Quantum Hall Systems

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. The Hall conductance of Integer Quantum Hall systems is described mathematically by the index of Fredholm operators. (For precise definitions, see below). In this paper we investigate the phase diagram of the Fredholm index for a few classes of operators. For the algebra of bounded operators, little can be said beyond the fact that the phase diagrams can be arbitrarily complicated. But for the algebra of Toeplitz operators, and other related classes of operators, we establish a kind of a Gibbs phase rule [1]. Typical of our results is the statement that if the system is governed by two parameters, then one should expect jumps by one at phase boundaries and jumps by up to 2 at triple points, while jumps by more than two should never be observed. We relate this behavior to experimental results, conjectures and open problems that arise in the context of the Quantum Hall Effect (QHE) [2]. In Section 1 we define Fredholm operators and their indices, and explore the different sorts of phase diagrams that can arise. In Section 2 we recall how ∗On leave from the Department of Mathematics, University of Texas, Austin, TX 78712 USA