A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states

Abstract

The one-to-one relation between the winding number and the number of robust zero-energy edge states, known as bulk-boundary correspondence, is a celebrated feature of 1d systems with chiral symmetry. Although this property can be explained by the K-theory, the underlying mechanism remains elusive. Here, we demonstrate that, even without resorting to advanced mathematical techniques, one can prove this correspondence and clearly illustrate the mechanism using only Cauchy's integral and elementary algebra. Furthermore, our approach to proving bulk-boundary correspondence also provides clear insights into a kind of system that doesn't respect chiral symmetry but can have robust left or right zero-energy edge states. In such systems, one can still assign the winding number to characterize these zero-energy edge states.

arXiv commenta new section, various improvements made

References in corpus (26)

• Topological Insulators
• Topological insulators and superconductors
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Topological Insulators in Three Dimensions
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Classification of topological quantum matter with symmetries
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Periodic table for topological insulators and superconductors
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Topological Origin of Zero-Energy Edge States in Particle-Hole Symmetric Systems
• Stability of the quantum spin Hall effect: effects of interactions, disorder, and Z_2 topology
• The Zak phase and the existence of edge states in graphene
• Edge states and the bulk-boundary correspondence in Dirac Hamiltonians
• Bulk-boundary correspondence of topological insulators from their Green's functions
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Bulk-edge correspondence for two-dimensional topological insulators
• Boundary Green functions of topological insulators and superconductors
• Symmetry protected topological phases in non-interacting fermion systems
• Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models
• Bulk-edge correspondence in graphene with/without magnetic field: Chiral symmetry, Dirac fermions and Edge states
• An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model
• Winding number and Zak phase in multi-band SSH models
• The Bulk-Edge Correspondence in Three Simple Cases
• Topology of critical chiral phases: multiband insulators and superconductors

Cited by in corpus (1)

• Topological edge states of the hexagonal linear chain