An example of the fractal structures in spin ice, together with a famous example of a fractal (the Mandelbrot set) on top of a photograph of water ice. Image: Jonathan N. Hallén, Cavendish Laboratory, University of Cambridge.
An example of the fractal structures in spin ice, together with a famous example of a fractal (the Mandelbrot set) on top of a photograph of water ice. Image: Jonathan N. Hallén, Cavendish Laboratory, University of Cambridge.

The nature and properties of materials depend strongly on dimension. Life in a one-dimensional or two-dimensional world would be very different from the three dimensions we’re commonly accustomed to. With this in mind, it is perhaps not surprising that fractals – objects with fractional dimension – have garnered significant attention since their discovery. Despite their apparent strangeness, fractals arise in surprising places – from snowflakes and lightning strikes to natural coastlines.

Researchers at the University of Cambridge in the UK, the Max Planck Institute for the Physics of Complex Systems in Dresden, Germany, the Universidad Nacional de La Plata in Argentina and the University of Tennessee have now uncovered an altogether new type of fractal appearing in a class of magnets called spin ices.

This was a surprising discovery because the fractals were seen in a clean three-dimensional crystal, where they conventionally would not be expected. Even more remarkably, the fractals are visible in dynamical properties of the crystal, and hidden in static ones. As the researchers report in a paper in Science, these features motivated the appellation of ‘emergent dynamical fractal’.

The fractals were discovered in crystals of the material dysprosium titanate, where the electron spins behave like tiny bar magnets. These spins cooperate through ice rules that mimic the constraints that protons experience in water ice. For dysprosium titanate, this leads to some very special properties.

Jonathan Hallén, a PhD student at the University of Cambridge and lead author of the paper, explains that “at temperatures just slightly above absolute zero, the crystal spins form a magnetic fluid.” This is no ordinary fluid, however.

“With tiny amounts of heat, the ice rules get broken in a small number of sites, and their north and south poles, making up the flipped spin, separate from each other, traveling as independent magnetic monopoles.”

The motion of these magnetic monopoles led to the discovery of the fractals. “We knew there was something really strange going on,” said Claudio Castelnovo, a professor at the University of Cambridge. “Results from 30 years of experiments didn’t add up.

“After several failed attempts to explain the noise results, 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.”

This latest analysis of the magnetic noise showed that the monopole’s world needed to look less than three-dimensional – 2.53 dimensional to be precise! Roderich Moessner, director of the Max Planck Institute for the Physics of Complex Systems, and Castelnovo proposed that the quantum tunneling of the spins themselves could depend on what the neighboring spins were doing.

“When we fed this into our models, fractals immediately emerged,” said Hallén. “The configurations of the spins were creating a network that the monopoles had to move on. The network was branching as a fractal with exactly the right dimension.”

But why had this been missed for so long? “This wasn’t the kind of static fractal we normally think of,” Hallén explained. “Instead, at longer times the motion of the monopoles would actually erase and rewrite the fractal.” This made the fractal invisible to many conventional experimental techniques.

Working closely with Santiago Grigera of the Universidad Nacional de La Plata and Alan Tennant of the University of Tennessee, the researchers succeeded in unravelling the meaning of the previous experimental works.

“The fact that the fractals are dynamical meant they did not show up in standard thermal and neutron-scattering measurements,” said Grigera and Tennant. “It was only because the noise was measuring the monopoles motion that it was finally spotted.”

“Besides explaining several puzzling experimental results that have been challenging us for a long time, the discovery of a mechanism for the emergence of a new type of fractal has led to an entirely unexpected route for unconventional motion to take place in three dimensions,” said Moessner.

Overall, the researchers are interested to see what other properties of these materials can be predicted or explained in light of the new understanding provided by this work, including ties to intriguing properties like topology.

According to Moessber, spin ice is one of the most accessible instances of a topological magnet. “The capacity of spin ice to exhibit such striking phenomena makes us hopeful that it holds promise of further surprising discoveries in the cooperative dynamics of even simple topological many-body systems,” he said.

This story is adapted from material from the University of Cambridge, with editorial changes made by Materials Today. The views expressed in this article do not necessarily represent those of Elsevier. Link to original source.