 
Summary: Hofstadter butterfly as quantum phase diagram
D. Osadchy and J. E. Avrona)
Department of Physics, Technion, 32000 Haifa, Israel
Received 20 June 2001; accepted for publication 29 August 2001
The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely
many phases, labeled by their integer Hall conductance, and a fractal structure.
We describe various properties of this phase diagram: We establish Gibbs phase
rules; count the number of components of each phase, and characterize the set of
multiple phase coexistence. © 2001 American Institute of Physics.
DOI: 10.1063/1.1412464
I. INTRODUCTION
Azbel1
recognized that the spectral properties of twodimensional, periodic, quantum systems
have sensitive dependence on the magnetic flux through a unit cell. A simple model conceived by
Peierls and put to the eponymous Harper as a thesis problem, gained popularity with D. Hofs
tadter's Ph.D. thesis,2
where a wonderful diagram, reminiscent of a fractal butterfly, provided a
source of inspiration and a tool for spectral analysis.39
The Hofstadter butterfly can also be viewed as the quantum zero temperature phase diagram
for the integer quantum Hall effect. It is a fractal phase diagram with infinitely many phases.10,11
