On top of a photograph of water ice, an example of fractal structures in spin ice and a famous Mandelbrot set
The dimensions of materials play a role in the properties of goods and services. Considering the differences between living in a one-dimensional or two-dimensional world and the three-dimensional world we know of, fractals have become a topic of interest since their discovery. From snowflakes to lightning strikes, to natural coastlines, fractals are easily discovered.
Researchers from the University of Cambridge, the Max Planck Institute for the Physics of Complex Systems in Dresden, the University of Tennessee, and the Universidad Nacional de La Plata have discovered a completely new type of fractal, which is visible in clean three-dimensional crystals, where they otherwise would not be seen.
The fractals were discovered in the crystals of the material dysprosium titanate, where the electron spins behave like tiny bar magnets. These spins cooperate by following ice rules that mimic those encountered in water ice.
Jonathan Hallén of the University of Cambridge is a Ph.D. student and the lead author of the study. "At temperatures just slightly below absolute zero, the crystal spins form a magnetic fluid."
The ice rules are broken in a small number of places and their north and south poles, separate from each other, traveling as independent magnetic monopoles.
The motion of these magnetic monopoles led to the discovery of this discovery. Professor Claudio Castelnovo, also from the University of Cambridge, says, "We knew there was something quite strange going on."
Castelnovo continued, referring to a new paper on the monopoles that was published earlier this year. "We finally had a eureka moment, realizing that the monopoles must be living in a fractal world and not moving freely in three dimensions, as had always been assumed."
The current study of magnetic noise suggests that the monopole's world must be less than three-dimensional, or rather 2.53-dimensional, according to Professor Roderich Moessner, Director of the Max Planck Institute for the Physics of Complex Systems in Germany.
"When we fed this into our models, fractals emerged." The monopoles had to move on a network that was branching as a fractal with exactly the right dimension."
Why was this information so long left out?
Hallén said that "this wasn't the type of static fractal we'd normally associate with." Instead, at longer periods the monopoles' movements would erase and rewrite the fractal."
This enabled many conventional experimental techniques to see the fractal.
Researchers completed their research in collaboration with Professors Santiago Grigera of the Universidad Nacional de La Plata and Alan Tennant of the University of Tennessee.
"The fact that the fractals are dynamic meant that they did not show up in standard thermal and neutron scattering tests," said Grigera and Tennant. "It was only because the noise was measuring the monopoles' movement that it was finally spotted."
“Besides explaining several obscure experimental findings that have been challenging us for a long time, the discovery of a new kind of fractal has led to an entirely unexpected way for unconventional motion to take place in three dimensions,” Moessner adds.
Overall, the researchers are interested in exploring what other abilities of these materials might be predicted or explained in the wake of their new findings, including ties to intriguing properties like topology. "The ability of spin ice to exhibit such striking phenomena makes us hopeful that it will lead to further surprising discoveries in even simple topological many-body systems," said Moessner.
Jonathan N. Hallén, Santiago A. Grigera, Claudio Castelnovo, and Roderich Moessner, all 15 December 2022, Science. DOI: 10.1126/science.add1644