Infinite Geometric Frustration in a Cubic Dipole Cluster

TL;DR

This paper uncovers a new form of magnetic frustration in a cubic dipole cluster, revealing a continuum of degenerate ground states and providing a complete enumeration of equilibrium configurations using advanced algebraic geometry methods.

Contribution

It introduces a novel infinite degeneracy of ground states in a cubic dipole system and employs new computational methods to classify all equilibria.

Findings

Discovery of a continuum of degenerate ground states.

Complete enumeration of 9536 unstable equilibria.

Identification of a Goldstone mode enabling frictionless magnetic couplings.

Abstract

The geometric arrangement of interacting (magnetic) dipoles is a question of fundamental importance 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 represents 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.