ar X iv : m at h - ph / 0 30 80 30 v 1 2 5 A ug 2 00 3 Colored Hofstadter butterflies

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 periodic potentials) while Fig. 4 with strong magnetic fields (and weak periodic 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 µ 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 Hall conductance, i.e. represent integers 1. Warm colors represent positive multiples and cold colors represent negative ones: Orange represents 2, red 1, white 0, blue −1 etc. Remark: It is problematic to represent integers by colors with good contrast between nearby integers. This is related to the fact that colors are not ordered on the line but rather are represented by the simplex (r, g, b) with r+g+b = 1. (Pure colors are located on the boundary of the simplex). The assignment in the figures becomes problematic for large, positive or negative, integers: Large positive integers are not represented anymore by warm colors but rather by yellow and green. I shall also present an open problem. Namely, how do these diagrams change if one replaces the magnetic induction B by the magnetic field H as the ther-modynamic coordinate. 1 The quantum unit of conductance, e 2 /h, is 1/2π, in natural units where e = = 1.