Ferrotoroidic moment as a quantum geometric phase.

We present a geometric characterization of the ferrotoroidic moment tau in terms of a set of Abelian Berry phases. We also introduce a fundamental complex quantity z munu, which provides an alternative way to calculate tau and its moments and is derived from the tensor T munu=2 under summation operator jrj muSj nu. This geometric framework defines a natural computational approach for density functional and many-body theories.