 
Summary: J. Phys. A: Math. Gen. 32 (1999) 60976113. Printed in the UK PII: S03054470(99)012780
Quantum response at finite fields and breakdown of Chern
numbers
J E Avron and Z Kons
Department of Physics, Technion, 32000 Haifa, Israel
Email: avron@physics.technion.ac.il and konsz@physics.technion.ac.il
Received 25 January 1999, in final form 10 May 1999
Abstract. We show that the response to an electric field, in models of the integral quantum Hall
effect, is analytic in the field and has isolated essential singularity at zero field. We also study
the breakdown of Chern numbers associated with the response of Floquet states. We argue, and
give evidence, that the breakdown of Chern numbers in Floquet states is a discontinuous transition
at zero field. This follows from an observation, of independent interest, that Chern numbers for
finitedimensional Floquet operators are generically zero. These results rule out the possibility
that the breakdown of the Hall conductance is a phase transition at finite fields for a large class of
models.
1. Introduction
The principal motivation for the present work is the question: is the breakdown of the integer
quantum Hall effect a (quantum) phase transition? Since the Hall conductance in the adiabatic
limit is identified with a Chern number, the question can also be phrased as: is the breakdown
of Chern numbers a phase transition?
