The index of a pair of pure states and the interacting integer quantum Hall effect

TL;DR

This paper introduces a new index for pairs of pure states on C*-algebras and demonstrates its application in expressing the Hall conductance of interacting 2D electronic systems, revealing topological and continuity properties.

Contribution

It generalizes the index of projections to pairs of states and links this to the Hall conductance in interacting quantum systems, providing a topological perspective.

Findings

Hall conductance expressed as an index of state pairs

Establishment of integrality and continuity of Hall conductance

Connection between topological index and magnetic flux insertion

Abstract

We introduce the index $\mathcal{N}(\omega_1,\omega_2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega$ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal{N}(\omega,\omega_D)$, where $\omega_D$ is obtained from $\omega$ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal{N}$.