Honeycomb Schrödinger Operators in the Strong Binding Regime

Charles L. Fefferman; James P. Lee‐Thorp; Michael I. Weinstein

doi:10.1002/cpa.21735

journal article Dec 18, 2017

Honeycomb Schrödinger Operators in the Strong Binding Regime

Charles L. Fefferman James P. Lee‐Thorp Michael I. Weinstein

Communications on Pure and Applied Mathematics Vol. 71 No. 6 pp. 1178-1270 · Wiley

View at Publisher Save 10.1002/cpa.21735

Abstract

In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet‐Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two‐band tight‐binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break 𝒫𝒯 symmetry and (b) the existence of topologically protected edge states—states that propagate parallel to and are localized transverse to a line defect or “edge”—for a large class of rational edges, and that are robust to a class of large transverse‐localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight‐binding model emerges in an extreme parameter limit.© 2017 Wiley Periodicals, Inc.

Topics

No keywords indexed for this article. Browse by subject →

References

61

[1]

10.1111/j.1467-9590.2012.00558.x

[2]

Ashcroft N. W. (1976)

[3]

10.1016/0003-4916(78)90143-4

[4]

10.1364/ol.33.002251

[5]

10.1103/physrevlett.110.033902

[6]

Berkolaiko G.;Comech A.Symmetry and Dirac points in graphene spectrum. Preprint 2014. arxiv:1412.8096 [math‐ph]

[7]

Colin de Verdière Y. "Sur les singularités de van Hove génériques. Analyse globale et physique mathématique (Lyon, 1989)" Mém. Soc. Math. France (N.S.) (1991) 10.24033/msmf.356

[8]

Cycon H. L. (1987)

[9]

10.1103/physrevb.84.195452

[10]

10.2478/nsmmt-2013-0007

[11]

Eastham M. S. P. (1973)

[12]

10.1007/s00220-002-0698-z

[13]

10.1007/s00220-005-1369-7

[14]

10.1073/pnas.1407391111

[15]

10.1088/2053-1583/3/1/014008

[16]

10.1007/s40818-016-0015-3

[17]

Topologically Protected States in One-Dimensional Systems

C. Fefferman, J. Lee-Thorp, M. Weinstein

Memoirs of the American Mathematical Society 10.1090/memo/1173

[18]

Honeycomb lattice potentials and Dirac points

Charles Fefferman, Michael Weinstein

Journal of the American Mathematical Society 10.1090/s0894-0347-2012-00745-0

[19]

Wave Packets in Honeycomb Structures and Two-Dimensional Dirac Equations

Charles L. Fefferman, Michael I. Weinstein

Communications in Mathematical Physics 10.1007/s00220-013-1847-2

[20]

10.1137/s0036139994263859

[21]

10.1137/s0036139995285236

[22]

The rise of graphene

A. K. Geim, K. S. Novoselov

Nature Materials 10.1038/nmat1849

[23]

Gerard C. "Resonance theory for periodic Schrodinger operators" Bull. Soc. Math. France (1990) 10.24033/bsmf.2134

[24]

10.1006/jfan.1997.3154

[25]

10.1007/s00220-009-0910-5

[26]

10.1007/s00220-012-1444-9

[27]

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Gian Michele Graf, Marcello Porta

Communications in Mathematical Physics 10.1007/s00220-013-1819-6

[28]

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

V. V. Grushin

Mathematical Notes 10.1134/s0001434609110194

[29]

Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry

F. D. M. Haldane, S. Raghu

Physical Review Letters 10.1103/physrevlett.100.013904

[30]

Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential

B. I. Halperin

Physical Review B 10.1103/physrevb.25.2185

[31]

Chern number and edge states in the integer quantum Hall effect

Yasuhiro Hatsugai

Physical Review Letters 10.1103/physrevlett.71.3697

[32]

Hempel R. (2003)

[33]

10.1017/cbo9781139031080

[34]

10.1007/bfb0094264

[35]

10.1038/ncomms9260

[36]

Photonic topological insulators

Alexander B. Khanikaev, S. Hossein Mousavi, Wang-Kong Tse et al.

Nature Materials 10.1038/nmat3520

[37]

10.1007/978-3-0348-8573-7

[38]

10.1090/bull/1528

[39]

10.1007/s00220-007-0316-1

[40]

Ma T.;Khanikaev A. B.;Mousavi S. H.;Shvets G.Topologically protected photonic transport in bi‐anisotropic meta‐waveguide. Preprint 2014. arxiv:1401.1276 [physics.optics]

[41]

10.1088/0305-4470/32/10/015

[42]

10.1103/physrevb.80.153412

[43]

The electronic properties of graphene

A. H. Castro Neto, F. Guinea, N. M. R. Peres et al.

Reviews of Modern Physics 10.1103/revmodphys.81.109

[44]

10.1007/s00220-008-0640-0

[45]

10.1038/nmat3783

[46]

Analogs of quantum-Hall-effect edge states in photonic crystals

S. Raghu, F. D. M. Haldane

Physical Review A 10.1103/physreva.78.033834

[47]

10.1103/physrevlett.111.103901

[48]

10.1038/nature12066

[49]

Reed M. (1978)

[50]

10.1016/0003-4916(76)90038-5

Showing 50 of 61 references

Cited By

72

Instability of Quadratic Band Degeneracies and the Emergence of Dirac Points

J. Chaban, Michael I. Weinstein · 2025

SIAM Journal on Mathematical Analys...

Modeling of Electronic Dynamics in Twisted Bilayer Graphene

Tianyu Kong, Diyi Liu · 2024

SIAM Journal on Applied Mathematics

Bistritzer–MacDonald dynamics in twisted bilayer graphene

Alexander B. Watson, Tianyu Kong · 2023

Journal of Mathematical Physics

Metrics

72

Citations

61

References

Details

Published: Dec 18, 2017
Vol/Issue: 71(6)
Pages: 1178-1270
License: View

Authors

C

Charles L. Fefferman

Department of Mathematics Princeton University Princeton NJ USA

J

James P. Lee‐Thorp

Courant Institute 251 Mercer St., New York NY 10012 USA

M

Michael I. Weinstein

Department of Applied Physics and Applied Mathematics and Department of Mathematics Columbia University New York NY 10027 USA

Cite This Article

Charles L. Fefferman, James P. Lee‐Thorp, Michael I. Weinstein (2017). Honeycomb Schrödinger Operators in the Strong Binding Regime. Communications on Pure and Applied Mathematics, 71(6), 1178-1270. https://doi.org/10.1002/cpa.21735

Honeycomb Schrödinger Operators in the Strong Binding Regime

You May Also Like