eBook ISBN: | 978-1-4704-3707-7 |
Product Code: | MEMO/247/1173.E |
List Price: | $75.00 |
MAA Member Price: | $67.50 |
AMS Member Price: | $45.00 |
eBook ISBN: | 978-1-4704-3707-7 |
Product Code: | MEMO/247/1173.E |
List Price: | $75.00 |
MAA Member Price: | $67.50 |
AMS Member Price: | $45.00 |
-
Book DetailsMemoirs of the American Mathematical SocietyVolume: 247; 2017; 118 pp
The authors study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. They then show that the introduction of an “edge”, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized “edge states”. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The authors' model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.
-
Table of Contents
-
Chapters
-
1. Introduction and Outline
-
2. Floquet-Bloch and Fourier Analysis
-
3. Dirac Points of 1D Periodic Structures
-
4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States
-
5. Main Theorem—Bifurcation of Topologically Protected States
-
6. Proof of the Main Theorem
-
A. A Variant of Poisson Summation
-
B. 1D Dirac points and Floquet-Bloch Eigenfunctions
-
C. Dirac Points for Small Amplitude Potentials
-
D. Genericity of Dirac Points - 1D and 2D cases
-
E. Degeneracy Lifting at Quasi-momentum Zero
-
F. Gap Opening Due to Breaking of Inversion Symmetry
-
G. Bounds on Leading Order Terms in Multiple Scale Expansion
-
H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction
-
-
Additional Material
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Requests
The authors study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. They then show that the introduction of an “edge”, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized “edge states”. These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The authors' model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene. The states the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.
-
Chapters
-
1. Introduction and Outline
-
2. Floquet-Bloch and Fourier Analysis
-
3. Dirac Points of 1D Periodic Structures
-
4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States
-
5. Main Theorem—Bifurcation of Topologically Protected States
-
6. Proof of the Main Theorem
-
A. A Variant of Poisson Summation
-
B. 1D Dirac points and Floquet-Bloch Eigenfunctions
-
C. Dirac Points for Small Amplitude Potentials
-
D. Genericity of Dirac Points - 1D and 2D cases
-
E. Degeneracy Lifting at Quasi-momentum Zero
-
F. Gap Opening Due to Breaking of Inversion Symmetry
-
G. Bounds on Leading Order Terms in Multiple Scale Expansion
-
H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction