By C. Menini, F. van Oystaeyen
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Extra info for Abstract Algebra - A Comprehensive Trtmt
In other words, the only way to cross from the topological to the trivial phase is to go through a localization-delocalization quantum transition. As we shall see, it is exactly this divergence of the localization length which triggers an abrupt change in the quantized values of the bulk topological invariant. While the bulk analysis, just by itself, can be carried in the regime of strong disorder, the bulk-boundary correspondence will be established under the following assumption: / σ (Hω ). 2), that is JHω J = − Hω , and therefore also the flat band Hamiltonian Qω = 1 − 2Pω = sgn(Hω ) satisfies JQω J = −Qω and Qω2 = 1.
16) are sometimes also called Dirac points, which is not appropriate for the following reasons. t. the product γ1 · · · γd . This splits it into two chiral sectors and, in each of those chiral sectors, one gets the classical Weyl operator k, σ when the “time” direction is separated out. Here, γ and σ denote the Dirac and Pauli matrices. This pattern can be recognized in any dimension, and in general, the Weyl operator involves an odd number of Clifford generators and does not have a chiral symmetry, but rather a chirality that will be introduced below.
21). This implies Ch1 (u) = Ch1 (u0 ) for u ∈ C 1 (A1 ). 4) as a winding number. In particular, this shows that Ch1 (u) ∈ Z. An alternative way to verify the integrality of Ch1 (u) is to prove an index theorem. This has the advantage that one can also prove that the pairing is well-defined and integral in the regime of a mobility bulk gap, namely, when the Fermi level lies in a region of the essential spectrum which is dynamically Anderson localized. This type of extension is crucial for the understanding of the quantum Hall effect  and will be discussed further in Chap.
Abstract Algebra - A Comprehensive Trtmt by C. Menini, F. van Oystaeyen