(d-2)-Dimensional Edge States of Rotation Symmetry Protected Topological States.
TL;DR: Fourfold rotation-invariant gapped topological systems with time-reversal symmetry in two and three dimensions with strongly interacting systems through the explicit construction of microscopic models having robust (d-2)-dimensional edge states are studied.
Abstract: Theorists have discovered topological insulators that are insulating in their interior and on their surfaces but have conducting channels at corners or along edges.
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TL;DR: This work provides a comprehensive framework for generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.
Abstract: Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.
916 citations
TL;DR: Using a recently developed formalism called topological quantum chemistry, a high-throughput search of ‘high-quality’ materials in the Inorganic Crystal Structure Database is performed and it is found that more than 27 per cent of all materials in nature are topological.
Abstract: Using a recently developed formalism called topological quantum chemistry, we perform a high-throughput search of 'high-quality' materials (for which the atomic positions and structure have been measured very accurately) in the Inorganic Crystal Structure Database in order to identify new topological phases. We develop codes to compute all characters of all symmetries of 26,938 stoichiometric materials, and find 3,307 topological insulators, 4,078 topological semimetals and no fragile phases. For these 7,385 materials we provide the electronic band structure, including some electronic properties (bandgap and number of electrons), symmetry indicators, and other topological information. Our results show that more than 27 per cent of all materials in nature are topological. We provide an open-source code that checks the topology of any material and allows other researchers to reproduce our results.
782 citations
TL;DR: A second-order topological insulator in d dimensions is an insulator which has no d-1 dimensional topological boundary states but has d-2 dimensional topology boundary states, which constitutes the bulk topological index.
Abstract: A second-order topological insulator in d dimensions is an insulator which has no d-1 dimensional topological boundary states but has d-2 dimensional topological boundary states. It is an extended notion of the conventional topological insulator. Higher-order topological insulators have been investigated in square and cubic lattices. In this Letter, we generalize them to breathing kagome and pyrochlore lattices. First, we construct a second-order topological insulator on the breathing Kagome lattice. Three topological boundary states emerge at the corner of the triangle, realizing a 1/3 fractional charge at each corner. Second, we construct a third-order topological insulator on the breathing pyrochlore lattice. Four topological boundary states emerge at the corners of the tetrahedron with a 1/4 fractional charge at each corner. These higher-order topological insulators are characterized by the quantized polarization, which constitutes the bulk topological index. Finally, we study a second-order topological semimetal by stacking the breathing kagome lattice.
627 citations
TL;DR: An algorithm based on symmetry indicators is used to search a crystallographic database and finds thousands of candidate topological materials, which could be exploited in next-generation electronic devices.
Abstract: Over the past decade, topological materials—in which the topology of electron bands in the bulk material leads to robust, unconventional surface states and electromagnetism—have attracted much attention. Although several theoretically proposed topological materials have been experimentally confirmed, extensive experimental exploration of topological properties, as well as applications in realistic devices, has been restricted by the lack of topological materials in which interference from trivial Fermi surface states is minimized. Here we apply our method of symmetry indicators to all suitable nonmagnetic compounds in all 230 possible space groups. A database search reveals thousands of candidate topological materials, of which we highlight 241 topological insulators and 142 topological crystalline insulators that have either noticeable full bandgaps or a considerable direct gap together with small trivial Fermi pockets. Furthermore, we list 692 topological semimetals that have band crossing points located near the Fermi level. These candidate materials open up the possibility of using topological materials in next-generation electronic devices. An algorithm based on symmetry indicators is used to search a crystallographic database and finds thousands of candidate topological materials, which could be exploited in next-generation electronic devices.
607 citations
TL;DR: A second-order topological insulator in an acoustical metamaterial with a breathing kagome lattice, supporting one-dimensional edge states and zero-dimensional corner states is demonstrated, and shape dependence allows corner states to act as topologically protected but reconfigurable local resonances.
Abstract: Higher-order topological insulators1–5 are a family of recently predicted topological phases of matter that obey an extended topological bulk–boundary correspondence principle. For example, a two-dimensional (2D) second-order topological insulator does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D topological insulator, but instead has topologically protected zero-dimensional (0D) corner states. The first prediction of a second-order topological insulator1, based on quantized quadrupole polarization, was demonstrated in classical mechanical6 and electromagnetic7,8 metamaterials. Here we experimentally realize a second-order topological insulator in an acoustic metamaterial, based on a ‘breathing’ kagome lattice9 that has zero quadrupole polarization but a non-trivial bulk topology characterized by quantized Wannier centres2,9,10. Unlike previous higher-order topological insulator realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically protected but reconfigurable local resonances. A second-order topological insulator in an acoustical metamaterial with a breathing kagome lattice, supporting one-dimensional edge states and zero-dimensional corner states is demonstrated.
591 citations
References
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TL;DR: In this paper, the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topologically insulators have been observed.
Abstract: Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on $\mathrm{Hg}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$, ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, and ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.
15,562 citations
TL;DR: Topological superconductors are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors and are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time reversal symmetry.
Abstract: Topological insulators are new states of quantum matter which cannot be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi2Te3 and Bi2Se3 crystals. Theoretical models, materials properties, and experimental results on two-dimensional and three-dimensional topological insulators are reviewed, and both the topological band theory and the topological field theory are discussed. Topological superconductors have a full pairing gap in the bulk and gapless surface states consisting of Majorana fermions. The theory of topological superconductors is reviewed, in close analogy to the theory of topological insulators.
11,092 citations
TL;DR: In this paper, the authors studied three-dimensional generalizations of the quantum spin Hall (QSH) effect and introduced a tight binding model which realized the WTI and STI phases, and discussed its relevance to real materials including bismuth.
Abstract: We study three-dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where a single ${Z}_{2}$ topological invariant governs the effect, in three dimensions there are 4 invariants distinguishing 16 phases with two general classes: weak (WTI) and strong (STI) topological insulators. The WTI are like layered 2D QSH states, but are destroyed by disorder. The STI are robust and lead to novel ``topological metal'' surface states. We introduce a tight binding model which realizes the WTI and STI phases, and we discuss its relevance to real materials, including bismuth.
3,357 citations
TL;DR: In this paper, a method for determining the optimally localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid is presented, which is suitable for use in connection with conventional electronic-structure codes.
Abstract: We discuss a method for determining the optimally localized set of generalized Wannier functions associated with a set of Bloch bands in a crystalline solid. By ''generalized Wannier functions'' we mean a set of localized orthonormal orbitals spanning the same space as the specified set of Bloch bands. Although we minimize a functional that represents the total spread Sigma(n)(r(2))(n) - (r)(n)(2) of the Wannier functions in real space, our method proceeds directly from the Bloch functions as represented on a mesh of k points, and carries out the minimization in a space of unitary matrices U-mn((k)) describing the rotation among the Bloch bands at each k point. The method is thus suitable for use in connection with conventional electronic-structure codes. The procedure also returns the total electric polarization as well as the location of each Wannier center. Sample results for Si, GaAs, molecular C2H4, and LiCl will be presented.
3,155 citations
TL;DR: In this article, the O(3) nonlinear sigma model of the large-spin one-dimensional Heisenberg antiferromagnet was shown to be the optimal model for weak easy-axis anisotropy.
Abstract: The continuum field theory describing the low-energy dynamics of the large-spin one-dimensional Heisenberg antiferromagnet is found to be the O(3) nonlinear sigma model. When weak easy-axis anisotropy is present, soliton solutions of the equations of motion are obtained and semiclassically quantized. Integer and half-integer spin systems are distinguished.
2,491 citations