Topological Invariants of Edge States for Periodic Two-Dimensional Models

@article{Avila2013TopologicalIO,
  title={Topological Invariants of Edge States for Periodic Two-Dimensional Models},
  author={Julio Avila and Hermann Schulz-Baldes and Carlos Villegas-Blas},
  journal={Mathematical Physics, Analysis and Geometry},
  year={2013},
  volume={16},
  pages={137-170}
}
Transfer matrix methods and intersection theory are used to calculate the bands of edge states for a wide class of periodic two-dimensional tight-binding models including a sublattice and spin degree of freedom. This allows to define topological invariants by considering the associated Bott–Maslov indices which can be easily calculated numerically. For time-reversal symmetric systems in the symplectic universality class this leads to a ${\mathbb Z}_2$-invariant for the edge states. It is shown… Expand

Figures from this paper

Topological Invariants and Corner States for Hamiltonians on a Three-Dimensional Lattice
We consider periodic Hamiltonians on a three-dimensional (3-D) lattice with a spectral gap not only on the bulk but also on two edges at the common Fermi level. By using K-theory applied for theExpand
Krein signatures of transfer operators for half-space topological insulators
We propose a complementary point of view on the topological invariants of two-dimensional tight-binding models restricted to half-spaces. The transfer operators for such systems are $J$-unitary on aExpand
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 withExpand
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-particleExpand
Topologically Protected States in One-Dimensional Systems
We study a class of periodic Schr\"odinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points". We then show that the introduction of an "edge", viaExpand
Topological edge states in two-gap unitary systems: a transfer matrix approach
We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate inExpand
Dimensional Reduction and Scattering Formulation for Even Topological Invariants
Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the CayleyExpand
INVARIANTS OF TOPOLOGICAL INSULATORS AS GEOMETRIC OBSTRUCTIONS
We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i. e. the time-reversal operator squares to −1. We investigate the existence of periodic andExpand
A non-commutative framework for topological insulators
We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutativeExpand
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 edgeExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 27 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-particleExpand
Quantum spin-Hall effect and topologically invariant Chern numbers.
TLDR
It is shown that the topology of the band insulator can be characterized by a 2 x 2 matrix of first Chern integers, and the nontrivial QSHE phase is identified by the nonzero diagonal matrix elements of the Chern number matrix (CNM). Expand
EDGE CURRENT CHANNELS AND CHERN NUMBERS IN THE INTEGER QUANTUM HALL EFFECT
A quantization theorem for the edge currents is proven for discrete magnetic half-plane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under aExpand
Z2 topological order and the quantum spin Hall effect.
TLDR
The Z2 order of the QSH phase is established in the two band model of graphene and a generalization of the formalism applicable to multiband and interacting systems is proposed. Expand
Chern number and edge states in the integer quantum Hall effect.
  • Hatsugai
  • Physics, Medicine
  • Physical review letters
  • 1993
TLDR
The relation between two different interpretations of the Hall conductance as topological invariants is clarified and it is found that vortices are given by the edge states when they are degenerate with the bulk states. Expand
Numerical study of symmetry effects on localization in two dimensions.
  • Ando
  • Physics, Medicine
  • Physical review. B, Condensed matter
  • 1989
TLDR
Effects of symmetry on localization on two-dimensional square lattices are studied numerically and the critical randomness and exponent for a metal-insulator transition are determined in the presence of strong spin-orbit interactions. Expand
Topological analysis of the quantum Hall effect in graphene: Dirac-Fermi transition across van Hove singularities and edge versus bulk quantum numbers
Inspired by a recent discovery of a peculiar integer quantum Hall effect (QHE) in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with two interests. First,Expand
Quantized Hall conductance, current carrying edge states, and the existence of extended states in a two-dimensional disordered potential
When a conducting layer is placed in a strong perpendicular magnetic field, there exist current-carrying electron states which are localized within approximately a cyclotron radius of the sampleExpand
Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications toExpand
Flat Bands of a Tight-Binding Electronic System with Hexagonal Structure
A constructive method to find flat bands is demonstrated for a model describing tight-binding electrons on the hexagonal lattice in which each atomic site has three orbitals. The system may representExpand
...
1
2
3
...