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Colored Hofstadter butterflies J. E. Avron

Summary: Colored Hofstadter butterflies
J. E. Avron
Department of Physics, Technion, 32000 Haifa, Israel
September 4, 2003
ABSTRACT I explain the thermodynamic significance, the duality and
open problems associated with the two colored butterflies shown in figures
1 and 4.
1 Overview
My aim is to explain what is known about the thermodynamic significance
of the two colored butterflies shown in figures 1 and 4 and what remains
open. Both diagrams were made by my student, D. Osadchy [14], as part
of his M.Sc. thesis. I shall explain their interpretation as the T = 0 phase
diagrams of a two dimensional gas of charged, though non-interacting,
fermions. Fig. 1 is associated with weak magnetic fields (and strong pe-
riodic potentials) while Fig. 4 with strong magnetic fields (and weak peri-
odic potentials). The two cases are related by duality. The duality, which
is further discussed below, is manifest if colors are disregarded.
The horizontal coordinate in both figures is the chemical potential
and the vertical coordinate is proportional to the magnetic induction B in
fig. 1 and 1/B in fig. 4. The colors represent the quantized values of the


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


Collections: Physics