DC Conductivity as a geometric phase
Abstract: The notion of a topological invariant is at the heart of a number of physical phenomena of recent interest, for example the integer and fractional quantum Hall effects or topological insulators. The first such invariant was introduced into physics by Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) to describe quantization of the Hall conductance. In this talk I will show that the Drude weight, the strength of the zero frequency conductivity, is also a topological invariant whose form is similar to the TKNN invariant. The many-body term of the Drude weight turns out to be a line-integral around a rectangle, one side of which is the total momentum, the other the total position. The conjecture of Kohn, according to which an insulator is a system in which the wavefunction is localized in the many-body space, is explicitly demonstrated and refined as follows: if a wavefunction is an eigenstate of the total current, the corresponding system is delocalized, the Drude weight is finite, therefore the system is conducting. Wavefunctions which have continuous distributions of total momentum give rise to insulation. These results can also be understood in terms of a generalization of off-diagonal long range order.