The Bott index of two unitary operators and the integer quantum Hall effect

Abstract

The Bott index of two unitary operators on an infinite dimensional Hilbert space is defined. Homotopic invariance with respect to multiplicative unitary perturbations of the type identity plus trace class and the "logarithmic" law for the index are proven. The index and its properties are then extended to the case of a pair of invertible operators. An application to the physics of two dimensional quantum systems proves that the index is equal to the Chern number therefore showing that the transverse Hall conductance is an integer.

On the Bott index of unitary matrices on a finite torus

“…A novel proof of the equality of the two indices is given in Sect. 5.2 through a mapping to a differential equation following an analogous proof for the infinite two-dimensional case recently presented in [44]. Three perspectives for future developments are given in Sect.…”

“…Equation (44) follows from (47) with A = U V U −1 V −1 . The trace of a matrix is less equal than the norm of the matrix itself times the dimension of the space the matrix is acting upon, then:…”

“…5, I will provide here a novel proof of the Bott index -Chern number correspondence, showing the equality among the Bott index of the pair of unitary matrices given in Eq. ( 72) and the Chern number Ch(P) taking advantage of an approach developed in [44] for bounded operators. The operators (72) have been already suggested in the context of the integer quantum Hall effect by Kitaev in appendix C of [52], and more recently, with a suitable modification, in the context of infinite-dimensional Hilbert spaces by the authors of [53,54].…”

Gaplessness of Landau Hamiltonians on Hyperbolic Half-planes via Coarse Geometry

“…We mention [13] which studies a certain Bott index for non translation-invariant lattice models on compact surfaces. For lattice models on Z 2 , standard half-spaces, and finiterange Hamiltonians, an argument in the spirit of Section 2 was sketched in §6 of [32], for the study of the QHE Chern numbers. The papers [17,18] apply coarse cohomology ideas, but in a very different way, for the study of other types of topological phases.…”