Non-Commutative Tools for Topological Insulators

Abstract

This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have non-trivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, relation that can be used to investigate the robustness of the edge states against disorder. The paper focuses on the motivations behind creating such tools and on how to use them.

Final version (some arguments were corrected)

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Cited by in corpus (64)

• Non-Hermitian Topological Invariants in Real Space
• Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials
• Monolayer Topological Insulators: Silicene, Germanene and Stanene
• Time-reversal-symmetry-broken quantum spin Hall effect
• Amorphous topological insulators constructed from random point sets
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• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models
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• Locality of the anomalous Hall conductivity
• Stabilization of Quantum Spin Hall Effect by Designed Removal of Time-Reversal Symmetry of Edge States
• A noncommutative framework for topological insulators
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• Topological phases of Bi(111) bilayer in an external exchange field
• Universal topological marker
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• Interaction effects on topological phase transitions via numerically exact quantum Monte Carlo calculations
• Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter
• Topological mechanics in quasicrystals
• Amorphous Bi${}_2$Se${}_3$ structural, electronic, and topological nature by first-principles
• Topologically Protected Extended States in Disordered Quantum Spin-Hall Systems without Time-Reversal Symmetry
• Photo-induced electronic and spin topological phase transitions in monolayer bismuth
• A Topological Spin Chern Pump
• Embedded Topological Insulators
• Robustness of topological order against disorder
• Topological marker approach to an interacting Su-Schrieffer-Heeger model
• Probing Chern number by opacity and topological phase transition by a nonlocal Chern marker
• Local Chern Marker for Periodic Systems
• Eight fold quantum Hall phases in a time reversal symmetry broken tight binding model
• Non-adiabatic topological spin pumping
• Local formula for the $Z_2$ invariant of topological insulators
• Entanglement spectrum and entropy in Floquet topological matter
• Topological superconductivity in Fibonacci quasicrystals
• Dielectric and optical markers originating from quantum geometry
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Optical absorption measurement of spin Berry curvature and spin Chern marker
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
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• Topological phase transitions with and without energy gap closing
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• Theory of local ${\mathbb{Z}}_2$ topological markers for finite and periodic two-dimensional systems
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• Quantum geometric origin of the Meissner effect and superfluid weight marker
• Robust propagating in-gap modes due to spin-orbit domain walls in graphene
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• Topological zero-modes of the spectral localizer of trivial metals
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• Kosterlitz-Thouless transition in disordered two-dimensional topological insulators
• A toy model for quantum spin Hall effect
• Entanglement, subsystem particle numbers and topology in free fermion systems
• Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators
• Adiabatic charge transport through non-Bloch bands
• Regularized universal topological markers for Dirac systems
• Real-space representation of the second Chern number
• Extended Haldane model -- a modern gateway to topological insulators
• Topological transition induced by selective random defects on a honeycomb lattice
• Out-of-plane spin polarization of edge currents in Chern insulator with Rashba spin-orbit interaction
• Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator
• Disorder-Induced Topological Phases in a Two-Dimensional Chern Insulator with Strong Magnetic Disorder