The bulk-edge correspondence in three simple cases

@article{Shapiro2017TheBC,
  title={The bulk-edge correspondence in three simple cases},
  author={Jacob Shapiro},
  journal={Reviews in Mathematical Physics},
  year={2017}
}
  • Jacob Shapiro
  • Published 29 October 2017
  • Mathematics, Physics
  • Reviews in Mathematical Physics
We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate the mechanism behind the bulk-edge principle without the overhead of the more general proofs which are available. We also give a new formula for the [Formula: see text]-index of our time-reversal invariant systems inspired by Moore and Balents. 
View PDF on arXiv
16 Citations
Highly Influential Citations
1
Background Citations
7

Figures from this paper

16 Citations

Bulk-interface correspondences for one dimensional topological materials with inversion symmetry

The interface between two materials described by spectrally gapped Hamiltonians is expected to host an in-gap interface mode, whenever a certain topological invariant changes across the interface. We…

Chiral flow in one-dimensional Floquet topological insulators

We propose a bulk topological invariant for one-dimensional Floquet systems with chiral symmetry which quantifies the particle transport on each sublattice during the evolution. This chiral flow is…

Effects of geometric frustration in Kitaev chains

We study the topological phase transitions of a Kitaev chain frustrated by the addition of a single long-range hopping. In order to study the topological properties of the resulting legged-ring…

Topology and Localization: Mathematical Aspects of Electrons in Strongly-Disordered Media

Topological insulators are usually studied in physics under the assumption of translation invariance, which allows for the usage of Bloch decomposition. Mathematically vector bundle theory over the…

Non-Hermitian Floquet Phases with Even-Integer Topological Invariants in a Periodically Quenched Two-Leg Ladder

This work introduces an experimentally realizable two-leg ladder model subjecting to both time-periodic quenches and non-Hermitian effects, which belongs to an extended CII symmetry class, and constructs a generalized version of the mean chiral displacement, which could be employed as a dynamical probe to the topological invariants of non- hermitian Floquet phases in the C II symmetry class.

Characterization of edge states in perturbed honeycomb structures

  • A. Drouot
  • Physics, Mathematics
    Pure and Applied Analysis
  • 2019
This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by work of Haldane-Raghu , we continue the study of a linear PDE initiated in papers of…

Real-space representation of the winding number for a one-dimensional chiral-symmetric topological insulator

The winding number has been widely used as an invariant for diagnosing topological phases in one-dimensional chiral-symmetric systems. We put forward a real-space representation for the winding…

Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence

Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding…

Topological squashed entanglement: Nonlocal order parameter for one-dimensional topological superconductors

Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the…

Squashed entanglement: order parameter for topological superconductors

Identifying entanglement-based order parameters characterizing topological systems has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that squashed…

References

SHOWING 1-10 OF 22 REFERENCES

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle…

The K-Theoretic Bulk–Edge Correspondence for Topological Insulators

We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge…

Controlled Topological Phases and Bulk-edge Correspondence

In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with…

Edge states and the bulk-boundary correspondence in Dirac Hamiltonians

We present an analytic prescription for computing the edge dispersion E(k) of a tight-binding Dirac Hamiltonian terminated at an abrupt crystalline edge. Specifically, we consider translationally…

Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics

Illustration of key concepts in dimension d = 1.- Topological solid state systems: conjectures, experiments and models.- Observables algebras for solid state systems.- K-theory for topological solid…

LETTER TO THE EDITOR: Simultaneous quantization of edge and bulk Hall conductivity

The edge Hall conductivity is shown to be an integer multiple of e 2 /h which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity…

Equality of Bulk and Edge Hall Conductance Revisited

Abstract: The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to different…

Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when…

Periodic table for topological insulators and superconductors

Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a…

Classification of topological insulators and superconductors in three spatial dimensions

We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial…