Richard Hofstadter, 1916–1970

Remembering Richard [Obituary, Richard A.Newton]

Fixed points of the dissipative Hofstadter model.

The phase diagram of a dissipative particle in a periodic potential and a magnetic field is studied in the weak barrier limit and in the tight binding regime. For the case of half flux per plaquette, and for a wide range of values of the dissipation, the physics of the model is determined by a nontrivial fixed point. A combination of exact and variational results is used to characterize this fi...

Hofstadter Butterfly and a Hidden Apollonian Gasket

The Hofstadter butterfly, a quantum fractal made up of integers describing quantum Hall states, is shown to be related to an integral Apollonian gasket with D3 symmetry. This mapping unfolds as the self-similar butterfly landscape is found to describe a close packing of (Ford) circles that represent rational flux values and is characterized in terms of an old ((300BC) problem of mutually tangen...

Phase Diagram of the Dissipative Hofstadter Model

A quantum particle moving in a uniform magnetic field and periodic potential (the Hofstadter model) has an energy band structure which varies in a discontinuous fashion as a function of the magnetic flux per lattice unit cell. In a real system, randomness of various kinds should “smooth out” this behavior in some way. To explore how this happens, we have studied the dissipative quantum mechanic...

Effects of screening on the Hofstadter butterfly.

We study, within the Hartree approximation, the effects of the electronelectron interaction on the energy spectrum of a two-dimensional electron gas in a perpendicular homogeneous magnetic field and a lateral superlattice potential with square symmetry. Due to the strong screening effects, the bandwidth of the Landau bands oscillates strongly with their filling. For short enough periods and str...