- Broida 1640
Douglas Hofstadter Center for Research on Concepts and Cognition Indiana University, Bloomington
In condensed-matter physics, the term “Hofstadter butterfly” denotes a fractal graph that displays the allowed energy levels of electrons in a crystal (also known as Bloch electrons) when the crystal is immersed in a uniform magnetic field. The graph is plotted as a function of a very natural dimensionless variable a, which is proportional to the magnetic field. The spectrum at any rational value of a= p/q turns out to consist of a set of q energy bands, while the spectrum at any irrational value ofaconsists of an infinite set of isolated points (a Cantor set). This strange and unexpected behavior is related to the expansion of the real numberaas a kind of continued fraction.
I had the happy fortune to stumble across this intricate pattern when in graduate school in the mid-1970s, and it rapidly became quite well known, but it remained purely a theoretical prediction until the past year, when empirical evidence was at last found to confirm the fractal nature of the spectrum.
Behind my discovery roughly four decades ago lies a tangled tale of one person’s idealistic quest for beauty in physics, his battle against ideas that he found ugly and implausible, his numerous changes of field, of advisor, and of graduate school, his fluctuating level of self-belief, and his bad and good luck. This is a scientific and personal saga that hopefully will be of interest to physicists, to physics graduate students, and even to outsiders to physics.