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

TL;DR

This paper introduces the Bott index for pairs of unitary operators, proves its key properties, and demonstrates its equivalence to the Chern number, linking mathematical index theory to the integer quantum Hall effect.

Contribution

It extends the Bott index concept to invertible operators and establishes its equivalence to the Chern number in quantum physics.

Findings

The Bott index is homotopically invariant under certain perturbations.

The index satisfies a logarithmic law.

The index equals the Chern number, confirming the integer quantum Hall conductance.

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.