Equality of bulk and edge Hall conductances for continuous magnetic random Schrödinger operators

Taarabt, Amal

doi:10.48550/arxiv.1403.7767

Cited by 10 publications

(14 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The former is a topological quantity that can be computed from the bulk media, while the latter is related to the number of edge modes supported by the structure. A variety of tools have been developed for the study of the bulk-edge correspondence in different settings, including K-theory, functional analysis, and microlocal analysis, etc [7,8,11,13,15,16,22,24,35,36]. The third one is the stability of the interface modes supported by the topological materials.…”

Section: Introductionmentioning

confidence: 99%

Mathematical theory for topological photonic materials in one dimension

Lin,

Zhang

2021

Preprint

View full text Add to dashboard Cite

This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence and stability of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, the associated Zak phase (or Berry phase) may not be quantized. We investigate the existence of an interface mode which is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both timereversal and inversion symmetry, the Zak phase is quantized, taking only two values 0, π. We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm-Liouville operators. The methods and results can be extended to general topological Sturm-Liouville systems in one dimension.

show abstract

Section: Introductionmentioning

confidence: 99%

Mathematical theory for topological photonic materials in one dimension

Lin,

Zhang

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The bulk-edge correspondence has been proved in a growing number of discrete settings [EGS05, ASV13, GP13, PS16, Ba17, Sh17, Br18a, GS18, GT18, ST18]; see [FC13,AOP16] for a good introduction. There is fewer work on continuous systems: see [KS04a,KS04b,Ta14] for the quantum Hall effect; [FSF12,Ba17,Ba18] for an analysis on Dirac operators; and [BR18] for a K-theory approach. Here, we prove the bulk-edge correspondence in a new continuous setting.…”

Section: Introductionmentioning

confidence: 99%

The bulk-edge correspondence for continuous honeycomb lattices

Drouot

2019

Communications in Partial Differential Equations

View full text Add to dashboard Cite

We study bulk/edge aspects of continuous honeycomb lattices in a magnetic field. We compute the bulk index of Bloch eigenbundles: it equals 2 or −2, with sign depending on nearby Dirac points and on the magnetic field. We then prove the existence of two topologically protected unidirectional waves propagating along line defects. This shows the bulk/edge correspondence for our class of Hamiltonians.

show abstract

“…The extensive physics literature on topologically robust edge states goes back to investigations of the quantum Hall effect; see, for example, [20,21,41,44] and the rigorous mathematical articles [7,8,30,40]. In [19,34] a proposal for realizing photonic edge states in periodic electromagnetic structures which exhibit the magneto-optic effect was made.…”

Section: Introduction and Outlinementioning

confidence: 99%

Edge States in Honeycomb Structures

2016

View full text Add to dashboard Cite

An edge state is a time-harmonic solution of a conservative wave system, e.g. Schrödinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge". Topologically protected edge states are edge states which are stable against spatially localized (even strong) deformations of the edge. First studied in the context of the quantum Hall effect, protected edge states have attracted huge interest due to their role in the field of topological insulators. Theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. In this paper we consider a rich family of continuum PDE models for which we rigorously study regimes where topologically protected edge states exist. Our model is a class of Schrödinger operators on R 2 with a background two-dimensional honeycomb potential perturbed by an "edge-potential". The edge potential is a domain-wall interpolation, transverse to a prescribed "rational" edge, between two distinct periodic structures. General conditions are given for the bifurcation of a branch of topologically protected edge states from Dirac points of the background honeycomb structure. The bifurcation is seeded by the zero mode of a one-dimensional effective Dirac operator. A key condition is a spectral no-fold condition for the prescribed edge. We then use this result to prove the existence of topologically protected edge states along zigzag edges of certain honeycomb structures. Our results are consistent with the physics literature and appear to be the first rigorous results on the existence of topologically protected edge states for continuum 2D PDE systems describing waves in a non-trivial periodic medium. We also show that the family of Hamiltonians we study contains cases where zigzag edge states exist, but which are not topologically protected.

show abstract

Equality of bulk and edge Hall conductances for continuous magnetic random Schrödinger operators

Cited by 10 publications

References 21 publications

Mathematical theory for topological photonic materials in one dimension

Mathematical theory for topological photonic materials in one dimension

The bulk-edge correspondence for continuous honeycomb lattices

Edge States in Honeycomb Structures

Contact Info

Product

Resources

About