In this paper we link the physics of topological nonlinear σ models with that of Chern-Simons insulators. We show that corresponding to every 2n-dimensional Chern-Simons insulator there is a (n-1)-dimensional topological nonlinear σ model with the Wess-Zumino-Witten term. Breaking internal symmetry in these nonlinear σ models leads to nonlinear σ models with the θ term. [This is analogous to the dimension reduction leading from 2n-dimensional Chern-Simons insulators to (2n-1) and (2n-2)-dimensional topological insulators protected by discrete symmetries.] The correspondence described in this paper allows one to derive the topological term in a theory involving fermions and order parameters (we shall referred to them as "fermion-σ models") when the conventional gradient-expansion method fails. We also discuss the quantum number of solitons in topological nonlinear σ model and the electromagnetic action of the (2n-1)-dimensional topological insulators. Throughout the paper we use a simple model to illustrate how things work.
We show how quantum oscillation measurements of surface states in an insulator may allow to diagnose a strong topological insulator and distinguish it from its weak or topologically trivial counterpart. The criterion is defined by the parity of the number of fundamental frequencies in the surface-state quantum oscillation spectrum: an even number of frequencies implies a weak or a topologically trivial insulator, whereas an odd number points to a strong topological insulator. We also discuss various aspects and issues related to applying this criterion in practice.
We present a theory on the quantum phase diagram of AB-stacked MoTe$_2$/WSe$_2$ using a self-consistent Hartree-Fock calculation performed in the plane-wave basis, motivated by the observation of topological states in this system. At filling factor $ν=2$ (two holes per moiré unit cell), Coulomb interaction can stabilize a $\mathbb{Z}_2$ topological insulator by opening a charge gap. At $ν=1$, the interaction induces three classes of competing states, spin density wave states, an in-plane ferromagnetic state, and a valley polarized state, which undergo first-order phase transitions tuned by an …
Recent developments of experimental techniques have given us unprecedented opportunities of studying topological insulators in high dimensions, while some of the dimensions are "synthetic", in the sense that the effective lattice momenta along these synthetic dimensions are controllable periodic tuning parameters. In this work, we study interaction effects on topological insulators with synthetic dimensions. We show that although the free fermion band structure of high dimensional topological insulators can be precisely simulated with the "synthetic techniques", the generic interactions in the…
The electronic structure at the interface between a topological band insulator and a Mott insulator is studied within layer dynamical mean field theory. To represent the bulk phases of these systems, we use the generalized Bernevig-Hughes-Zhang model that incorporates the Hubbard-like onsite Coulomb energy U in addition to the spin-orbit coupling term that causes band inversion. The topological and Mott insulating phases are realized by appropriately choosing smaller and larger values of U, respectively. As expected, the interface is found to be metallic because of the localized edge state. Wh…
A definition of topological phases of density matrices is presented. The topological invariants in case of both noninteracting and interacting systems are extended to nonzero temperatures. Influence of electron interactions on topological insulators at finite temperatures is investigated. A correlated topological insulator is described by the Kane-Mele model, which is extended by the interaction term of the Falicov-Kimball type. Within the Hartree-Fock and the Hubbard I approximations the thermodynamic and topological phase diagrams are determined, where the long-range order is included. The r…
Fractional topological insulators are electronic systems that carry fractionally charged excitations, conserve charge and are symmetric to reversal of time. In this review we introduce the basic essential concepts of the field, and then survey theoretical understanding of fractional topological insulators in two and three dimensions. In between, we discuss the case of "two and a half dimensions", the fractional topological insulators that may form on the two dimensional surface of an unfractionalized three dimensional topological insulator. We focus on electronic systems and emphasize properti…
We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry $\mathcal{C}^\prime$, a generalized particle-hole symmetry that gives the Hopf insulator a $\mathbb{Z}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry $\mathcal{C}^\prime$ has the same $\mathbb{Z}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as "Chern-insulator$ \rtimes S^1$" and "Chern-insulator$\rtimes T^2$" respectively. We also discuss the relati…
We provide the first unbiased evidence for a higher-order topological Mott insulator in three dimensions by numerically exact quantum Monte Carlo simulations. This insulating phase is adiabatically connected to a third-order topological insulator in the noninteracting limit, which features gapless modes around the corners of the pyrochlore lattice and is characterized by a $\mathbb{Z}_{4}$ spin-Berry phase. The difference between the correlated and non-correlated topological phases is that in the former phase the gapless corner modes emerge only in spin excitations being Mott-like. We also sho…
We study boundary conditions of topological sigma models with the goal of generalizing the concepts of anomalous symmetry and symmetry protected topological order. We find a version of 't Hooft's anomaly matching conditions on the renormalization group flow of boundaries of invertible topological sigma models and discuss several examples of anomalous boundary theories. We also comment on bulk topological transitions in dynamical sigma models and argue that one can, with care, use topological data to draw sigma model phase diagrams.
Topological insulators are new states of quantum matter with metallic edge/surface states. In this paper, we pointed out that there exists a new type of particle-hole symmetry-protected topological insulator - topological hierarchy insulator (THI), a composite topological state of a (parent) topological insulator and its defect-induced topological mid-gap states. A particular type of THI is topological fractal insulator, that is a THI with self-similar topological structure. In the end, we discuss the possible experimental realizations of THIs.
Indications for a line of continuous phase transitions at finite temperatures connected with the apparent metal-insulator transition in 2d disordered systems
In this manuscript, we study the interplay between symmetry and topology with a focus on the $Z_2$ topological index of 2D/3D topological insulators and high-order topological insulators. We show that in the presence of either a two-fold-rotational symmetry or a mirror symmetry, a gauge-invariant quantity can be defined for arbitrary 1D lines in the Brillouin zone. Such 1D quantities provide a new pathway to compute the $Z_2$ index of topological insulators. In contrast to the generic setup, where the $Z_2$ index generally involves 2D planes in the Brillouin zone with a globally-defined smooth…
This research highlight (originally published at http://www.ntu.edu.sg/ias/newsletters/Documents/APPN/APPNv5n2May2016-lowres.pdf) introduces recent developments in Topological Insulators and Holography, and provides an overview of how one can construct a holographic bulk topological insulator from a boundary Chern insulator with small gap.
Topological Kondo insulators are a new class of topological insulators in which metallic surface states protected by topological invariants reside in the bulk band gap at low temperatures. Unlike other three-dimensional topological insulators, a truly insulating bulk state, which is critical for potential applications in next-generation electronic devices, is guaranteed by many-body effects in the topological Kondo insulator. Furthermore, the system has strong electron correlations that can serve as a testbed for interacting topological theories. This topical review focuses on recent advances …
SmB6 was recently proposed to be both a strong topological insulator and a topological crystalline insulator. For this and related cubic topological Kondo insulators, we prove the existence of four different topological phases, distinguished by the sign of mirror Chern numbers. We characterize these phases in terms of simple observables, and we provide concrete tight-binding models for each phase. Based on theoretical and experimental results for SmB6 we conclude that it realizes the phase with C^+_{k_z=0}=+2, C^+_{k_z=pi}=+1, C^+_{k_x=k_y}=-1, and we propose a corresponding minimal model.
We use the method of bulk-boundary correspondence of topological invariants to show that disordered topological insulators have at least one delocalized state at their boundary at zero energy. Those insulators which do not have chiral (sublattice) symmetry have in addition the whole band of delocalized states at their boundary, with the zero energy state lying in the middle of the band. This result was previously conjectured based on the anticipated properties of the supersymmetric (or replicated) sigma models with WZW-type terms, as well as verified in some cases using numerical simulations a…
Conventional topological insulators and superconductors have topologically protected nodal points on their boundaries, and the recent interests in nodal-line semimetals only concerned bulk band structures. Here, we present a novel four-dimensional topological insulator protected by an anti-unitary reflection symmetry, whose boundary band has a single $PT$-symmetric nodal line with double topological charges. Inspired by the recent experimental realization of the four-dimensional quantum Hall effect, we also propose a cold-atom system which realizes the novel topological insulator with tunable …
We show that the oblique confinement states in the Cardy-Rabinovici model are superinsulators, not topological insulators. This is because their only bulk excitations are strings that completely prevent the separation of charge-anticharge pairs, causing a strictly infinite resistance (below a critical temperature and/or voltage). This is very different from the typical activated behaviour of topological insulators, caused by a finite bulk energy gap.
Persistent Coherence and Spin-Polarization of Topological Surface States on Topological Insulators
拓扑绝缘体上的无带隙表面态可以避免非磁性杂质的弹性散射，这使得它们成为低功率电子应用的有希望的候选者。然而，对于广泛的应用，这些状态应该保持相干，并在环境温度下显着自旋极化。在这里，我们研究的相干性和自旋结构的拓扑状态的表面上的模型拓扑绝缘体，Bi 2 Se 3，在高温下的自旋和角度分辨光电子能谱。我们发现一个非常弱的加宽和基本上没有衰减的自旋极化的拓扑表面状态到室温。我们的研究结果表明，拓扑绝缘体表面的拓扑态可以作为室温电子器件的基础。
We have simulated a half-filled $1D$ $p$-wave periodic Anderson model with numerically exact projector quantum Monte Carlo technique, and the system is indeed located in the Haldane-like state as detected in previous works on the $p$-wave Kondo lattice model, though the soluble non-interacting limit corresponds to the conventional $Z_{2}$ topological insulator. The site-resolved magnetization in an open boundary system and strange correlator for the periodic boundary have been used to identify the mentioned topological states. Interestingly, the edge magnetization in the Haldane-like state is …
Axion electrodynamics, first proposed in the context of particle physics, manifests itself in condensed matter physics in the topological field theory description of 3d topological insulators and gives rise to magnetoelectric effect, where applying magnetic (electric) field $\mathbf{B}(\mathbf{E})$ induces polarization (magnetization) $\mathbf{p}(\mathbf{m})$. We use linear response theory to study the associated topological current using the Fu-Kane-Mele model of 3d topological insulators in the presence of time-dependent uniform weak magnetic field. By computing the dynamical current suscept…
The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.
We calculate supercurrent across a two-dimensional topological insulator subjected to an external magnetic field. When the edge states of a narrow two-dimensional topological insulator are hybridized, an external magnetic field can close the hybridization gap, thus driving a quantum phase transition from insulator to semimetal states of the topological insulator. We find a sign reversal of the supercurrent at the quantum phase transition revealing intrinsic properties of topological insulators via Josephson effect.
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 paper, 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 f…
We construct a three-dimensional (3D), time-reversal symmetric generalization of the Chalker-Coddington network model for the integer quantum Hall transition. The novel feature of our network model is that in addition to a weak topological insulator phase already accommodated by the network model framework in the pre-existing literature, it hosts strong topological insulator phases as well. We unambiguously demonstrate that strong topological insulator phases emerge as intermediate phases between a trivial insulator phase and a weak topological phase. Additionally, we found a non-local transfo…
The discovery of the quantum spin Hall effect and topological insulators more than a decade ago has revolutionized modern condensed matter physics. Today, the field of topological states of matter is one of the most active and fruitful research areas for both experimentalists and theorists. The physics of topological insulators is typically well described by band theory and systems of non-interacting fermions. In contrast, several of the most fascinating effects in condensed matter physics merely exist due to electron-electron interactions, examples include unconventional superconductivity, th…
In recent years, the role of crystal symmetries in enriching the variety of TIs have been actively investigated. Higher-order TIs are a new type of topological crystalline insulators that exhibit gapless boundary states whose dimensionality is lower than those on the surface of conventional TIs. In this paper, relying on a concrete tight-binding model, we show that higher-order TIs can be smoothly connected to conventional TIs in a lower dimension without the bulk-gap closing or symmetry breaking. Our result supports the understanding of higher- order TIs as a stacking of lower-dimensional TIs…
We predict pseudo topological insulators that have been previously overlooked. We determine some conditions under which robust pseudo topological edge states appear and illustrate our idea on the Su-Schrieffer-Heeger (SSH) model with extra chiral symmetry breaking potentials. We discuss that pseudo topological insulating phase transition occurs without band gap closing.
Holographic classification of Topological Insulators and its 8-fold periodicity
使用一般的克利福德代数在任何空间维度的属性，我们明确分类狄拉克哈密顿与零模式的时间反演，粒子洞对称性和手征的离散对称性的保护。假设拓扑绝缘体的边界态是狄拉克费米子，我们由此全息再现了Kitaev和Ryu发现的拓扑绝缘体的周期表。et. al，不使用拓扑不变量，也不使用K-理论。此外，我们发现候选Z_2拓扑绝缘体在类A1，AII中的尺寸0，4 mod 8和类C，D中的尺寸2，6 mod 8。
Topological insulators (TIs) are expected to be a promising platform for novel quantum phenomena, whose experimental realizations require sophisticated devices. In this Technical Review, we discuss four topics of particular interest for TI devices: topological superconductivity, quantum anomalous Hall insulator as a platform for exotic phenomena, spintronic functionalities, and topological mesoscopic physics. We also discuss the present status and technical challenges in TI device fabrications to address new physics.
