Spin models on non-Euclidean hyperlattices: Griffiths phases without extrinsic disorder

We study short-range ferromagnetic models residing on planar manifolds with global negative curvature. We show that the local metric properties of the embedding surface induce droplet formation from the boundary, resulting in the stability of a Griffiths phase at a temperature lower than that of the bulk transition. We propose that this behaviour is independent of order parameter and hyperlattice specifics, and thus is universal for such non-Euclidean spin models. Their temperature–curvature phase diagrams are characterized by two distinct bulk and boundary transitions; each has mean-field critical behaviour and a finite correlation length related to the curvature of the embedding surface. The implications for experiments on superconducting hyperlattice networks are also discussed.