Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect

Abstract

For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.

35 pages, 2 figures

References in corpus (5)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Exponential localization of Wannier functions in insulators
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Classification of Topological Insulators and Superconductors
• Topology and Phases in Fermionic Systems

Cited by in corpus (56)

• Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder
• Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator
• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice
• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• K-Theory and Pseudospectra for Topological Insulators
• Chiral-Symmetric Higher-Order Topological Phases of Matter
• Topological Insulators and C^*-Algebras: Theory and Numerical Practice
• Theory of spin Bott index for quantum spin Hall states in non-periodic systems
• Real-space representation of winding number for one-dimensional chiral-symmetric topological insulator
• Topological states in quasicrystals
• Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons
• The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors
• A noncommutative framework for topological insulators
• Matrix embeddings on flat $R^3$ and the geometry of membranes
• Extraction of many-body Chern number from a single wave function
• The $Z_2$ Index of Disordered Topological Insulators with Time Reversal Symmetry
• On the Bott index of unitary matrices on a finite torus
• Floquet topological insulator phase in a Weyl semimetal thin film with disorder
• Disorder induced transitions in resonantly driven Floquet Topological Insulators
• Localisation, delocalisation, and topological transitions in disordered 2D quantum walks
• Long-range Kitaev Chains via Planar Josephson Junctions
• Amorphous Bi${}_2$Se${}_3$ structural, electronic, and topological nature by first-principles
• Stability of flat-band edge states in topological superconductors without inversion center
• Time-dependent topological systems: A study of the Bott index
• An Exact Chiral Amorphous Spin Liquid
• The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields
• Local Chern Marker for Periodic Systems
• Local formula for the $Z_2$ invariant of topological insulators
• Topological charge pumping in quasiperiodic systems characterized by Bott index
• Topological invariants for interacting systems: from twisted boundary condition to center-of-mass momentum
• Single-point spin Chern number in a supercell framework
• Nonlinear Hall effect on a disordered lattice
• Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higher
• Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• Wave-packet propagation in a finite topological insulator and the spectral localizer index
• Programmable Hamiltonian engineering with quadratic quantum Fourier transform
• Even spheres as joint spectra of matrix models
• Photocontrol of spin scalar chirality in centrosymmetric itinerant magnets
• Almost commuting self-adjoint matrices --- the real and self-dual cases
• Almost commuting unitary matrices related to time reversal
• Characterization of higher-order topological superconductors using Bott indices
• Adiabatic charge transport in extended SSH models
• Predicting topological invariants and unconventional superconducting pairing from density of states and machine learning
• Fluctuations and Correlations of Local Topological Order Parameters in Disordered Two-dimensional Topological Insulators
• Regularized universal topological markers for Dirac systems
• Adiabatic charge transport through non-Bloch bands
• Three-Dimensional Quantum Hall Effect in Topological Amorphous Metals
• Disorder-Induced Topological Phases in a Two-Dimensional Chern Insulator with Strong Magnetic Disorder
• Coarse geometric approach to topological phases: Invariants from real-space representations
• Projected branes as platforms for crystalline, superconducting, and higher-order topological phases
• Localization and topological signatures under periodic twisting
• Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator
• Topological Invariant for Bosonic Bogoliubov-de Gennes Systems with Disorder