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Fredholm indices and the phase diagram of quantum Hall systems

Summary: Fredholm indices and the phase diagram of quantum
Hall systems
J. E. Avron and L. Saduna)
Department of Physics, Technion, 32000 Haifa, Israel
Received 17 March 2000; accepted for publication 6 September 2000
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. 2001 American Institute of Physics. DOI: 10.1063/1.1331317
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


Source: Avron, Joseph - Physics Department, Technion, Israel Institute of Technology


Collections: Physics