Two-Dimensional Matter: Order, Curvature and Defects

Understand

Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces.

• We review here the fundamental role payed by both the topology of the underlying surface and its detailed curvature.
• Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics controls the detailed structured of ordered phases.
• Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature.