  There has been much work in the recent past in developing the idea of quantum
geometry to characterize and understand the structure of many-particle states.
For mean-field states, the quantum geometry has been defined and analysed in
terms of the quantum distances between two points in the space of single
particle spectral parameters (the Brillioun zone for periodic systems) and the
geometric phase associated with any loop in this space. These definitions are
in terms of single-particle wavefunctions. In recent work, we had proposed a
formalism to define quantum distances between two points in the spectral
parameter space for any correlated many-body state. In this paper we argue
that, for correlated states, the application of the theory of optimal transport
to analyse the geometry is a powerful approach. This technique enables us to
define geometric quantities which are averaged over the entire spectral
parameter space. We present explicit results for a well studied model, the one
dimensional t-V model, which exhibits a metal-insulator transition, as evidence
for our hypothesis.
