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

摘要

We have shown that Hofstadter's butterfly spectrum, which is usually conceived as peculiar to two-dimensional systems in magnetic fields, can occur, unexpectedly, 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 Harper's equations enables us to calculate the Hall conductivity in terms of Kohmoto-Halperin-Wu's topological invariants, where each of a,,, a,, is found to be quantized. (C) 2001 Elsevier Science B.V. All rights reserved. [References: 9]