Ice on curved surfaces: defect rings and differential local dynamics

Abstract: Ice systems are prototypes of locally constrained dynamics. This is exemplified in Coulomb-liquid phases where a large space of configurations is sampled, each satisfying local ice rules. Dynamics proceeds through flipping' rings, i.e., through reversing arrows running along the edges of a polygon. We examine the role of defect rings in such phases, with square-ice as a testing ground. When placed on a curved surface, the underlying square lattice will form defects such as triangles or pentagons. We show that triangular defects are statistically moreflippable' than the background. In contrast, pentagons and larger polygons are less flippable. In fact, flippability decreases monotonically with ring size, as seen from a Pauling-like argument. As an explicit demonstration, we wrap the square ice model on a sphere. We start from an octahedron and perform repeated rectifications, producing a series of clusters with sphere-like geometry. They contain a fixed number of defect triangles in an otherwise square lattice. We numerically enumerate all ice-rule-satisfying configurations. Indeed, triangles are flippable in a larger fraction of configurations than quadrilaterals. The obtained flippabilities are in broad agreement with the Pauling-like estimates. As a minimal model for dynamics, we construct a Hamiltonian with quantum tunnelling terms that flip rings. The resulting ground state is a superposition of all ice configurations. The dominant contribution to its energy comes from localized resonance within triangles. Our results suggest local dynamics as a promising observable for experiments in spin ice and artificial ice systems. They also point to hierarchical dynamics in materials such as ice V that contain rings of multiple sizes.