Quantum Hall effect and the relative index for projections

Abstract

We define the relative index, Index(P,Q), for a pair of infinite-dimensional projections on a Hilbert space to be the integer that is the natural generalization of dim(P)-dim(Q) in finite-dimensional vector spaces. We show that the Hall conductance for independent electrons in the plane is the relative index where P and Q project on the states below the Fermi energy for Hamiltonians that differ by a quantum flux and the Fermi energy is appropriately placed. This approach is closely related to, and sheds light on, Bellissard’s interpretation of the Hall conductance as an index.