Butterfly spectrum and integer quantum Hall effect in three dimensions - amapping between 2D and 3D Hofstadter problems

We have shown that Hofstadter's butterfly spectrum, which is usually concei ved as peculiar to two-dimensional systems in magnetic fields, can occur, u nexpectedly, in three-dimensional lattices as well under certain criteria. Curiously, the butterfly arises due to the third-direction transfer rather than as a remnant of a 2D butterfly. A formal mapping between 2D and 3D Har per's equations enables us to calculate the Hall conductivity in terms of K ohmoto-Halperin-Wu's topological invariants, where each of a,,, a,, is foun d to be quantized. (C) 2001 Elsevier Science B.V. All rights reserved.