Quantum geometry of correlated many-body states

Abstract

We provide a definition of the quantum distances of correlated many fermion wave functions in terms of the expectation values of certain operators that we call exchange operators. We prove that the distances satisfy the triangle inequalities. We apply our formalism to the one-dimensional t − V model, which we solve numerically by exact diagonalisation. We compute the distance matrix and illustrate that it shows clear signatures of the metal-insulator transition.

Theory of Optimal Transport and the Structure of Many-Body States

“…Consistent with the interpretation of the quantum distances in one band models discussed above, we found from our numerical results for the one dimensional t − V model 1,2 that deep in the metallic regime (V << 1), the distances classify the quasi momenta inside the Fermi sea and those outside it into two different categories. The distances between any two points both inside or outside the Fermi sea are very small (∼ 0) and those between two points, where one lies inside the Fermi sea and the other outside it, are very large (1). The points inside the Fermi sea we label as k in and points outside the Fermi sea we label as k out .…”

“…Developing good techniques to visualize them and characterize their structure can contribute to the understanding of a large number of physical systems. In recent work 1,2 , we have proposed a mathematically consistent definition of distances between two points in the spectral parameter space for a general many-fermion state. Our definition generalises the previous definition which was in terms of the single-particle wavefunctions and hence was only valid for mean-field states.…”

“…Motivated by the above discussion and our recent work 1,2 , in this paper we attempt to develop a formalism to bring out the detailed geometry of translationally invariant, correlated states. Our previous numerical results 1,2 indicate that while it is possible to give a mathematically consistent definition of quantum distances between two points in the Brillioun zone (BZ), the differential metric may not exist for correlated states. Thus, we develop the formalism in terms of quantum distances rather than the differential quantum metric.…”

“…Our previous work 1,2 and this one constitutes our attempt to implement our definition of quantum distances for correlated states in the physical context of the work of Kohn and others 4-8 towards a quantum geometric characterization of the insulating state. All our results in these papers are obtained by exact diagonalization of the one dimensional t − V model [19][20][21] .…”