Fredholm Indices and the Phase Diagram of Quantum Hall Systems

TL;DR

This paper explores the relationship between Fredholm indices and the phase diagram of quantum Hall systems, providing a mathematical framework that could elucidate the topological nature of quantum Hall phases.

Contribution

It introduces a generic phase diagram of Fredholm indices for bounded and Toeplitz operators, linking mathematical indices to quantum Hall phase transitions.

Findings

Fredholm indices relate to quantized Hall conductance

Phase diagram of indices characterizes quantum Hall phases

Potential relevance to disordered quantum Hall systems

Abstract

The quantized Hall conductance in a plateau is related to the index of a Fredholm operator. In this paper we describe the generic ``phase diagram'' of Fredholm indices associated with bounded and Toeplitz operators. We discuss the possible relevance of our results to the phase diagram of disordered integer quantum Hall systems.