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 topological invariants for all condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields, whenever the Fermi level lies in a region of Anderson localized spectrum.

Controlled Topological Phases and Bulk-edge Correspondence

“…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 .…”

Virtual topological insulators with real quantized physics

“…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.…”

Electron Dynamics: Concrete Physical Models

“…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.…”

“…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.…”