The amazingly precise quantization of Hall conductance in a two-dimensional electron gas can be understood in terms of a topological invariant known as the Chern number.

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.Introduction.-Azbel [1] recognized that the spectral properties of two-dimensional, 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. Hofstadter Ph.D. thesis [2], where a wonderful diagram, reminiscent of a fractal butterfly, provided a source of inspiration and a tool for spectral analysis [3][4][5][6][7][8][9].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 [11,12]. The diagram leads to certain natural questions: Count the number of components of a given phase; Classify which phases coexist and where. It also leads to the general question: What form does the Gibbs phase rule [14,13] take for quantum phase transitions.Fractal phase diagrams and/or infinitely many phases appear in dynamical systems [17,18]. In classical lattice systems fractal phase diagrams [14,19,20,13] are commonly viewed as a pathology due to either long range interactions, or, as is the case for spin glasses, loss of translation invariance. The Hofstadter model, when viewed as a statistical mechanical model, is both short range and translation invariant in a natural way. But, it is quantum and the translation group is non-commutative. It suggests that fractal phase diagram may be more common in quantum phase transitions than in classical phase transitions.The Hofstadter Model.-The model conceived by Peierls has two versions. For the sake of concreteness we shall focus here on the tight binding version. On the lattice Z 2 , define magnetic shifts