Knot-like structures that arise from various fields called Hopfions have been the subject of intense study for fifty years, but until now have yet to be made manifest in a physical system.
Unique knot-shaped solitons named Hopfions emerge as a basic polarization field in confined ferroelectric nanoparticles, new research published in the journal Nature reveals.
The finding is of particular significance as, in addition to being of importance in fields of physics such as cosmology, astrophysics, and high-energy physics, Hopfions can now be considered fundamental to the study of the electromagnetic properties of nanocomposites — materials that comprise of nanoparticles weaved into a matrix of ‘standard’ or macro substances.
In addition to this, magnetic Hopfions could be of particular importance in the development of spintronic technology — devices that use an intrinsic magnetic property of electrons called spin to store and transmit information.
The research also marks something of a first in physics, as, despite the fact that the search for Hopfions has been undertaken for nearly half a century, an actual realization has yet to materialize.
What are Hopfions?
The concept of a Hopfion may seem initially quite abstract possessing an intrinsic non-physical nature, the topology — the properties of a geometric object — of fields has been of interest to scientists ever since the birth of electromagnetism, and the introduction of an integral to link two curves in 1833.
At the turn of the 20th Century, tangles and knots in topological fields rose to prominence when such structures were observed in solar currents, arising as a result of magnetic forces.
Hopfions are simply knot shaped structures developing within fields, as a result of which, many researchers will specify if they are discussing electromagnetic or gravitational Hopifions.
Thus far, despite their ubiquity in various areas of science and particularly physics, Hopfions have been underutilized. Part of the reason for this is due to the fact that, as Hopfions are characterized by non-linear partial differential equations, the mathematics required to study them extremely difficult.
What is important to note is that Hopfions — knots in a three-dimensional continuous unit vector field that can’t be unknotted without cutting — aren’t particles. They are a class of string-like solitons or solitary waves — a self-reinforcing wave packet — that is able to maintain its shape whilst propagating at a constant velocity. Scientists observe examples of solitons in a number of varied places, including in the flow of water or as light pulses within optical fibers.
Solitons can hold their shape for surprisingly large distances, a result of a delicate balancing act between a wave’s tendency to disperse, and the properties of the medium through which it propagates. This quality means that Hopfions should also possess many of the properties of a particle and can be considered a ‘quasi-particle.’
Previous studies have identified Hopfions as emerging as stable states of magnetic materials, and research has focused on systems that are both constrained and unconstrained. The team in question, led by Valeriji Vinokour, Argonne National Laboratory, Lemont, United States, used a constrained system discovering that it has a considerable effect in the generation of Hopfions.
To arrive at their findings, the physicists considered a spherical nanoparticle in which the total energy of the system couldn’t be described as by isotropically distributed. In other words, the energy took different values across the system. Such systems are generally considered a good way of studying nanodots — bundles of nanoparticles with dimensions of around 100 nm or less.
To this constrained system, the team introduced small perturbations, leading to the formation of depolarization charges and mono-domains’ at the surface of the nanoparticle. The presence of these depolarized regions results in the system not being energetically stable and creates a depolarization electric field. In an attempt to regain stability, the system transforms itself into a structure with the vanishing depolarization charges.
Magnetic field lines reconnect redistributing a field-induced depolarization charge at the points of their termination, guaranteeing the near-perfect screening of the applied field. A polarization vortex is formed and this ‘escapes’ into a third spatial dimension as a result of continuous deformation of the vector field.
This is where the constraints enforced by the team become important. In an unrestricted 3D space, the process described above would simply result in a uniform polarisation. In a confined space, this should result in a return to an earlier, unfavorable state. To avoid this, polarization ‘backflows’ across the axis of the polarisation vortex. The resultant field is a 3D knotted solution — the Hopfion.
Knotty solutions and future applications
The team also note, that in larger nanoparticles, the polarization winds in a similar way creating Hopfions, but then further structures ‘bud’ off these formations — referred to by the team as ‘Hopfioninos’ — this development can eventually, with increasing particle size, result in a chaotic texture.
The team’s finding could considerably increase the potential uses of nanocomposites and in particular, demonstrates that confinement of ferromagnetic material has a radical effect on its electromagnetic properties.
One of the most exciting potential uses of nanocomposites involves biomedical applications like tissue growth and cellular therapies, as well as a wide series of uses in engineering.
As a result, a seemingly abstract element of physics could vastly improve our quality of life.
Sources and Further Reading
Luk’yanchuk. I, Tikhonov. Y, Vinokur. V. M,  ‘Hopfions emerge in ferroelectrics,’ Nature Communications, [https://www.nature.com/articles/s41467-020-16258-w#article-comments]
‘Introduction to Hopfions’ http://hopfion.com/
Rybakov. F. N, Kiselev. N. S, Borisov. A. B, et al, ‘Magnet hopfions in solids,’ , Condensed matter, [https://arxiv.org/abs/1904.00250]
Hayat. H, Kohary. K, Wright. C.D, , ‘Emerging Nanoscale Phase-Change Memories: A Summary of Device Scaling Studies,’ Reference Module in Materials Science and Materials Engineering, [https://doi.org/10.1016/B978-0-12-803581-8.04144-8]