Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties,including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.     

Model Hamiltonian for the Quantum Anomalous Hall State in Iron-Halogenide

We examine quantum anomalous Hall(QAH) insulators with intrinsic magnetism displaying quantized Hall conductance at zero magnetic fields.The spin-momentum locking of the topological edge stats promises QAH insulators with great potential in device applications in the field of spintronics.Here,we generalize Haldane’s model on the honeycomb lattice to a more realistic two-orbital case without the artificial real-space complex hopping.Instead,we introduce an intraorbital coupling,stemming directly ...     

Model characterization of gapless edge modes of topological insulators using intermediate Brillouin-zone functions

We characterize gapless edge modes in translation invariant topological insulators. We show that the edge mode spectrum is a continuous deformation of the spectrum of a certain gluing function defining the occupied state bundle over the Brillouin zone. Topologically nontrivial gluing functions, corresponding to nontrivial bundles, then yield edge modes exhibiting spectral flow. We illustrate our results for the case of chiral edge states in two-dimensional Chern insulators, as well as helical edges in quantum spin Hall states.     

Studies on the electronic structures of three-dimensional topological insulators by angle resolved photoemission spectroscopy

Three-dimensional (3D) topological insulators represent a new state of quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. The unusual surface states of topological insulators rise from the nontrivial topology of their electronic structures as a result of strong spin-orbital coupling. In this review, we will briefly introduce the concept of topological insulators and the experimental method that can directly probe their unique electronic structure: angle resolved photoemission spectroscopy (ARPES). A few examples are then presented to demonstrate the unique band structures of different families of topological insulators and the unusual properties of the topological surface states. Finally, we will briefly discuss the future development of topological quantum materials.     

Theoretical prediction of topological insulators in two-dimensional ternary transition metal chalcogenides (MM'X4, M = Ta,Nb, or V; M'= Ir,Rh, or Co; X = Se or Te)

Ternary transition metal chalcogenides (TTMCs) have attracted interest due to the discovery of their Weyl semimetallic property and the recent synthesis of layered TTMCs which are regarded as potential candidates for two-dimensional (2D) topological insulators. Here, employing first-principles calculations, we predicted the emergence of non-trivial band topologies in the monolayer MM'X₄ family (M= V, Nb, or Ta; M' = Co, Rh, or Ir; and X = Se or Te) within hybrid functional calculations. Five of eighteen 2D materials were found to be topological insulators, while four of them are magnetic thin films. The nontrivial topologies were verified via the calculated Z₂ topological invariant and topologically protected edge states. Further calculations showed a strain-induced phase transition in VCoTe₄ from a magnetic phase to a nonmagnetic topological insulating phase. Our comprehensive study revealed a diverse family of monolayer ternary transition metal chalcogenides adding new members to the current catalog of 2D topological insulators and 2D magnetic materials.     

Electronic and Spin Structures of Intrinsic Antiferromagnetic Topological Insulators of the MnBi2Te4(Bi2Te3)m Family and Their Magnetic Properties (Brief Review)

JETP Letters - Magnetic topological insulators (TIs) are narrow-gap semiconductor materials combining a nontrivial band structure and the magnetic order. Unlike their nonmagnetic analogs, magnetic...     

Spin-Hall insulator

Recent theories predict dissipationless spin current induced by an electric field in doped semiconductors. Nevertheless, the charge current is still dissipative in these systems. In this work, we theoretically predict the dissipationless spin-Hall effect, without any accompanying charge current, in some classes of band insulators, including zero-gap semiconductors such as HgTe and narrow-gap semiconductors such as PbTe. This effect is similar to the quantum-Hall effect in that all the states below the gap contribute and there occurs no dissipation. However, the spin-Hall conductance is not quantized even in two dimensions. This is the first example of a nontrivial topological structure in a band insulator without any magnetic field.     

Dynamics of Electronic States and Magnetoabsorption in 3D Topological Insulators in a Quantizing Magnetic Field

Quantum states have been calculated analytically; the dynamics of a wave packet in a magnetic field has been investigated, and the optical absorption coefficient has been calculated for surface states in 3D topological insulators of the Bi₂Te₃ family. We have detected a qualitative effect of the hexagonal warping of the spectrum on the structure of wavefunctions at the Landau levels, its manifestation in the features of the wave packet dynamics in a quantizing magnetic field, as well as in the frequency dependence of the optical absorption coefficient, in which new peaks that are absent in the isotropic model of the spectrum appear depending on the polarization of the incident wave. The effects considered here can be manifested in the optical and transport experiments with topological insulators, which makes it possible to determine the parameters of their band structure.     

Topological magnetic insulators with corundum structure

Topological insulators are new states of quantum matter in which surface states residing in the bulk insulating gap are protected by time-reversal symmetry. When a proper kind of antiferromagnetic long-range order is established in a topological insulator, the system supports axionic excitations. In this Letter, we study theoretically the electronic states in a transition metal oxide of corundum structure, in which both spin-orbit interaction and electron-electron interaction play crucial roles. A tight-binding model analysis predicts that materials with this structure can be strong topological insulators. Because of the electron correlation, an antiferromagnetic order may develop, giving rise to a topological magnetic insulator phase with axionic excitations.     

Topological hinge modes in Dirac semimetals

Dirac semimetals (DSMs) are an important class of topological states of matter. Here, focusing on DSMs of band inversion type, we investigate their boundary modes from the effective model perspective. We show that in order to properly capture the boundary modes, k-cubic terms must be included in the effective model, which would drive an evolution of surface degeneracy manifold from a nodal line to a nodal point. Sizable k-cubic terms are also needed for better exposing the topological hinge modes in the spectrum. Using first-principles calculations, we demonstrate that this feature and the topological hinge modes can be clearly exhibited in β-CuI. We extend the discussion to magnetic DSMs and show that the time-reversal symmetry breaking can gap out the surface bands and hence is beneficial for the experimental detection of hinge modes. Furthermore, we show that magnetic DSMs serve as a parent state for realizing multiple other higher-order topological phases, including higher-order Weyl-point/nodal-line semimetals and higher-order topological insulators.     

Topological crystalline insulators

The recent discovery of topological insulators has revived interest in the band topology of insulators. In this Letter, we extend the topological classification of band structures to include certain crystal point group symmetry. We find a class of three-dimensional "topological crystalline insulators" which have metallic surface states with quadratic band degeneracy on high symmetry crystal surfaces. These topological crystalline insulators are the counterpart of topological insulators in materials without spin-orbit coupling. Their band structures are characterized by new topological invariants. We hope this work will enlarge the family of topological phases in band insulators and stimulate the search for them in real materials.     

Topological Aspects of Triplet Superconductors

In this paper, using the Φ-mapping theory, it is shown that two kinds of topological defects, i.e., the vortex lines and the monopoles exist in the helical configuration of magnetic field in triplet superconductors. And the inner topological structure of these defects is studied. Because the knot solitons in the triplet superconductors are characterized by the Hopf invariant, we also establish a relationship between the Hopf invariant and the linking number of knots family, and reveal the inner topological structure of the Hopf invariant.     

Topological solitons in 8-spinor mie electrodynamics

We investigate the effective 8-spinor field model suggested earlier as the generalization of nonlinear Mie electrodynamics. We first study in pure spinorial model the existence of topological solitons endowed with the nontrivial Hopf invariant Q _H, which can be interpreted as the lepton number. Electromagnetic field being included as the perturbation, we estimate the energy and the spin of the localized charged configuration.     

Fragile Mott insulators

We prove that there exists a class of crystalline insulators, which we call "fragile Mott insulators," which are not adiabatically connected to any sort of band insulator provided time-reversal and certain point-group symmetries are respected, but which are otherwise unspectacular in that they exhibit no topological order nor any form of fractionalized quasiparticles. Different fragile Mott insulators are characterized by different nontrivial one-dimensional representations of the crystal point group. We illustrate this new type of insulators with two examples: the d Mott insulator discovered in the checkerboard Hubbard model at half-filling and the Affleck-Kennedy-Lieb-Tasaki insulator on the square lattice.     

Topological Insulators on the Ruby Lattice with Rashba Spin-Orbit Coupling

We investigate a tight-binding model of the ruby lattice with Rashba spin-orbit coupling. We calculate the band structure of the lattice and evaluate the Z₂ topological indices. According to the Z₂ topological indices and the band structure, we present the phase diagrams of the lattice with different filling fractions. We find that topological insulators occur in some range of parameters at 1/6, 1/3, 1/2, 2/3 and 5/6 filling fractions. We analyze and discuss the characteristics of these topological insulators and their edge states.     

二维介电光子晶体中的赝自旋态与拓扑相变

基于背散射抑制且对缺陷免疫的传输性质,光子拓扑绝缘体为电磁传输调控提供了一种新颖的思路.类比电子体系中的量子自旋霍尔效应,本文设计出一种简单的二维介电光子晶体,以实现自旋依赖的光子拓扑边界态.该光子晶体是正三角环形硅柱子在空气中排列而成的蜂窝结构.将硅柱子绕各自中心旋转60°,可实现二重简并的偶极子态和四极子态之间的能带翻转.这两对二重简并态的平均能流密度围绕原胞中心的手性可充当赝自旋自由度,其点群对称性可用来构建赝时间反演对称.根据k·p微扰理论,给出了布里渊区中心附近的有效哈密顿量以及对应的自旋陈数,由此证实能带翻转的实质是拓扑相变.数值计算结果揭示,在拓扑非平庸和平庸的光子晶体分界面上可实现单向传输且对弯曲、空穴等缺陷免疫的拓扑边界态.本文中的光子晶体只由电介质材料组成并且晶格结构简单,实现拓扑相变时无需改变柱子的填充率或位置,只需转动一个角度.因此,这种结构在拓扑边界态的应用中更为有效.     

反铁磁半金属 EuCd2Pn2（Pn = As,Sb）中磁交换诱导的外尔态

由于丰富的拓扑量子效应及巨大的潜在应用价值,拓扑材料逐渐成为凝聚态物理前沿的研究材料体系。其中,作为与石墨烯具有相似电子结构的材料,三维拓扑半金属吸引了越来越多的研究兴趣。目前已知的拓扑半金属大多为非磁性的,而磁性拓扑半金属数量有限,与非磁性拓扑半金属相比较,研究开展的还比较少。磁性与拓扑之间的相互作用能够导致非常规的物理性质,如反常霍尔效应甚至量子反常霍尔效应等。此外,在一些具有特殊磁结构的拓扑半金属中,施加外磁场能够调制其自旋结构,从而影响其拓扑能带结构。在该综述中,笔者将详细介绍利用外磁场在 EuCd2Pn2 (Pn = As, Sb) 反铁磁半金属材料中通过调制自旋结构从而改变晶体结构对称性来诱导拓扑相变。此外,笔者也将简单介绍包括 GdPtBi 和 MnBi2Te4 在内的几个相关材料。该综述中讨论的外磁场调控的磁交换诱导的拓扑相变不仅有望应用于拓扑器件,也有助于为理解磁性与拓扑态之间的紧密关联提供新的线索,对于设计新的磁性拓扑材料有启发意义。综述最后,笔者对发展磁性拓扑半金属做了一些简单展望。     

Z_2拓扑不变量与拓扑绝缘体

文章回顾了几种Z2拓扑数的计算方法,并详细介绍了一种用非阿贝尔贝里联络表示绝缘体Z2不变量的计算方法.这种方法可以确定出一般能带绝缘体的拓扑性质,而不需要限定波函数的规范.利用这种新方法,文章作者计算了二维石墨烯(graphene)系统的Z2拓扑数,得到了和以前研究相一致的结论.     

Topological insulator as an optical modulator for pulsed solid‐state lasers

Topological insulators are states of quantum matter that have narrow topological nontrivial energy gaps and a large third‐order nonlinear optical response. The optical absorption of topological insulators can become saturated under strong excitation. In this work, with Bi₂Se₃ as an example, it was demonstrated that a topological insulator can modulate the operation of a bulk solid‐state laser by taking advantage of its saturable absorption. The result suggests that topological insulators are potentially attractive as broadband pulsed modulators for the generation of short and ultrashort pulses in bulk solid‐state lasers, in addition to other promising applications in physics and computing.