Butterflies and topological quantum numbers

The Hofstadter model illustrates the notion of topological quantum numbers and how they account for the quantization of the Hall conductance. It gives rise to colorful fractal diagrams of butterflies where the colors represent the topological quantum numbers. 1 The Hall effect in four acts The first act in the Hall saga begins with a mistake made by James Clerk Maxwell, (1831-1879). In the first edition of his book “Treatise on Electricity and Magnetism”, which appeared in 1873, Maxwell discussed the deflection of a current carrying wire by a magnetic field. Maxwell then says: It must be carefully remembered that the mechanical force which urges a conductor . . . acts, not on the electric current, but on the conductor which carries it. If the reader is puzzled that is OK, he should be. In 1878 Edwin H. Hall was a student at Johns Hopkins University reading Maxwell for a class by Henry A. Rowland. Hall was puzzled by this passage and approached Rowland. Rowland told him that ...he doubted the truth of Maxwell statement and had sometimes before made a hasty experiment . . . though without success. A schematic diagram of the scheme proposed by Rowland is shown in Fig. 1. Possibly, because of this failure, Hall made a fresh start, and tried to make accurate measurements of the changes in the resistance—a much harder experiment. This experiment failed, in accordance with Maxwell. Hall then decided to repeat the experiments made by Rowland, and following a suggestion of Rowland, replaced the original thick metal bar with a thin gold leaf and found that the magnetic field deflected the galvanometer needle. This work earned Hall a position at Harvard. Maxwell died in the year that Hall’s paper came out. In the second edition of Maxwell’s book, which appeared posthumously in 1881, there is polite footnote by the editor saying: Mr. Hall has discovered that a steady magnetic field does slightly alter the distribution of currents in most conductors so that the statement in brackets must be regarded as only approximately true. It turned out that the magnitude, and even the sign of the Hall voltage depends on the conductor.