Non-Commutative Chern Numbers for Generic Aperiodic Discrete Systems

Abstract

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc. (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.

44 pages, 4 figures. v2: typos corrected and references updated. v3: Minor changes, to appear in J. Phys. A (Mathematical and Theoretical)

References in corpus (22)

• 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
• 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 quantum chemistry
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Topological classification of crystalline insulators through band structure combinatorics
• Observation of Topological Phase Transitions in Photonic Quasicrystals
• Topological modes bound to dislocations in mechanical metamaterials
• Amorphous topological insulators constructed from random point sets
• Topological Pumping over a Photonic Fibonacci Quasicrystal
• Topological Insulators in Random Lattices
• Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport
• Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -
• Measurement of a topological edge invariant in a microwave network
• Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal
• Virtual Topological Insulators with Real Quantized Physics
• Hofstadter butterfly of a quasicrystal
• Disordered Crystals from First Principles I: Quantifying the Configuration Space
• On the Role of Symmetries in the Theory of Photonic Crystals
• Application of semifinite index theory to weak topological phases

Cited by in corpus (40)

• Observation of topological edge modes in a quasi-periodic acoustic waveguide
• Amorphous photonic topological insulator
• Topological Amorphous Metals
• Structural Disorder Induced Second-order Topological Insulators in Three Dimensions
• Topological Weaire-Thorpe models of amorphous matter
• Topological states on fractal lattices
• Local invariants identify topology in metals and gapless systems
• Continuous bulk and interface description of topological insulators
• Topological Edge Modes by Smart Patterning
• Amorphous topological matter: theory and experiment
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Topological Gaps by Twisting
• Symmetry-Protected Topological Phases in a Rydberg Glass
• Topological phase transitions in glassy quantum matter
• Topological Classification Table Implemented with Classical Passive Meta-Materials
• Structural and electronic properties of realistic two-dimensional amorphous topological insulators
• Bulk Spectrum and K-theory for Infinite-Area Topological Quasicrystal
• Topological Anderson insulators in an Ammann-Beenker quasicrystal and a snub-square crystal
• The K-Theoretic Bulk-Boundary Principle for Dynamically Patterned Resonators
• Obstructed insulators and flat bands in topological phase-change materials
• Average Symmetry Protected Higher-order Topological Amorphous Insulators
• Origin and localization of topological band gaps in gyroscopic metamaterials
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Density-driven higher-order topological phase transitions in amorphous solids
• Rational Approximations of Quasi-Periodic Problems via Projected Green's Functions
• Topological Gaps in Quasi-Periodic Spin Chains: A Numerical and K-Theoretic Analysis
• Localization via Quasi-Periodic Bulk-Bulk Correspondence
• Structural spillage: an efficient method to identify non-crystalline topological materials
• Disordered Crystals from First Principles II: Transport Coefficients
• Calculations of Chern number: equivalence of real-space and twisted-boundary-condition formulae
• Index theory and topological phases of aperiodic lattices
• Quantum Hall effect and Landau levels without spatial long-range correlations
• A groupoid approach to interacting fermions
• Topological Lattice Defects by Groupoid Methods and Kasparov's KK-Theory
• Localised module frames and Wannier bases from groupoid Morita equivalences
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• Three-Dimensional Quantum Hall Effect in Topological Amorphous Metals
• Regularized universal topological markers for Dirac systems
• Robustness of topological phases on aperiodic lattices
• A Universal Chern Model on Arbitrary Triangulations