# Intrinsic and extrinsic geometries of correlated many-body states

## TL;DR

This paper investigates the quantum geometry of correlated fermion ground states using intrinsic and extrinsic distance-based approaches, revealing geometric signatures of metallic and insulating phases in a one-dimensional lattice system.

## Contribution

It introduces two geometric frameworks for analyzing quantum states and applies them to a fermionic lattice model, highlighting phase-dependent geometric features.

## Key findings

- Intrinsic curvature varies sharply at Fermi points in the metallic phase.
- Embedded points form two distinct clusters in the metallic regime.
- In the insulating phase, the clusters merge, indicating a change in geometry.

## Abstract

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.