Infinite geometric frustration in a cubic dipole cluster

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 eight interacting dipoles in a cubic cluster is investigated in detail. Instead of discrete equilibria we find a 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 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. Magnetism has fascinated mankind for millenia [1]. Today, even the smallest magnets can hardly be overestimated in their relevance for magnetic storage technology. A fascinating example for the interplay of magnetic particles is their self-arrangement in cubic lattice clusters (see, e.g., [2,3]). Its macroscopic analog is the toy known as "magnetic cube puzzle" shown in Fig. 1(a), a stable arrangement of spherical magnets in a simple cubic cluster. How are these magnetic spheres oriented in such an ordered cluster? For the minimal arrangement within this class, namely, a cube consisting of eight magnets [see Fig. 1(b)], the answer is intriguing: There are infinitely many orientations. We find the ground state to be a continuum of energetically degenerate states-an extreme form of magnetic frustration. The phenomenon of frustration arises when the system cannot simultaneously minimize all dipole-dipole interaction energies (see [4] for a recent review). As this continuum is the ground state of the cube system, the question arises: Are there any other equilibrium orientations? Through our application of methods from numerical algebraic geometry (see Supplementary 4 [5]) we are able to construct and classify the complete set of equilibrium states. This set comprises thousands of unstable discrete dipole orientations in addition to the continuous states. We stress here that we find all equilibrium configurations (stable and unstable) unlike commonly used relaxation methods.The study of equilibrium states of dipolar hard and soft spheres has a long history in the context of magnetic colloids. Early works [6-8] on these so-called ferrofluids investigate the phase behavior (colloidal crystal structures, chain formation, string fluids, etc.). More recently, thermodynamic properties of two-dimensional (2D) monolayer systems have been studied [9] and a full phase diagram in terms of dipole strength and packing fraction was given [10]. Further, the self-assembly and transition from rings to chains controlled by an external field was investigated (see, e.g., [11][12][13]). Isolated planar dipole clusters in exte...

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