Curvature of quantum rings

TL;DR

This paper introduces a geometric framework for analyzing quantum spin rings using a quantum mechanical distance, revealing non-trivial geometric properties such as Gauss curvature despite their simple visual appearance.

Contribution

It develops a novel geometric approach to quantum spin networks, specifically spin rings, by defining a distance that uncovers complex curvature properties.

Findings

Quantum spin rings exhibit non-trivial Gauss curvature.

The geometry derived from quantum distances differs from visual geometry.

The approach links quantum network properties to classical differential geometry.

Abstract

We develop a geometric approach to spin networks with Heisenberg or XX coupling. Geometry is acquired by defining a distance on the discrete set of spins. The key feature of the geometry of such networks is their Gauss curvature $\kappa$, viewed here as the ability to isometrically embed the chain in the standard Riemannian manifold of curvature $\kappa$. Here we focus on spin rings. Even though their visual geometry is trivial, it turns out that the geometry they acquire from the quantum mechanical distance is far from trivial.