Geometry and curvature of spin networks

We develop a geometric approach to spin networks with Heisenberg or XX coupling. Geometry is acquired by defin-ing a distance on the discrete set of spins. The key feature of the geometry of such networks is their Gauss curvature κ, viewed here as the ability to isometrically embed the chain in the standard Riemannian manifold of curvature κ. 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. Index Terms — Spin chains, coarse geometry, curvature, Riemannian spaces, Feynman path integral