2016
DOI: 10.1016/j.jfa.2016.06.001
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Non-commutative odd Chern numbers and topological phases of disordered chiral systems

Abstract: Abstract. An index theorem for higher Chern characters of odd Fredholm modules over crossed product algebras is proved, together with a local formula for the associated cyclic cocycle. The result generalizes the classic NoetherGohberg-Krein index theorem, which in its simplest form states that the winding number of a complex-valued function over the circle is equal to the index of the associated Toeplitz operator. When applied to the non-commutative Brillouin zone, this generalization allows to define topologi… Show more

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Cited by 44 publications
(65 citation statements)
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“…Example 4.5. In the same way as above, the odd cyclic cocycle T ω d gives a homomorphism TP(X; AIII) → R, which is explicitly written by ProdanSchulz-Baldes [PSB14] as Example 4.6. For a 2-dimensional Quantum Spin-Hall system (quantum systems with type AII symmetry) without any other symmetry, the bulk and edge indices take value in KAII 2 (R) = KQ 2 (R) ∼ = Z 2 .…”
Section: Bulk-edge Correspondencementioning
confidence: 99%
“…Example 4.5. In the same way as above, the odd cyclic cocycle T ω d gives a homomorphism TP(X; AIII) → R, which is explicitly written by ProdanSchulz-Baldes [PSB14] as Example 4.6. For a 2-dimensional Quantum Spin-Hall system (quantum systems with type AII symmetry) without any other symmetry, the bulk and edge indices take value in KAII 2 (R) = KQ 2 (R) ∼ = Z 2 .…”
Section: Bulk-edge Correspondencementioning
confidence: 99%
“…We believe that progress in this direction within the chiral symmetry class can be achieved based on the non-commutative winding number introduced in Refs. 37,38 . It also seems that virtual quantum spin-Hall insulators in 1 + 1 dimensions can be straightforwardly constructed starting from two copies of the already existing 1 + 1-dimensional Chern insulators and inserting S z -non-conserving terms of Rashba type.…”
Section: Discussionmentioning
confidence: 99%
“…norm-deformations cannot be used in this context. However, under the Aizenman-Molchanov condition, the righthand side of (2.32) (or of (2.39)) varies continuously with the deformations of the models [24,25]. As such, using both sides of (2.32) and (2.39), one can indeed establish the quantization and invariance of the bulk topological numbers under the Anderson localization assumption.…”
Section: Remark 215mentioning
confidence: 92%
“…The theory of the topological bulk invariants for the phases classified by Z or 2Z in Table 1.1 was developed in [8,24,25] for the regime of strong disorder. The recent work [11] by Bourne and Rennie extended these result to continuum models.…”
Section: Topological Invariantsmentioning
confidence: 99%