DC Conductivity as a Geometric Phase

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.

References: 
1. arXiv:1309.2962 
2. Phys. Rev. B 87 235123 (2013) 
3. J. Phys. Soc. Japan 81 124711 (2012) 
4. J. Phys. Soc. Japan 81 023701 (2012).