We give a perspective on the Hofstadter butterfly (fractal energy spectrum in
magnetic fields), which we have shown to arise specifically in
three-dimensional(3D) systems in our previous work. (i) We first obtain the
`phase diagram' on a parameter space of the transfer energies and the magnetic
field for the appearance of Hofstadter's butterfly spectrum in anisotropic
crystals in 3D. (ii) We show that the orientation of the external magnetic
field can be arbitrary to have the 3D butterfly. (iii) We show that the
butterfly is beyond the semiclassical description. (iv) The required magnetic
field for a representative organic metal is estimated to be modest (∼40
T) if we adopt higher Landau levels for the butterfly. (v) We give a simpler
way of deriving the topological invariants that represent the quantum Hall
numbers (i.e., two Hall conductivity in 3D, σxy,σzx, in
units of e2/h).Comment: 8 pages, 8 figures, eps versions of the figures will be sent on
request to [email protected]