Infinite geometric frustration in a cubic dipole cluster

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Infinite geometric frustration in a cubic dipole cluster

Johannes Schönke, Tobias M. Schneider, and Ingo Rehberg

Phys. Rev. B 91, 020410(R) – Published 29 January 2015

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.

Received 24 July 2014
Revised 5 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.020410

Authors & Affiliations

Johannes Schönke¹, Tobias M. Schneider^1,2, and Ingo Rehberg³

¹MPRG-ECPS, MPI für Dynamik und Selbstorganisation, Am Faßberg 17, 37077 Göttingen, Germany
²Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, Switzerland
³Physikalisches Institut, Experimentalphysik V, Universität Bayreuth, 95440 Bayreuth, Germany

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Issue

Vol. 91, Iss. 2 — 1 January 2015

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Images

Figure 1
(a) The “magnetic cube puzzle”: A stable arrangement of 216 spherical magnets in a $6 \times 6 \times 6$ simple cubic cluster. (b) The minimal simple cubic configuration of spherical magnets, in a $2 \times 2 \times 2$ cluster. The ground state of this arrangement is not a single equilibrium configuration with only one discrete orientation for each dipole but rather a continuum of infinitely many configurations.
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Figure 2
(a) Rendering of eight dipoles as arrows located at the corners of a cube. The colored and translucent arrows show selected orientations in the ground state of the system which is a continuum of infinitely many configurations. The continuum corresponds to rotations of the dipoles in planes perpendicular to the cube's volume diagonal of the respective corner in analogy to the mechanical system in Fig. 2. Consecutive arrows are seperated by 30°. Along the continuum we repeatedly pass through two particularly notable configurations: First, the color red shows two counter-rotating rings of four dipoles each lying in the upper and lower faces of the cube. Secondly, the color green shows two groups of four dipoles lying in two diagonal planes perpendicular to each other. Thirdly, the color blue shows two rings similar to red but now in the left and right faces of the cube. Pairs of dipoles in opposing corners are always parallel. (b) Magnetic energy landscape for eight dipoles located at the corners of a cube. The energy is shown as a function of the two orientation angles of one dipole, covering all possible orientations. The other seven dipoles adjust to the respective minimum energy configuration. The white line through the “valley” marks the continuous ground state. (c) Rendering of eight bevel gears located at the corners of a cube. The rotation axes point to the cube's center and every gear interlocks with its three edge neighbors. The motion of this mechanical system, indicated by the red arrows, corresponds to the motion of the ground state continuum in the magnetic system shown in Figs. 1 and 2. This analogy opens up new engineering possibilities to construct frictionless magnetic couplings.
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Figure 3
Magnetic energy spectrum of all 183 families of discrete equilibrium configurations for eight dipoles located at the corners of a cube. Each family (black filled circles) with its unique energy may contain up to 96 members due to polarity and cube symmetries. The red filled circles mark positions of exemplarily chosen configurations which are displayed in their respective inset. The upper left inset shows a fully unstructured configuration with no apparent symmetries, therefore the family has 96 members. The lower right inset shows a more structured configuration with some obvious symmetries, so this family has only six distinguishable members. The upper right inset shows the maximum energy configuration which is the most ordered one with all dipoles oriented to the cube center; all symmetries of the cube are retained and only the polarity flip gives a new configuration, so this family has two members. The additional two blue circles mark the energetic positions of the ground state continuum and the second real continuum.
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