Curvature of quantum rings

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.