Using a genetic algorithm to study properties of minimum energy states and geometrical
frustration in artificial “spin ice” systems by A. Leon and J. Pozo, Facultdad de
Ingenieria, Universidad Diego Portales, Chile
Summary: The minimum energy state of a two-dimensional array of ferro-magnetic
islands with intrinsic moments is reported in this paper. Genetic and Monte Carlo
algorithms are used to arrive at the base (minimum-energy) state. The results of the
genetic and Monte Carlo simulations are compared with experimental results for a similar
system, with the conclusion that the experimental protocols used to minimize the energy
of these systems are insufficient.
Introduction: Frustration is defined as a “competition between interactions” leading to a
system that cannot minimize its total energy while simultaneously minimizing the energy
of each individual interaction. The authors give the example of a “spin-ice” system of
magnetic ions for which each 4-spin interaction site favors a “2-in, 2-out” tetrahedral
arrangement, where not all sites can be simultaneously satisfied; thus the system is
referred to as “frustrated.” A similar system is water ice, where each oxygen atom favors
a tetrahedral coordination of 4 hydrogens, with two hydrogens close to the oxygen and
two farther away. Due to the similarity of the interaction rules, systems of magnetic ions
with the aforementioned favored coordination are referred to as “spin-ice” systems. In
practice, magnetically frustrated systems are difficult to investigate without altering the
systems. Several experimental techniques are used to circumvent this practical difficulty,
notably the work reported by Wang et al. on minimum energy states for a ferromagnetic
system.
Model and Simulation: The system is described as a two-dimensional array of “singledomain ferromagnetic islands with intrinsic magnetic moments.” “Single-domain”
implies the magnetic moment of a given island exists in only one direction at a given time.
The islands are 80ns in width, 220nm in length, and 25nm height, implying a threedimensional system for which the magnetic spins are only varying in two dimensions;
the system was modeled to emulate the experiments performed by Wang et al.. A
diagram of a small section of the simulated system of 1024 islands is given below.
Figure 1: Bird’s-eye diagram of a section of the ferromagnetic system. Each rectangle represents a
ferromagnetic island, while each arrow represents the spin associated with a given island. An example of a
vertex is circled to the right of the diagram. 1024 islands were studied in the simulations.
There are 4 possible vertex types, each with a specific degeneracy and energy. The
favored vertex type, Type I, consists of two spins on opposite sides of the vertex pointing
inward and the other two spins pointing away from the vertex. The 4 vertex types, listed
in order of increasing energy from top to bottom are shown with their degenerate states
below.
Figure 2: The 4 possible vertex types, listed in order of increasing energy.
Also important, in addition to the vertex type, is the effective dipolar interaction between
two nearby islands. The equations for the dipole interactions between islands are
presented, followed by a description of the genetic and Monte Carlo algorithms used to
minimize the energy of the system.
To obtain the minimum-energy configuration using the genetic algorithm, 100 individual
configurations are chosen randomly, and the total energy of each system is calculated by
summing each dipole interaction. A survival probability is assigned to each
configuration, with a 100% survival probability assigned to the lowest energy state. 29
more configurations, chosen based on survival probability, undergo mutation and
recombination to yield 99 new configurations which combine with the lowest energy
configuration to form the next generation.
For the Monte-Carlo algorithm, a random configuration is generated using boxes of three
possible colors to represent the area between the islands, with the ordering of the colored
boxes determining the spin types of the adjacent islands. Subsequent configurations are
generated using the same method, with the acceptance probability equal to unity if the
change in energy is less than zero and following a Boltzmann distribution if the energy
change is positive.
Results: Both the genetic and Monte Carlo algorithms lead to the same minimum-energy
state, containing 100% Type I vertices; because there are two ways to arrange a Type I
vertex, the degeneracy of the base state is equal to two. The experimental protocol of
Wang was then simulated by subjecting a random ferromagnetic system to a fluctuating
magnetic field and observing the response of the system energy. It was determined that
the system was unable to achieve the minimum-energy configuration attained by both
algorithms, and thus that the experimental protocols currently used to minimize the
system energy are insufficient.
Questions/Concerns: I cannot reconcile their assertion that they are studying “frustrated”
systems when the results of both of their simulations yield a non-frustrated (100%
minimum-energy interaction) system. Furthermore, if the real (non-simulated) systems
are in fact frustrated, and the simulations indicate that non-frustrated systems are
accessible, then perhaps the simulations are missing several nuances that exist outside of
modeling. For example, the authors only consider the dipole and vertex interactions, but
there may be other types of interactions not accounted for in the simulations. I was also
surprised that the simulations apparently calculated every possible dipole interaction, as
the authors say nothing of a “cutoff” distance beyond which islands do not appreciably
interact.