe the system and later derived an equation for the relative motion of the hole and electron, which is very similar to the equations discussed in the chapter.
We present a theoretical study of soliton formation in long-chain polyenes, including the energy of formation, length, mass, and activation energy for motion. The results provide an explanation of the mobile neutral defect observed in undoped (CH)x. Since the soliton formation energy is less than that needed to create band excitation, solitons play a fundamental role in the charge-transfer doping mechanism.
DOI:https://doi.org/10.1103/PhysRevLett.16.1193
We report experimental information which elucidates important features of the Bi1-xSbx alloy system. By means of observations of the Shubnikov-de Haas effect on undoped and doped samples we have followed the transition from a band structure near the Fermi level which is essentially that of pure Bi to one which resembles that of pure Sb. Supplementary information has been obtained from transport measurements. In both Bi and Sb, the conduction band is at the L point in the Brillouin zone. A mirror-image band lies not far below. The upper L-point band is the conduction band throughout the alloy system. Our model of the transition involves two processes: (1) The descent of the T-point band (the valence band in Bi) relative to the L-point bands, a transition to six valence maxima at the nearby H points, and their subsequent rise past the Fermi level to become the valence bands in Sb; (2) the approach, crossing, and separation of the L-point bands. At 4.2°K, the descent of the T-point maximum below the T-point minimum and the L-point crossing both occur near x=8%; consequently, there is a transition from semimetallic to direct-gap (L-point) semiconducting behavior. The most direct determination of the L-point gap is obtained from the nonparabolic behavior of the very small transverse mass components. Strong corroborative evidence is obtained from transport measurements in the temperature range 4°-30°K.
The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential U. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small Uℏomegac.
The electronic band structure of bismuth is studied by means of a pseudopotential approach. The Lin-Kleinman pseudopotential was adopted; its parameters were adjusted slightly to bring the band structure into agreement with two known energy differences in bismuth. With this pseudopotential, the band structure along symmetry lines and planes is calculated and the effective masses of the carriers are studied. The band structure is in good agreement with optical data and with effective-mass anisotropies, but the magnitude of the effective masses may differ from experiment by a factor of 3. Using the experimental effective masses and g factor of the holes, we infer the energy-level scheme at T near EF. Also, we have tentatively identified a higher-lying band which has been experimentally observed. A very efficient method of calculating nonlocal and spin-orbit coupling terms in k.pi perturbation theory is presented.
In this paper a new and more comprehensive characterization of the insulating state of matter is developed. This characterization includes the conventional insulators with energy gap as well as systems discussed by Mott which, in band theory, would be metals. The essential property is this: Every low-lying wave function Phi of an insulating ring breaks up into a sum of functions, Phi=Sigma-∞∞PhiM, which are localized in disconnected regions of the many-particle configuration space and have essentially vanishing overlap. This property is the analog of localization for a single particle and leads directly to the electrical properties characteristic of insulators. An Appendix deals with a soluble model exhibiting a transition between an insulating and a conducting state.
A condensed-matter analog of (2+1)-dimensional electrodynamics is constructed, and the consequences of a recently discovered anomaly in such systems are discussed.
The Adler-Bell-Jackiw (ABJ) axial anomaly is derived from the physical
point of view as the production of Weyl particles and it is used to show
the absence of the net production of particles for lattice regularized
chirally invariant theories with locality. An analogy or a simulation is
pointed out between the Weyl fermion theory and gapless semiconductors
where two energy bands have pointlike degeneracies. For such materials,
in the presence of parallel electric and strong magnetic fields, there
exists an effect similar to the ABJ anomaly that is the movement of the
electrons in the energy-momentum space from the neighborhood of one
degeneracy point to another one. The longitudinal magneto-conduction
becomes extremely strong.
Supported by a grant of the US department of energy under Contract No.
DE-AC02-76ER03130 A011 - Task. A.
Effect of the spin-orbit interaction is studied for the random potential scattering in two dimensions by the renormalization
group method. It is shown that the localization behaviors are classified in the three different types depending on the symmetry.
The recent observation of the negative magnetoresistance of MOSFET is discussed.
The electronic band structure of arsenic is studied by means of a pseudopotential approach. The pseudopotential has been chosen by (a) fitting the "atomic" pseudopotential of Ge to a four-parameter curve and (b) adjusting the parameters slightly so as to allow comparison of the various theoretical curves with the experimental data. The energy bands are determined by diagonalizing a fairly large (~90×90) secular determinant. Spin-orbit coupling is included a posteriori. Group-theoretical analysis is carried out throughout the Brillouin zone and use is made of the classification of the levels as well as the compatibility relations. It is found that the holes are located at T, the center of the hexagonal face. The electrons are probably distributed in six equivalent pockets, each one located along a binary axis in a pseudohexagonal face, near its center L.
The conductivity and low-field Hall coefficient of high-purity (ND-NA∼5×1014 cm-3) and lightly doped (2×1015≤ND≤2×1017 cm-3) n-type gray tin subjected to oriented uniaxial compressions have been measured between 1.4 and 100 °K. Stress (χ) exceeding 3 × 109 dyn/cm2 was achieved in both [001] and [111] orientations. Density-of-states expressions are developed to account for the severe band anisotropies imposed by the strain in the normally degenerate Γ8+ conduction and valence bands, and these are employed to determine the band splittings at k⃗=0 from the Hall coefficient of the high-purity samples above 15 °K. Shear deformation potentials of b=-2.3±0.5 eV and d≃-4.1 eV are obtained by this procedure. The Hall coefficient of three high-purity samples below 10 °K is analyzed to find the stress-dependent impurity-ionization energy ED(χ), and from the measured ED(χ) for the highest-purity sample an independent determination of b=-2.4 eV is obtained if ED(χ) is interpreted as reflecting donor-to—conduction-band activation. However, the measured ED(χ) for this sample is also found to be consistent with activation from the donor ground state into a D- band. The stress dependence of the impurity mobility in two of these samples is explained in terms of Sladek's model for exchange jumping between filled and unfilled impurity sites. The piezoresistance of lightly doped samples is attributed to the increased effectiveness of ionized impurity scattering caused by a stress enhancement of the Γ8+ density-of-states mass.
Relativistic all-electron full-potential first-principles calculations have been performed in order to study the symmetry of the energy levels around the valence band maximum in the zinc blende II-VI semiconductors β-HgS, HgSe, and HgTe. It is demonstrated that in general, an inverted band-structure does not necessarily lead to a zero fundamental energy gap for systems with zinc blende symmetry. Specifically, β-HgS is found to have at the same time an inverted band structure, and a small, slightly indirect, fundamental energy gap. Possibly, the energy levels around the valence band maximum order differently in each of these systems.
A first-principles relativistic orthogonalized-plane-wave calculation has been used to determine the energy eigenvalues of grey tin at seven key points of the reduced zone. An extended zone k⃗·p⃗ method has been used as an interpolation scheme to map out the band structure in the remainder of the zone. Optical constants and derivative optical constants have been calculated from the k⃗·p⃗ parameters. The calculated normal incidence reflectivity is compared to experiment. A detailed critical-point analysis of the calculated optical spectra is presented. Valence-band mass parameters, effective masses, and g factors at several points in the zone have been obtained and are compared to available experimental data.
We have investigated the band structure of zinc-blende (ZB) Hg chalcogenides using a corrected local density approximation method. We find that the band gaps of HgS, HgSe, and HgTe are 0.30, −0.24, and −0.31 eV, respectively. That is, HgS has a positive band gap, whereas HgSe and HgTe have inverted band structures. The chemical trend of the band gaps is explained by the atomic energy levels and sizes, as well as by the related deformation potentials for these compounds. We also show systematically how the band gap of the inverted band structure can open up when the Td symmetry of the ZB structure is reduced under strain or in the presence of a surface or interface.
It is shown that the quantization of the Hall conductivity of two-dimensional metals which has been observed recently by Klitzing, Dorda, and Pepper and by Tsui and Gossard is a consequence of gauge invariance and the existence of a mobility gap. Edge effects are shown to have no influence on the accuracy of quantization. An estimate of the error based on thermal activation of carriers to the mobility edge is suggested.
When a conducting layer is placed in a strong perpendicular magnetic field, there exist current-carrying electron states which are localized within approximately a cyclotron radius of the sample boundary but are extended around the perimeter of the sample. It is shown that these quasi-one-dimensional states remain extended and carry a current even in the presence of a moderate amount of disorder. The role of the edge states in the quantized Hall conductance is discussed in the context of the general explanation of Laughlin. An extension of Laughlin's analysis is also used to investigate the existence of extended states in a weakly disordered two-dimensional system, when a strong magnetic field is present.
Various transport measurements are performed to assess the alloying and size effects in sub-100 nm Bi1−xSbx (0 ⩽ x ⩽ 0.15) nanowires. Temperature-dependent resistance measurements exhibit non-monotonic trends as x increases, and a theoretical model is presented to explain the features which are related to the unusual band structure of Bi1−xSbx systems. Magnetoresistance measurements of these Bi1−xSbx nanowires show interesting size-dependent behaviors similar to those in Bi nanowires. © 2001 American Institute of Physics.
The electronic energy band structures of SnTe, GeTe, and PbTe are
calculated including spin-orbit interactions. The resulting band
structures are used to discuss some of the electronic properties of
these crystals and to calculate the intervalley deformation potential
for electron-phonon scattering in SnTe. The value obtained for the
deformation potential is consistent with the value used to explain the
superconducting properties of SnTe.
A simple model is proposed which unites many data on alloys of bismuth
containing small amounts of antimony. Contrary to the usual
interpretation of these data, we propose a crossing of the conduction
and valence bands at L at about 6 at.% antimony (i.e., the direct gap,
which is responsible for the very small effective masses in bismuth,
vanishes at this composition).
We have developed a semi-empirical perturbation scheme for the purpose of relating the energy band structures of diamond-type and zinc-blende-type crystals. Our sole objective is to predict the approximate location of the valence and conduction band edges in the reduced zone for some of the better known zinc-blende-type crystals. In this paper we explain how the perturbation scheme operates. Since a more complete account will appear in The Physical Review (Herman 1955), we will restrict ourselves here only to the essentials. A brief discussion of the effect of the spin-orbit interaction upon the band structure of zinc-blende-type crystals is also included in the present paper.
Describes growth of the first thin ( approximately 1 mu m) epitaxial films of pure bismuth-antimony alloys using molecular beam epitaxy techniques. These structures were grown at elevated temperatures on single-crystal barium fluoride substrates of (111) orientation. Electron microscope observations show the films to be featureless and defect-free on the scale of 0.1 mu m. The films grow with their trigonal axis parallel to the (111) axis of the substrate, and Laue-backscattering pictures show that they are epitaxial. Mobilities of alloys with x=0 are of the order of 2 m2 V-1 s-1 at room temperature and increase to over 10 at 20 K and 100 at liquid helium temperatures. These values are far superior to those of other bismuth films grown to data, and approach mobilities observed in single-crystal bismuth. The dependence of energy gap and c axis lattice constant on x is different from that in bulk alloys, which may be due to the effects of strain arising from the 3.6% lattice mismatch between sample and substrate.
The electronic structures of the two thermoelectric materials and are studied using density-functional theory with the spin - orbit interaction included. The electron states in the gap region and the chemical bonding can be described in terms of interaction between the atomic p orbitals within the `quintuple' layer. For , we find both the valence-band maximum as well as the conduction-band minimum, each with a nearly isotropic effective mass, to occur at the zone centre in agreement with experimental results. For , we find that the six valleys for the valence-band maximum are located in the mirror planes of the Brillouin zone and they have a highly anisotropic effective mass, leading to an agreement between the de Haas - van Alphen data for the p-doped samples and the calculated Fermi surface. The calculated conduction band, however, has only two minima, instead of the six minima indicated from earlier experiments. The calculated Seebeck coefficients for both p-type and n-type materials are in agreement with the experiments.
This study is focused on the investigation of the transport properties of BiSb alloys. Electrical resistivity, thermoelectric power and thermal conductivity were measured in a direction perpendicular or parallel to the trigonal axis within the temperature range 4.2–300 K on various alloy compositions containing up to 18.2 at.% antimony. The temperature dependences of the three coefficients are described in detail. Low temperature behaviour depends strongly on the crystal purity, particularly in the semiconducting range. A qualitative explanation of these results is given in terms of an impurity band which merges with the conduction band. Thermoelectric properties are discussed and compared with previous studies.
We study the electrodynamics of a PbTe-type bulk semiconductor with a domain wall. We show the existence of states bound to the wall. In the presence of static electric and magnetic fields a current with abnormal parity and a nonzero induced electric charge are shown to exist. These systems are a physical realization of the parity anomaly of 2 + 1 dimensional QED.
The topological aspects of wavefunctions for electrons in a two dimensional periodic potential with a magnetic field are discussed. Special attention is paid to the linear response formula for the Hall conductance σxy. It is shown that the quantized value of σxy is related to the number of zeros of wavefunctions in the magnetic Brillouin zone. A phase of wavefunctions cannot be determined in a unique and smooth way over the entire magnetic Brillouin zone unless the magnetic subband carries no Hall current.
The macroscopic electric polarization of a crystal is often defined as the dipole of a unit cell. In fact, such a dipole moment is ill defined, and the above definition is incorrect. Looking more closely, the quantity generally measured is differential polarization, defined with respect to a "reference state" of the same material. Such differential polarizations include either derivatives of the polarization (dielectric permittivity, Born effective charges, piezoelectricity, pyroelectricity) or finite differences (ferroelectricity). On the theoretical side, the differential concept is basic as well. Owing to continuity, a polarization difference is equivalent to a macroscopic current, which is directly accessible to the theory as a bulk property. Polarization is a quantum phenomenon and cannot be treated with a classical model, particularly whenever delocalized valence electrons are present in the dielectric. In a quantum picture, the current is basically a property of the phase of the wave functions, as opposed to the charge, which is a property of their modulus. An elegant and complete theory has recently been developed by King-Smith and Vanderbilt, in which the polarization difference between any two crystal states-in a null electric field-takes the form of a geometric quantum phase. The author gives a comprehensive account of this theory, which is relevant for dealing with transverse-optic phonons, piezoelectricity, and ferroelectricity. Its relation to the established concepts of linear-response theory is also discussed. Within the geometric phase approach, the relevant polarization difference occurs as the circuit integral of a Berry connection (or "vector potential"), while the corresponding curvature (or "magnetic field") provides the macroscopic linear response.
We present a theoretical study of soliton formation in long-chain polyenes, including the energy of formation, length, mass, and activation energy for motion. The results provide an explanation of the mobile neutral defect observed in undoped ${(\mathrm{CH})}_{x}$. Since the soliton formation energy is less than that needed to create band excitation, solitons play a fundamental role in the charge-transfer doping mechanism.
For generic time-reversal-invariant systems with spin-orbit couplings, we clarify a close relationship between the Z2 topological order and the spin Chern number (SChN) in the quantum spin-Hall effect. It turns out that a global gauge transformation connects sectors with different SChNs (even integers) modulo 4, which implies that the SChN and Z2 topological orders yield the same classification. We present a method of computing the SChN and demonstrate it in single and double planes of graphene.
Attention is drawn to a relation between the recently discovered three-dimensional chiral anomaly and fermion zero modes. Application to planar systems of electrons in an external magnetic field is suggested.
We have developed a third-neighbor tight-binding model, with spin-orbit coupling included, to treat the electronic properties of Bi and Sb. This model successfully reproduces the features near the Fermi surface that will be most important in semimetal-semiconductor device structures, including (a) the small overlap of valence and conduction bands, (b) the electron and hole effective masses, and (c) the shapes of the electron and hole Fermi surfaces. The present tight-binding model treats these semimetallic properties quantitatively, and it should, therefore, be useful for calculations of the electronic properties of proposed semimetal-semiconductor systems, including superlattices and resonant-tunneling devices.
The mean-field theory of a T- and P-symmetric spin-liquid state is developed. The quasiparticle excitations in the spin-liquid state are shown to be spin-1/2 neutral fermions (the spinons) and charge e spinless bosons (the holons). The spin-liquid state is shown to be characterized by a nontrivial topological order. Although our discussions are based on the mean-field theory, the concept of the topological order and the associated universal properties (e.g., the quantum number of the quasiparticles) are expected to be valid beyond the mean-field theory. We also discuss the dynamical stability of the mean-field theory.
When the Fermi level lies in a gap, the Hall conductivity of three-dimensional electrons in a periodic potential is expressed in a topologically invariant form with a set of three integers. If the magnetic fluxes through the three independent areas of the periodic lattice are rational numbers, one obtains a Diophantine equation relating these numbers and the integers which characterize the Hall conductivity. Journal Article
We consider the change in polarization $\Delta${}P which occurs upon making an adiabatic change in the Kohn-Sham Hamiltonian of the solid. A simple expression for $\Delta${}P is derived in terms of the valence-band wave functions of the initial and final Hamiltonians. We show that physically $\Delta${}P can be interpreted as a displacement of the center of charge of the Wannier functions. The formulation is successfully applied to compute the piezoelectric tensor of GaAs in a first-principles pseudopotential calculation.
A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance sigmaxy in the absence of an external magnetic field. Massless fermions without spectral doubling ccur at critical values of the model parameters, and exhibit the so-called ``parity anomaly'' of (2+1)-dimensional field theories.
Berry's phase is defined for the dynamics of electrons in periodic solids and an explicit formula is derived for it. Because of the special torus topology of the Brillouin zone a nonzero Berry phase is shown to exist in a one-dimensional parameter space. Symmetry of the Bloch functions in the Brillouin zone leads to the quantization of Berry's phase. A connection is established between the latter and the Wyckoff positions in the crystal in the framework of band representations of space groups. Berry's phase can therefore be used for labeling energy bands in solids.
We consider the integer quantum Hall effect on a square lattice in a uniform rational magnetic field. The relation between two different interpretations of the Hall conductance as topological invariants is clarified. One is the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) integer in the infinite system and the other is a winding number of the edge state. In the TKNN form of the Hall conductance, a phase of the Bloch wave function defines U(1) vortices on the magnetic Brillouin zone and the total vorticity gives sigmaxy. We find that these vortices are given by the edge states when they are degenerate with the bulk states.