Bulk-Edge Correspondence for Chern Topological Phases: A Viewpoint from a Generalized Index Theorem

@article{Fukui2012BulkEdgeCF,
  title={Bulk-Edge Correspondence for Chern Topological Phases: A Viewpoint from a Generalized Index Theorem},
  author={Takahiro Fukui and Ken Shiozaki and Takanori Fujiwara and Satoshi Fujimoto},
  journal={Journal of the Physical Society of Japan},
  year={2012},
  volume={81},
  pages={114602}
}
We explore the bulk-edge correspondence for topological insulators (superconductors) without time-reversal symmetry from the viewpoint of the index theorem for open spaces. We assume generic Hamiltonians not only with a linear dispersion but also with higher order derivatives arising from generic band structures. Using a generalized index theorem valid for such systems, we show the equivalence between the spectral flow of the edge states and the Chern numbers specifying the bulk systems. 
Symmetry protected weak topological phases in a superlattice
We explore novel topological phases realized in a superlattice system based on the Wilson–Dirac model. Our main focus is on a two-dimensional analogue of weak topological insulator phases. We find
The Bulk-Edge Correspondence in Quantum Hall and Topological Insulator Systems
This text documents work done towards a master thesis in mathematical physics at the ETH Zurich physics department from October 2014 until April 2015. We study two-band and four-band systems and
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
Berry-Chern monopoles and spectral flows
This lecture note adresses the correspondence between spectral flows, often associated to unidirectional modes, and Chern numbers associated to degeneracy points. The notions of topological indices
An intrinsic connection between topological stabilities of Fermi surfaces and topological insulators/superconductors
A topology-intrinsic connection between the stabilities of Fermi surfaces (FSs) and topological insulators/superconductors (TIs/TSCs) is revealed. In particular, a one-to-one relation between the
Anomalous bulk-edge correspondence in continuous media
Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of
Notes on topological insulators
This paper is a survey of the ℤ2-valued invariant of topological insulators used in condensed matter physics. The ℤ-valued topological invariant, which was originally called the TKNN invariant in
Index of Dirac operators and classification of topological insulators
Real and complex Clifford bundles and Dirac operators defined on them are considered. By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford
Topological phases in nodeless tetragonal superconductors.
TLDR
It is demonstrated that 2D tetragonal superconductors show surprising topological features: non-trivial high Chern numbers, massive edge states, and zero-energy modes out of high symmetry points, even though the edge states remain topologically protected.
...
...

References

SHOWING 1-10 OF 29 REFERENCES
Explicit Gauge Fixing for Degenerate Multiplets: A Generic Setup for Topological Orders
We supply basic tools for the study of the topological order of a multiplet which is an eigenspace of a finite-dimensional normal operator with continuous parameters. We allow intrinsic degeneracies
Index theorem for topological heterostructure systems
We apply the Niemi-Semenoff index theorem to an s-wave superconductor junction system attached with a magnetic insulator on the surface of a three-dimensional topological insulator. We find that the
Topological stability of Majorana zero-modes in superconductor-topological insulator systems
We derive an index theorem for zero-energy Majorana fermion modes in a superconductor–topological insulator system in both two and three dimensions, which is valid for models with chiral symmetry as
Bulk-boundary correspondence of topological insulators from their respective Green’s functions
Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different
Topology of Andreev bound states with flat dispersion
A theory of dispersionless Andreev bound states on surfaces of time-reversal invariant unconventional superconductors is presented. The generalized criterion for the dispersionless Andreev bound
TOPOLOGICAL INSULATOR AND THE DIRAC EQUATION
We present a general description of topological insulators from the point of view of Dirac equations. The Z2 index for the Dirac equation is always zero, and thus the Dirac equation is topologically
Chiral topological superconductor from the quantum Hall state
The chiral topological superconductor in two dimensions has a full pairing gap in the bulk and a single chiral Majorana state at the edge. The vortex of the chiral superconducting state carries a
Generic wave-function description of fractional quantum anomalous Hall states and fractional topological insulators.
  • X. Qi
  • Physics
    Physical review letters
  • 2011
TLDR
This work provides the first explicit wave-function description of fractional topological insulators in the absence of spin conservation, and demonstrates that generic chiral topologically ordered states can be realized in lattice models, without requiring magnetic translation symmetry and Landau level structure.
Axial anomalies and index theorems on open spaces
Using an approach inspired by the theory of the anomalous divergence of the axial vector current, we derive trace formulas for the resolvents of Dirac operators on open spaces of odd dimension. These
...
...