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

## TL;DR

This paper introduces an optimal transport-based framework to analyze the quantum geometry of correlated many-body states, providing new insights into their structure and applying it to the one-dimensional t-V model with a metal-insulator transition.

## Contribution

It develops a novel optimal transport approach to define geometric quantities for correlated many-body states, extending previous single-particle concepts.

## Key findings

- Optimal transport effectively characterizes many-body quantum geometry.
- Explicit analysis of the one-dimensional t-V model demonstrates the approach.
- The method captures the metal-insulator transition in the model.

## Abstract

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.