Infinite Geometric Frustration in a Cubic Dipole Cluster

The geometric arrangement of interacting (magnetic) dipoles is a question of fundamental im-portance in physics, chemistry and engineering. Motivated by recent progress concerning the self-assembly of magnetic structures, the equilibrium orientation of 8 interacting dipoles in a cubic cluster is investigated in detail. Instead of discrete equilibria we find a new type of ground state consisting of infinitely many orientations. This continuum of energetically degenerate states repre-sents a yet unknown form of magnetic frustration. The corresponding dipole rotations in the flat potential valley of this Goldstone mode enable the construction of frictionless magnetic couplings. Using novel computer-assisted algebraic geometry methods, we moreover completely enumerate all equilibrium configurations. The seemingly simple cubic system allows for exactly 9536 unstable discrete equilibria falling into 183 distinct energy families. PACS numbers: 05.65.+b, 41.20.Gz, 75.10.-b Magnetism has fascinated mankind for millenia [1]. Today, even the smallest magnets can hardly be overes-timated in their relevance for magnetic storage technol-ogy. A fascinating example for the interplay of magneti