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Department of Physics

The Cavendish Laboratory
Fractal on Ice

The properties and behaviour of physical systems are strongly dependent on their dimensionality. Life in a one-dimensional or two-dimensional world would be very different from the three dimensions that we are commonly accustomed to. With this principle in mind, it is perhaps not surprising that fractals – objects with non-integer dimensions – have attracted significant attention since their discovery, both in academic as well as popular science contexts. Despite their apparent oddity, fractals occur in manifold settings and length scales in nature, ranging from snowflakes and lightning strikes to natural coastlines. Much effort has been expended to generate fractals for use in many-body physics.

Researchers at the University of Cambridge, the Max Planck Institute for the Physics of Complex Systems, the University of Tennessee, and the Universidad Nacional de La Plata have now uncovered an altogether new type of fractal appearing in a class of materials called spin ices. Published in Science journal, the team’s findings reveal that there are two reasons for the novelty. Firstly, the phenomenon occurs in a clean, perfect three-dimensional crystal, whereas a typical ingredient to induce fractal behaviour is the presence of disorder. Secondly, fractals in spin ice are borne out of the peculiar rules that govern the time evolution of the magnetisation in these systems. These features motivated the appellation of "emergent dynamical fractal".

Spin ice materials had already stood out in recent year for the unusual topological nature of their magnetic properties, and their ability to host emergent magnetic monopole excitations. It is indeed the dynamics of these magnetic monopoles, and their interplay with the crystal structure, that brings about – for the first time – the appearance of a fractal pattern in the bulk of a perfect crystal without disorder.

In more technical terms, the dynamical rules governing the monopole motion in spin ice are underpinned by a quantum mechanical process that depends on the magnetic state of nearby atoms. The researchers implemented this process in large-scale computer simulations and compared the results to high-resolution experimental measurements obtained at ultra-low temperatures. Being dynamical in nature, the fractals are not detectable through measurements of static properties. They do, however, produce a characteristic signal in the response and fluctuations of the magnetisation, which can be measured.  “Indeed, signatures of these fractals had been observed in experiments, some dating back to nearly two decades ago, and they had remained poorly understood to date. Besides the general interest and scientific curiosity of our findings, we thus also explain several puzzling results that have been challenging the scientific community”, shares Jonathan N. Hallén, the first author and current PhD student at the Cavendish Laboratory.

It will be interesting to see what other properties of these materials may be predicted or explained in light of the new understanding provided by our work. The capacity of spin ice to exhibit such striking phenomena holds promise of further surprising discoveries in the cooperative dynamics of even simple topological many-body systems.

“One may wonder whether the slow relaxation observed in these systems – arising from the emergent dynamical fractal behaviour – may be used to put forward a possible new paradigm for the appearance of glassiness in systems without disorder”, said Professor Claudio Castelnovo, Theory of Condensed Matter Physics, Cavendish Laboratory.


Image: 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 Credit: Jonathan Nilsson Hallén, Cavendish Laboratory, University of Cambridge