Quantum Hall conductances and localization in a magnetic field

The relation between extended and localized states in a magnetic field is investigated. A general form for the magnetic Bloch states in an arbitrary rational field (with p/q flux quanta through a unit cell, p and q relatively prime integers) is written, and their basic properties are studied. It is shown that the completeness properties of lattices of orbitals relative to a set of N magnetic subbands are connected with the value of the total quantum Hall conductance ${\sigma }_N$ (in units of $e^2$/h) carried by these subbands. In particular, lattices of orbitals can reproduce continuously all the magnetic Bloch states of N subbands if and only if ${\sigma }_N$=0, a case which may occur only for N multiples of q. This is also the only case where localized magnetic Wannier functions for the subbands can be constructed. In the light of these results a discussion is given of the almost-free-electron limit and the tight-binding approach of Harper’s equation.