# Quantum geometry of correlated many-body states

## TL;DR

This paper introduces a quantum distance measure for correlated many-fermion states, proves its mathematical properties, and demonstrates its effectiveness in identifying phase transitions in a model system.

## Contribution

It defines a new quantum distance based on exchange operators and applies it to detect metal-insulator transitions in the t-V model.

## Key findings

- Distance matrix reveals signatures of the metal-insulator transition
- Distances satisfy triangle inequalities
- Formalism applicable to correlated fermion 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.