We explore two approaches to characterise the quantum geometry of the ground state of correlated fermions in terms of the distance matrix in the spectral parameter space. (a) An intrinsic geometry approach, in which we study the intrinsic curvature defined in terms of the distance matrix. (b) An extrinsic geometry approach, in which we investigate how the distance matrix can be approximately embedded in finite dimensional Euclidean spaces. We implement these approaches for the ground state of a system of one-dimensional fermions on a 18-site lattice with nearest neighbour repulsion. The intrinsic curvature sharply changes around the Fermi points in the metallic regime but is more or less uniform in the insulating regime. In the metallic regime, the embedded points clump into two well seperated sets, one corresponding to modes in the Fermi sea and the other to the modes outside it. In the insulating regime, the two sets tend to merge. arXiv:1812.06374v1 [cond-mat.str-el]

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.