Disordered Topological Insulators: A Non-Commutative Geometry Perspective

Abstract

This review deals with strongly disordered topological insulators and covers some recent applications of a well established analytic theory based on the methods of Non-Commutative Geometry (NCG) and developed for the Integer Quantum Hall-Effect. Our main goal is to exemplify how this theory can be used to define topological invariants in the presence of strong disorder, other than the Chern number, and to discuss the physical properties protected by these invariants. Working with two explicit 2-dimensional models, one for a Chern insulator and one for a Quantum spin-Hall insulator, we first give an in-depth account of the key bulk properties of these topological insulators in the clean and disordered regimes. Extensive numerical simulations are employed here. A brisk but self-contained presentation of the non-commutative theory of the Chern number is given and a novel numerical technique to evaluate the non-commutative Chern number is presented. The non-commutative spin-Chern number is defined and the analytic theory together with the explicit calculation of the topological invariants in the presence of strong disorder are used to explain the key bulk properties seen in the numerical experiments presented in the first part of the review.

Topical Review for J. Phys. A

References in corpus (50)

• Electric Field Effect in Atomically Thin Carbon Films
• Two Dimensional Atomic Crystals
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Dissipationless Quantum Spin Current at Room Temperature
• First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
• Topological Surface States Protected From Backscattering by Chiral Spin Texture
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Nonlocal edge state transport in the quantum spin Hall state
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• The Quantum Spin Hall Effect: Theory and Experiment
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Large bulk resistivity and surface quantum oscillations in the topological insulator Bi2Te2Se
• Quantum Spin Hall Effect and Topologically Invariant Chern Numbers
• Gate-Voltage Control of Chemical Potential and Weak Anti-localization in Bismuth Selenide
• Quantum Spin Hall Effect and Enhanced Magnetic Response by Spin-Orbit Coupling
• Topological Anderson Insulator
• Coexistence of the topological state and a two-dimensional electron gas on the surface of Bi2Se3
• Bulk Fermi surface coexistence with Dirac surface state in Bi${}_2$Se${}_3$: a comparison of photoemission and Shubnikov-de Haas measurements
• Quantum spin Hall effect in a transition metal oxide Na2IrO3
• Strong surface scattering in ultrahigh mobility Bi2Se3 topological insulator crystals
• Exponential localization of Wannier functions in insulators
• Robustness of the Spin-Chern number
• SU(2) Non-Abelian Holonomy and Dissipationless Spin Current in Semiconductors
• On the $Z_2$ classification of Quantum Spin Hall Models
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Direct Evidence for the Dirac-Cone Topological Surface States in Ternary Chalcogenide TlBiSe2
• Transmission of topological surface states through surface barriers
• Single-Dirac-Cone topological surface states in TlBiSe2 class of Topological Insulators
• Topological surface states in three-dimensional magnetic insulators
• Tuning phase transition between quantum spin Hall and ordinary insulating phases
• Quantum Hall Effect of Dirac Fermions in Graphene: Disorder Effect and Phase Diagram
• Topological aspects of quantum spin Hall effect in graphene: Z${}_2$ topological order and spin Chern number
• Equality of the bulk and edge Hall conductances in a mobility gap
• Two-dimensional spin-filtered chiral network model for the Z_2 quantum spin-Hall effect
• Topological insulating behaviour in conducting property of crystalline Ge-Sb-Te
• Spin Hall effect of conserved current: Conditions for a nonzero spin Hall current
• Boundary criticality at the Anderson transition between a metal and a quantum spin Hall insulator in two dimensions
• Boundary criticality and multifractality at the 2D spin quantum Hall transition
• The edge spectrum of Chern insulators with rough boundaries
• Topological invariant in three-dimensional band insulators with disorder
• Topological meaning of Z${}_2$ numbers in time reversal invariant systems
• Spin Hall Effect of Excitons
• An edge index for the Quantum Spin-Hall effect
• Topological quantization of ensemble averages

Cited by in corpus (128)

• Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder
• Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport
• Scattering theory of topological insulators and superconductors
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• Disorder and metal-insulator transitions in Weyl semimetals
• K-Theory and Pseudospectra for Topological Insulators
• Vortexability: A Unifying Criterion for Ideal Fractional Chern Insulators
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• AIII and BDI Topological Systems at Strong Disorder
• Phase diagrams of disordered Weyl semimetals
• Anomalous Hall Effects of Light and Chiral Edge Modes on the Kagome Lattice
• Topological Anderson Insulator in electric circuits
• Topology in the Sierpiński-Hofstadter problem
• Three-dimensional phase diagram of disordered HgTe/CdTe Quantum spin-Hall wells
• Layer Hall effect induced by hidden Berry curvature in antiferromagnetic insulators
• Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models
• Topological Superconductivity in Skyrmion Lattices
• Hyperbolic Topological Band Insulators
• Local invariants identify topology in metals and gapless systems
• Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk
• Virtual Topological Insulators with Real Quantized Physics
• Real-space representation of winding number for one-dimensional chiral-symmetric topological insulator
• Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions
• Photonic ${\mathbb{Z}}_2$ topological Anderson insulators
• Disorder-Induced Multiple Transition involving Z2 Topological Insulator
• Quantized charge transport in chiral Majorana edge modes
• Detecting topological invariants in chiral symmetric insulators via losses
• Disorder-induced metal-insulator transitions in three-dimensional topological insulators and superconductors
• Topological phase transitions in (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ and (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$
• Real space topological invariant and higher-order topological Anderson insulator in two-dimensional non-Hermitian systems
• Localization, topology and quantized transport in disordered Floquet systems
• A noncommutative framework for topological insulators
• The topological Anderson insulator phase in the Kane-Mele model
• Quantum Anomalous Layer Hall Effect in the Topological Magnet MnBi2Te4
• Universal topological marker
• The non-commutative n-th Chern number
• Effect of Strong Disorder in a 3-Dimensional Topological Insulator: Phase Diagram and Maps of the Z2 Invariant
• Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter
• On the Bott index of unitary matrices on a finite torus
• Criticality of the metal-topological insulator transition driven by disorder
• Effects of random domains on the zero Hall plateau in quantum anomalous Hall effect
• Quantum transport in disordered systems under magnetic fields: A study based on operator algebras
• Notes on topological insulators
• ${\mathbb{Z}}_2$ invariants of topological insulators as geometric obstructions
• Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker
• Fermi arc reconstruction at junctions between Weyl semimetals
• Half-Quantized Hall Effect at the Parity-Invariant Fermi Surface
• Kibble-Zurek behavior in disordered Chern insulators
• Machine learning spectral indicators of topology
• Manifestly gauge independent formulations of the Z2 invariants
• The Global Phase Diagram of disordered Higher-order Weyl Semimetals
• Construction of real-valued localized composite Wannier functions for insulators
• Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
• Evolutions of helical edge states in disordered HgTe/CdTe quantum wells
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• Topologically Protected Extended States in Disordered Quantum Spin-Hall Systems without Time-Reversal Symmetry
• Quantum criticality at the Chern-to-normal insulator transition
• Chiral Interface States and Related Quantized Transport in Disordered Chern Insulators
• Effect of Strong Disorder on 3-Dimensional Chiral Topological Insulators: Phase Diagrams, Maps of the Bulk Invariant and Existence of Topological Extended Bulk States
• Anomalous magneto-transport in disordered structures: classical edge-state percolation
• Dimensional Tuning of Majorana Fermions and the Real Space Counting of the Chern Number
• Disordered Chern insulator with a two step Floquet drive
• Time-dependent topological systems: A study of the Bott index
• Quantum Engineering of Majorana Fermions
• Evolution of Berry curvature and reentrant quantum anomalous Hall effect in an intrinsic magnetic topological insulator
• Disorder effects in correlated topological insulators
• Robustness of topological order against disorder
• Topological marker approach to an interacting Su-Schrieffer-Heeger model
• Measuring Chern numbers in Hofstadter strips
• The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields
• Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker
• A Non-Commutative Formula for the Isotropic Magneto-Electric Response
• Phase diagrams of Weyl semimetals with competing intraorbital and interorbital disorders
• Robustness of chiral edge modes in fractal-like-lattices below two dimensions: A case study
• Identifying Axion Insulator by Quantized Magnetoelectric Effect in Antiferromagnetic ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ Tunnel Junction
• Delocalization of boundary states in disordered topological insulators
• Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism
• Spin fluctuations in quantized transport of magnetic topological insulators
• Fragility of surface states in non-Wigner Dyson topological insulators
• Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields
• Local formula for the $Z_2$ invariant of topological insulators
• Topological critical states and anomalous electronic transmittance in one dimensional quasicrystals
• Anomalous Hall metals from strong disorder in class A systems on partite lattices
• Nonlinear Hall effect on a disordered lattice
• Metal--topological-insulator transition in the quantum kicked rotator with Z2 symmetry
• Anatomy of topological Anderson transitions
• Characterization of the Quantized Hall Insulator Phase in the Quantum Critical Regime
• Optical absorption measurement of spin Berry curvature and spin Chern marker
• Dielectric and optical markers originating from quantum geometry
• Stability of Disordered Topological Superconducting Phases in Magnet--Superconductor Hybrid Systems
• Robust measurement of wave function topology on NISQ quantum computers
• Temperature-driven gapless topological insulator
• Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae
• Nonlocal Moments in the Chern Bands of Twisted Bilayer Graphene
• From topological phase to Anderson localization in a two-dimensional quasiperiodic system
• Floquet Exceptional Topological Insulator
• Anomalous levitation and annihilation in Floquet topological insulators
• Interacting Local Topological Markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states
• Topologically localized insulators
• Disorder-induced phase transitions in double Weyl semimetals
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• Evidence for Half-Quantized Chiral Edge Current in a C = 1/2 Parity Anomaly State
• Emergent $Z_2$ Topological Invariant and Robust Helical Edge States in Two-Dimensional Topological Metals
• Almost commuting self-adjoint matrices --- the real and self-dual cases
• Almost commuting unitary matrices related to time reversal
• Random-Flux-Induced Transition Sequence between Weak and Strong Topological Phases with Anisotropic Localization Properties
• Adiabatic charge transport in extended SSH models
• Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
• Topology of ultra-localized insulators and superconductors
• Kosterlitz-Thouless transition in disordered two-dimensional topological insulators
• Half-Quantized Hall Metal and Marginal Metal in Disordered Magnetic Topological Insulators
• Fate of Quadratic Band Crossing under quasiperiodic modulation
• Disorder and quantum transport of the helical quantum Hall phase in graphene
• Quantum simulation of quantum mechanical system with spatial noncommutativity
• Extended Haldane model -- a modern gateway to topological insulators
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors
• Regularized universal topological markers for Dirac systems
• Gapless Edge-Modes and Topology in the Qi-Wu-Zhang Model: A Real-Space Analysis
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator
• Photonic topological Anderson insulator in a two-dimensional atomic lattice
• Interplay of Disorder and Point-Gap Topology: Chiral Modes, Localization and Non-Hermitian Anderson Skin Effect in One Dimension
• Effects of dilution in a 2D topological magnon insulator
• Dynamics of transport by helical edge states
• Finite-rate quench in disordered Chern and $Z_2$ topological insulators
• Topological transition induced by selective random defects on a honeycomb lattice
• Disordered Parity Anomalous Semimetal
• Disorder-Induced Topological Phases in a Two-Dimensional Chern Insulator with Strong Magnetic Disorder
• Disorder averaging in random lattice models with periodic boundary conditions: Application to models with uncorrelated and correlated disorder