RAPID COMMUNICATIONSPHYSICAL REVIEW B 91, 020410(R) (2015)

Infinite geometric frustration in a cubic dipole cluster
Johannes Schönke,1 Tobias M. Schneider,1,2 and Ingo Rehberg3
1

2

MPRG-ECPS, MPI für Dynamik und Selbstorganisation, Am Faßberg 17, 37077 Göttingen, Germany
Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, Switzerland
3
Physikalisches Institut, Experimentalphysik V, Universität Bayreuth, 95440 Bayreuth, Germany(Received 24 July 2014; revised manuscript received 5 January 2015; published 29 January 2015)
The geometric arrangement of interacting (magnetic) dipoles is a question of fundamental importance inphysics, 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 offrictionless magnetic couplings. Using computer-assisted algebraic geometry methods, we moreover completely
enumerate all equilibrium configurations. The seemingly simple cubic system allows for exactly 9536 unstablediscrete equilibria falling into 183 distinct energy families.
DOI: 10.1103/PhysRevB.91.020410

PACS number(s): 75.10.−b, 05.65.+b, 41.20.Gz

Magnetism has fascinated mankind for millenia [1]. Today,even the smallest magnets can hardly be overestimated intheir relevance for magnetic storage technology. A fascinating
example for the interplay of magnetic particles is theirself-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 sphericalmagnets in a simple cubic cluster. How are these magnetic
spheres oriented in such an ordered cluster? For the minimalarrangement within this class, namely, a cube consisting ofeight magnets [see Fig. 1(b)], the answer is intriguing: There
are infinitely many orientations. We find the ground state to bea continuum of energetically degenerate states—an extreme
form of magnetic frustration. The phenomenon of frustrationarises 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, thequestion arises: Are there any other equilibrium orientations?
Through our application of methods from numerical algebraic
geometry (see Supplementary 4 [5]) we are able to constructand classify the complete set of equilibrium states. This set
comprises thousands of unstable discrete dipole orientationsin 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 havebeen studied [9] and a full phase diagram in terms of dipole
strength and packing fraction was given [10]. Further, theself-assembly and transition from rings to chains controlled
by an external field was investigated (see, e.g., [11–13]).
Isolated planar dipole clusters in external fields have been
considered in quadratic and ringlike configurations [14,15], as
well as the ground states of planar chains and rings and theirinteraction [16]. A three-dimensional (3D) isolated cluster of
uniformly distributed dipoles on a sphere is given in [17] and1098-0121/2015/91(2)/020410(5)equilibrium configurations with a hierarchy of chains, rings,
and tubes have been found in [18] and further detailed in [19].
In all these related works, a continuous equilibrium (ground)state of a dipole cluster has not been discovered. Further,
we characterize the complete set of all possible equilibrium
configurations of a cluster.
The precise problem addressed reads as follows: N freelyorientable dipoles of equal magnitudes are given together with
their fixed positions in space. Which equilibrium configurations are possible? How many of these equilibria are stable?
Which equilibrium represents the energetically favorable
ground state, i.e., has the lowest energy? Here, we consider theclassical dipole-dipole interaction with the magnetic energy
per dipole,
1  mi · mj − 3 (mi · eij ) (mj · eij ),N i<j|rij |3N

E=

where m1 , . . . ,mN are the variable dipole moments withequal magnitudes |mi | = 1, and rij denotes the fixed relative
position vector between dipole i and j with eij being the

FIG. 1. (Color online) (a) The “magnetic cube puzzle”: A stablearrangement of 216 spherical magnets in a 6 × 6 × 6 simple cubic
cluster. (b) The minimal simple cubic configuration of sphericalmagnets, in a 2 × 2 × 2 cluster. The ground state of this arrangement
is not a single equilibrium configuration with only one discreteorientation for each dipole but rather a continuum of infinitely many
configurations.

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PHYSICAL REVIEW B 91, 020410(R) (2015)

(b)

(c)

1

3/4
angle
polar

1/2

/

1/4

0
-1/4

0

1/4

1/2

3/4

1

azimuthal angle

5/4

3/2

7/4

/

FIG. 2. (Color online) (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 torotations of the dipoles in planes perpendicular to the cube’s volume diagonal of the respective corner in analogy to the mechanical system inFig. 2(c). 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 landscapefor 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(b) and 2(a). This analogy opens up new engineering
possibilities to construct frictionless magnetic couplings.

corresponding unit vector (cf. Supplementary 1 [5]). The equilibrium condition ∇E = 0 corresponding to stationary points
in the energy landscape is represented by a set of strongly
coupled polynomial equations for all dipole orientations of
the cluster (cf. Supplementary 2 [5]). Numerical algebraic
geometry methods described in Supplementary 4 [5] allow
one to construct the complete solution set and thereby find all
equilibrium configurations.
We investigate different elementary clusters of dipoles (seebelow) but report here in detail about one case which proves
to be special: the case of eight dipoles located at the cornersof a cube. As pointed out before, the ground state of thisarrangement is not a single configuration with only one discreteorientation for each dipole but rather a continuum of infinitely
many configurations. Below we refer to this ground statecontinuum in short as the “continuum”; its spatial structureis shown in Fig. 2(a). The continuum exhibits a reflection
symmetry through the three central planes (each parallel toa pair of cube faces)—the dipole moments (as axial vectors)
flip sign under reflection. If we rotate one of the dipoles along
the continuum all the other dipoles rotate accordingly. Sucha rotation is not affected by any magnetic counterforce since
we stay on the same level in the magnetic energy landscape.
This walk through the “ground state valley” is depicted inFig. 2(b). For the unit cube the energy of√ the continuum√
has the characteristic value of Ec = −2 + 2/16 + 3/18
and its net magnetic moment is zero. Furthermore, the toroid
moment with respect to the center of the cube is also zero (cf.
Supplementary 1 [5]).
The continuum described here sheds a new light onfrustration in magnetic systems, which has regained alot of attention in recent years because of tailor-designed

structures (e.g., “artificial spin ice”) showing new andexciting thermodynamic behavior [4]. Classically, frustration
has been studied with the following assumptions: Discrete
spins interact (anti)ferromagnetically with nearest neighbors,
mostly on an infinite lattice, though some interesting studies
on finite isolated magnetic clusters do also exist (e.g., [20]).
In these discrete systems frustration emerges necessarily froma countable number of different states which are energeticallydegenerated. Furthermore, there are quantum mechanicalmodels of 2D spin systems (XY model) with “antidipolar”
interactions [21] (the Hamiltonian has the opposite signcompared to the dipolar interaction) or general bilinear spin
interactions [22] (which would include dipolar interactions
but they are not considered in [22]). These models can possess
continuous states in tetrahedral arrangements [21,23] or on
the pyrochlore lattice [22]. In contrast to all these systems
we study classical dipoles which are freely orientable in 3D
space subject to not only nearest-neighbor but fully coupledinteractions in a finite system of N dipoles. For dipolar interactions, the resulting type of frustration in the cube has a new
quality: The finite system reported here has an uncountableinfinite number of different states which are energeticallydegenerated. In a sense, it is therefore “infinitely frustrated.”
Note that the continuous state is not a simple consequence of
the individual dipoles being freely (continuously) orientable.
In general, the anisotropic dipole-dipole interaction induces
discrete equilibrium configurations (see examples below) sothat the cube continuum is indeed exceptional.
A further intriguing aspect of the continuum is the existenceof an exact mechanical analog. The aforementioned possibility
to rotate one dipole along the continuum with the other dipoles
following accordingly, raises the question: Can we reproduce

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PHYSICAL REVIEW B 91, 020410(R) (2015)

2N

k=1

λk = Tr(H) = −4E3unstable
continuum2
magnetic energy E

the same dynamics through another type of interaction? It
is possible with eight bevel gears, as explained in Fig. 2(c).
The analogy between the mechanical and the magnetic systemallows for new ways to engineer couplings. Since there are nomagnetic counterforces to overcome along the continuum, one
can build a frictionless sevenfold magnetic coupling. Practicalapplications require robustness of the ground state continuumwith respect to perturbations of the perfect cubic arrangement
as well as to deviations from the point dipole approximation.
Experimental tests for two prototype realizations directly
confirm the robustness of our results. Details about these
constructions together with the analysis for the perturbed
arrangement are given in Supplementary 6 [5]).
The cube continuum is embedded in a richly structured
state space which contains a multitude of other equilibrium
configurations. We now provide the complete enumeration for
all possible equilibria in the cube. This is a highly nontrivial
problem since we aim at determining all solutions of astrongly coupled system of 32 polynomial equations (cf. Supplementary 2 [5]). The number of possible zero-dimensional(0D) solutions (corresponding to discrete isolated orientations)grows exponentially with the number of dipoles. For the cube
we have a simple upper bound of 224 = 16 777 216 possible 0D
solutions (cf. Supplementary 3 for the derivation [5]). Additionally, there may be higher dimensional solution manifolds;
the continuum described above is a 1D example. Fortunately,
this system size can be tackled with methods known under theterm “numerical algebraic geometry” which were developed in
the last two decades [24–26] (Supplementary 4 [5]). The result
is an astonishing number of 1 594 032 (generally complex) 0D
solutions. Besides that, there are four 1D continua, two of them
being complex, plus the ground state continuum describedabove and a second real continuum. Higher dimensional
solution manifolds do not exist (cf. Supplementary 4 [5]).
Extracting the physically meaningful real-valued subset of
0D solutions, we still end up with 9536 solutions. These can
now be sorted into energy families, i.e., all solutions withidentical energies belong to the same family. There are always
at least two configurations with identical energy because thepolarity symmetry (reversing the orientation of all dipoles) isagain a solution and leaves the energy unchanged. In addition,
there is the full symmetry group of the cube of order 48.
Depending on the symmetries of the respective solution, wetherefore may have up to 2 × 48 = 96 members in one energy
family. In general, there could be more members in a family if
two configurations which are not related through symmetries
have accidentally the same energy, although this does not
happen in the cube. The sorting gives rise to 183 families of
0D solutions. The energy spectrum of these families together
with some exemplary configurations is shown in Fig. 3. Aninteractive web gallery of all equilibrium configurations in the
cube can be found in [27].
The stability of any equilibrium configuration in our system
is determined by the 2N eigenvalues λk of the Hessian matrix
H, i.e., the matrix of all second-order partial derivatives of theenergy E with respect to the 2N degrees of freedom. A general
result for systems considered here is the relation

1

0

-1

ground state continuum
-2
0

20

40

60

80

100

120

140

160

180

family number

FIG. 3. (Color online) 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 andcube 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 noapparent 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; allsymmetries 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 statecontinuum and the second real continuum.

(see Supplementary 5 for the derivation [5]). It shows that apositive energy E > 0 is a sufficient condition for instability.
Since the sum of all eigenvalues is negative, there must be
at least one negative λk , which classifies a configuration as
unstable. From this we can conclude that a negative energy
E < 0 is a necessary condition for stability. For the cube thismeans that the second real continuum is unstable (cf. Fig. 3).
Calculating the eigenvalues, we actually find all 183 familiesof discrete equilibria to be (unstable) saddles in the energy
landscape, i.e., mixed positive and negative λk . The only
exception is the maximum energy family which is necessarily
unstable in any direction. This confirms that the ground statecontinuum is the only stable state.
Now we put the cube and its continuum into the context
of other regular dipole clusters. We consider the dipoles to belocated at the vertices of various regular geometric shapes.
Table I lists the number of solutions and energy families
for different arrangements. The ground state continuum of
the cube seems to be an exceptional property. So far we
did not find any other regular arrangement which has thisfeature. Simple planar arrangements like the line segment or
the equilateral triangle do not have continua; their ground states
are necessarily discrete. The two smaller (in terms of numbersof corners) platonic solids, i.e., tetrahedron and octahedron,
have continua, but these are unstable (see also [23]). Therefore,

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TABLE I. Number of discrete equilibrium configurations (DEC)for different arrangements of freely orientable dipoles. The dipoles
are positioned at the corners of the respective arrangement. N is thenumber of dipoles, Smax is a simple upper bound (cf. Supplementary
3 [5]) for the number of possible DECs, S is the actual number of(generally complex) DECs, Sreal is the number of real-valued DECs,and Freal is the number of energy families the real-valued DECs
split into. The last row serves solely as an illustration of exponential
complexity.
Arrangement

N

Smax = 23N

S

Sreal

Freal

Line segment
Triangle
Tetrahedron
Octahedron
Cube
Icosahedron

2
3
4
6
8
12

64
512
4096
262 144
16 777 216
68 719 476 736

8
96
420
37 608
1 594 032
?

8
48
116
1156
9536
?

4
8
10
43
183
?

their ground states are also discrete. Another common featureof the regular arrangements investigated is the existence of only
one stable configuration (modulo energetic degeneracies due
to symmetries). For larger clusters, we expect several stableconfigurations to coexist.

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In this study we report on a yet unknown type of ground state
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many energetically degenerate orientations. This result raises
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