Nobel awarded for using math of shapes to explain exotic matter

Three physicists applied topology to quantum properties of materials in extreme conditions

David Thouless, Duncan Haldane and Michael Kosterlitz

TOPOLOGICAL TRIO David Thouless, Duncan Haldane, and J. Michael Kosterlitz received the Nobel Prize in physics for their work on theoretical discoveries in exotic states of matter, which were based on the mathematics of topology.

From left: ©Trinity Hall, Cambridge, Photographer Kiloton Howard; Denise Applewhite/Office of Communications/Princeton University; Johanna Lassy/Aalto University

Bagels and pretzels have a lot in common with the physics of certain materials: The snacks illustrated the mathematics behind theoretical descriptions of exotic states of matter, work which won the 2016 Nobel Prize in physics on October 4. David Thouless of the University of Washington in Seattle, J. Michael Kosterlitz of Brown University and Duncan Haldane of Princeton University received the prize for their research, which predicted new types of materials and spurred interest in the field of topological materials.

Many physicists were surprised by the selection; speculation online predicted that the prize would be awarded for the first detection of gravitational waves (SN: 3/5/16, p. 6), announced on February 11. But the deadline for nominations fell before that date — and researchers typically wait decades for a Nobel nod.

For Thouless, Kosterlitz and Haldane, “this is a deserving recognition,” says M. Zahid Hasan, a physicist at Princeton. “It’s fairly abstract theoretical work. It’s not like finding a new particle, but it’s about how ordinary matter can behave in extraordinary ways.”

animation of mug turning into bagel
MORPHING MUG In the branch of mathematics known as topology, a mug and a bagel are equivalent, as they both have one hole. One shape can be morphed into another without breaking it apart or pasting it back together. LucasVB/Wikimedia Commons
The three scientists’ work was united by concepts from topology, a branch of mathematics that deals with the study of shapes. In topology, different shapes are distinguished by abrupt transitions. A bagel, for example, is distinct from a cinnamon bun, because the bagel has a single hole, explained Thors Hans Hansson, a physicist at Stockholm University and a member of the Nobel committee, who brought himself a topological lunch for the purposes of demonstration. And the Swedish style of pretzel, which has two holes, likewise differs from a bagel. But a bagel and a coffee cup, which each have a single hole, are topological twins — one shape can be gradually morphed into the other without cutting or pasting.

Using concepts from topology, the physicists explored unusual states of matter that occur under extreme conditions, including temperatures near absolute zero, where quantum mechanics becomes important to a material’s behavior. Such unusual quantum states include superconductors, which conduct electricity without resistance, and superfluids, which flow without friction. The prizewinning research predicted new types of phenomena that appear in two-dimensional sheets of materials or one-dimensional chains of atoms — behaviors that were later observed in the laboratory.

Such developments have led to an explosion of interest in the new types of topological materials, one of the hottest topics in physics (SN: 5/22/10, p. 22). “Suddenly, people are realizing that topological effects in quantum mechanics are just a tremendously rich subject,” Haldane said in a phone interview during the announcement of the prize. He said he was “surprised and very gratified” by the honor.

The winners “certainly were the first people to emphasize the role of topology in physical phenomena,” says experimental physicist Laurens Molenkamp of the University of Würzburg in Germany. “These are very capable persons who certainly deserve a Nobel Prize.”

Half of the prize of 8 million Swedish kronor (about $934,000) goes to Thouless, with the other half split between Kosterlitz and Haldane. All three physicists hail from the United Kingdom: Thouless was born 1934 in Bearsden, Scotland; Haldane was born 1951 in London; and Kosterlitz was born 1942 in Aberdeen, Scotland.

Kosterlitz got the news in a phone call he received while in a parking lot in Finland, where he was traveling. “Jesus, that’s incredible,” Kosterlitz remarked during the call, which was posted on the Nobel Prize Facebook page. “It just feels a little bit odd getting this news in an underground car park outside Helsinki.”

Thouless and Kosterlitz made discoveries about phase transitions in two-dimensional materials. While everyone is familiar with certain phase changes — ice melting into water, for example — exotic quantum materials can exhibit different, less intuitive phase transitions, when a material’s properties suddenly shift. The pair discovered a new phase change called the Kosterlitz-Thouless transition. At low temperatures, tornado-like vortices of swirling electrons are locked together. As the temperature is raised, these vortices suddenly separate and travel independently. This type of phase transition has been seen in the lab in very thin films of superfluid helium and superconductors.

In 1983, Thouless used topology to explain a mysterious phenomenon that had been observed in the lab, known as the quantum Hall effect. This effect appears in a thin layer of electrically conductive material, under extreme cold and a high magnetic field. Under such conditions, the conductivity of the layer, rather than varying gradually, can take on only certain values, which are integer multiples of each other. As the magnetic field changes, for example, the conductivity changes, but instead of shifting gradually, it undergoes discrete jumps.

Thouless showed how this effect was related to topology. In the cinnamon bun, bagel and pretzel analogy, each object can have an integer number of holes; there’s no way to add half a hole to the object. The number of holes can’t change smoothly, but only in big leaps — just as the conductivity changes in discrete jumps in the quantum Hall effect. In 1988, Haldane showed that a similar effect can occur even in the absence of a magnetic field.

Haldane also predicted new behavior in chains of atoms. Atoms in the chain each have a quantum property known as spin, which makes them behave like tiny magnets. Haldane showed that a chain of atoms with half-integer spin will act differently than a chain with integer spin.

illustration of graphene
Graphene, which is composed of single-atom-thick sheets of carbon (illustrated above), is one type of material that has shown topological properties. The prizewinning research laid the foundations for the growing field of physics research that uses topology to explain and predict the properties of such unconventional materials. E. Otwell
The trio “laid a foundation for the way that we think about materials and matter,” says Charles Kane of the University of Pennsylvania. “From our understanding of the phenomenon of superconductivity to our understanding of the electronic structure of materials, this has profound implications.”

The work has inspired new developments in topological materials. Materials known as topological insulators carry current on their surface, but are insulators inside. Potential applications range from quantum computing to new types of computer hard drives (SN: 8/23/14, p.8), and include materials that could carry light or electrical current without being disrupted by impurities (SN: 5/18/13, p. 8). The two-dimensional sheets of carbon known as graphene exhibit topological features as well.

The research has “combined beautiful mathematical and profound physics insights and achieved unexpected results that have been confirmed by experiment,” Hansson said during the Nobel announcement.  “It’s really beautiful and it’s deep.”

Staff writer Christopher Crockett contributed to this story.

Editor’s note: This story was edited for clarity on October 5, 2016.

Physics writer Emily Conover has a Ph.D. in physics from the University of Chicago. She is a two-time winner of the D.C. Science Writers’ Association Newsbrief award.

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