Math predicts weird materials; leads to 2016 physics Nobel | Science News for Students

Math predicts weird materials; leads to 2016 physics Nobel

Prize recognizes three researchers working in a type of math called topology
Oct 4, 2016 — 5:25 pm EST
Nobel physics

David Thouless, Duncan Haldane and J. Michael Kosterlitz (left to right) will share the 2016 Nobel Prize in physics. The award recognizes their theoretical discoveries on exotic states of matter. Their work was 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

Three physicists won the 2016 Nobel Prize in physics for discoveries about exotic states of matter. Their work was theoretical. That means it was based on mathematical calculations, not on tests in a lab or on observations in nature.

On October 4, the Nobel committee in Stockholm, Sweden announced the three winners. David Thouless is at the University of Washington in Seattle. J. Michael Kosterlitz works at Brown University in Providence, R.I. Duncan Haldane is at Princeton University in New Jersey.

Their work is “fairly abstract," notes Princeton’s M. Zahid Hasan. “It's not like finding a new particle,” this physicist explains. “It's about how ordinary matter can behave in extraordinary ways." Clearly, he says, this award is “deserving” for the trio’s important work.

All three worked in a branch of mathematics that deals with the study of shapes. It is known as topology. Their discoveries spurred interest in creating what have come to be called topological materials.

Potential uses for such materials include quantum computing and new types of computer hard drives.

Thinking about bread

In topology, different shapes are distinguished by abrupt transitions. Consider the holes found in some breads. A loaf of bread has no holes in it. To create a bagel shape, one needs to cut out a hole. A pretzel with two holes can be made from a looped cord of bread. But to morph from the bread loaf shape into a bagel or looped pretzel, a baker needs to introduce a hole or two. That makes each shape, from a topological point of view, quite different.

One of these shapes cannot be gradually turned into the other without cutting parts. But anything with the same basic shape — one hole, two holes or no holes — will be twins. That’s true, at least, from a topological point of view. So as hard as it may be to imagine, by this way of reasoning a bagel and a coffee cup are twins. After all, each has a single hole: The bagel’s is in the middle and the coffee cup’s is in its handle.

topological phase
This diagram illustrates a change known as a topological phase transition. Swirling vortices (arrows) form in a thin layer of material and are locked into pairs at low temperatures (left). As the temperature rises, the vortices suddenly split, free to go off on their own .
©Johan Jarnestad/The Royal Swedish Academy of Sciences

Using concepts from this type of math, the researchers worked to probe extreme states of matter. Most people know the basic states: solid, liquid, gas and plasma. With a phase change, a material may morph from one to another of these. But under extreme conditions — such as temperatures near absolute zero — materials can behave oddly. Now the physics of quantum mechanics takes over. Superconductors are one such quantum state. Here, electricity flows with no resistance.

Research by this year’s Nobel winners predicted new phenomena that might come to life in such a quantum world. And indeed, such oddities eventually did show up.  

In recent years, topological materials have become one of the hottest topics in physics. One widely studied example: topological insulators. These are materials that can serve simultaneously as both electrical insulators — blocking the flow of a current — and conductors of an electrical current.

Suddenly, people are realizing that topological effects “are just a tremendously rich subject,” said Haldane. The Princeton-based winner offered this assessment by phone during the announcement of his prize.

He and his co-winners “certainly were the first people to emphasize the role of topology in physical phenomena,” says Laurens Molenkamp. He’s an experimental physicist. He works at the University of Würzburg in Germany. The scientists named today, he says, “certainly deserve a Nobel Prize.”

What they did

A two-dimensional material has length and width but no height. Thouless and Kosterlitz made discoveries about phase changes in such 2-D materials. Everyone is familiar with certain phase changes. Ice melting into water is one of the best known. Exotic quantum materials also exhibit phase changes, where their properties suddenly shift. But theirs are far less easy to imagine than the change from ice to water.

For instance, this pair of scientists discovered a new phase change that occurs at very low temperatures. Named for them, this Kosterlitz-Thouless transition occurs with the behavior of tornado-like vortices of swirling electrons. Those swirls pair up and are locked together. If the temperature warms, however, these vortices will suddenly separate. Afterward, each will travel on its own. Eventually, experimental scientists witnessed this happening in very thin films of superfluid helium and with superconductors.

In 1983, Thouless used topology to explain another mysterious phenomenon that had been seen in tests. It is known as the quantum Hall effect. The effect develops within a thin layer of electrically conducting material. It was initially seen only when the temperature was extremely cold and a high magnetic field was present. Under such conditions, the electrical conductivity of a layer would change. But it didn't happen gradually. Instead, the changes took place in regular jumps. The move from one phase to another occurred at integer multiples. As the magnetic field changed, for example, the conductivity changed. But instead of shifting gradually, it happened in discrete jumps, perhaps changing to double or triple its initial value.

topology math
In the branch of mathematics known as topology, a mug and a bagel are equivalent. That is because they both have one hole. That lets one shape be morphed into another without breaking it apart or pasting it back together.
Wikimedia Commons

Thouless showed how this effect was related to topology. Consider the bread loaf, bagel and pretzel again. The loaf has no hole, the bagel has one and the slightly looped pretzel has two. Their number of holes varies by a whole number. It can’t vary by a fraction. There is no bread with one-and-a-half holes, or two-and-a-third holes. So any change in the number of holes can vary only in integer leaps — just as the conductivity changes 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 a new behavior in chains of atoms.

Atoms in the chain each have a quantum property. Scientists refer to this as spin. It makes the chains behave like tiny magnets. In this case, the spins can occur as an integer or a half integer.  A chain with integer spin will behave differently than one with half-integer spin, Haldane showed.

Why scientists care about these odd traits

The three new Nobelists “laid a foundation for the way that we think about materials and matter,” says Charles Kane. He’s a physicist at the University of Pennsylvania in Philadelphia. Their findings have profound implications, he says. The effects can be seen “from our understanding of the phenomenon of superconductivity to our understanding of the electronic structure of materials.”

Their work also has inspired new topological materials. For instance, those topological insulators may carry current on their surface, while barring the flow of electricity inside. Or they might prevent light from flowing through a piece of etched glass but allow that light to flow well along its surface.

The research behind such materials has “combined beautiful mathematical and profound physics insights,” said Thors Hans Hansson during the Nobel announcement. Hansson is a physicist at Stockholm University. He is also a member of the Nobel committee for physics. Importantly, he noted, the winners’ mathematical predictions were later “confirmed by experiment.”

The three men will share 8 million Swedish kronor (or about $934,000). But it won’t be divided equally. At a December 10 ceremony in Stockholm, Thouless will receive half of the total. Kosterlitz and Haldane will split the rest.

Their prize is named for Alfred Nobel. Best known as the inventor of dynamite, he was a wealthy man when he died on December 10, 1896. In his will, Nobel left much of his fortune to create prizes to those who have done their best for humanity in the fields of physics, chemistry, physiology or medicine, literature and peace.

Note: This story has been revised to more accurately describe the quantum effects of spin in chains of atoms.

Power Words

(for more about Power Words, click here)

absolute zero     The coldest possible temperature, also known as 0 kelvin. It is equal to minus 273.15 degrees Celsius (minus 459.67 degrees Fahrenheit).

abstract     Something that exists as an idea or thought but not concrete or tangible (touchable) in the real world. Beauty, love and memory are abstractions; cars, trees and water are concrete and tangible.

atom     The basic unit of a chemical element. Atoms are made up of a dense nucleus that contains positively charged protons and neutrally charged neutrons. The nucleus is orbited by a cloud of negatively charged electrons.

bagel     A dense bread roll made in the shape of a fat ring. Bakers first boil the dough, then bake it.

chemistry     The field of science that deals with the composition, structure and properties of substances and how they interact with one another. Chemists use this knowledge to study unfamiliar substances, to reproduce large quantities of useful substances or to design and create new and useful substances. (about compounds) The term is used to refer to the recipe of a compound, the way it’s produced or some of its properties.

conductor     (in physics and engineering) A material through which an electrical current can flow.

current     A fluid body — such as of water or air — that moves in a recognizable direction. (in electricity) The flow of electricity or the amount of electricity moving through some point over a particular period of time.

dynamite     A type of explosive.

electrical conductivity     The ability of some substance (such as water or metals) to transport an electrical charge or current.

electricity     A flow of charge, usually from the movement of negatively charged particles, called electrons.

electron     A negatively charged particle, usually found orbiting the outer regions of an atom; also, the carrier of electricity within solids.

field     An area of study, as in: Her field of research was biology. Also a term to describe a real-world environment in which some research is conducted, such as at sea, in a forest, on a mountaintop or on a city street. It is the opposite of an artificial setting, such as a research laboratory. (in physics) A region in space where certain physical effects operate, such as magnetism (created by a magnetic field), gravity (by a gravitational field) or mass (by a Higgs field).

hard drive     A device that reads and writes — and hence can store — digital data onto a rigid magnetic disk.

helium     An inert gas that is the lightest member of the noble gas series. Helium can become a solid at -458 degrees Fahrenheit (-272 degrees Celsius).

insight     The ability to gain an accurate and deep understanding of a situation just by thinking about it, instead of working out a solution through experimentation.

insulator     A substance or device that does not readily conduct electricity.

integer    A whole (not fractional) number. One, 7, 21 and 381 are all examples of integers. Two-thirds is not.

magnet     A material that usually contains iron and whose atoms are arranged so they attract certain metals.

magnetic field     An area of influence created by certain materials, called magnets, or by the movement of electric charges.

matter     Something which occupies space and has mass. Anything with matter will weigh something on Earth.

mechanics     The study of how things move.

particle     A minute amount of something.

phenomenon     Something that is surprising or unusual.

physical     (adj.) A term for things that exist in the real world, as opposed to in memories or the imagination. It can also refer to properties of materials that are due to their size and non-chemical interactions (such as when one block slams with force into another).

physics     The scientific study of the nature and properties of matter and energy. Classical physics is an explanation of the nature and properties of matter and energy that relies on descriptions such as Newton’s laws of motion. Quantum physics, a field of study which emerged later, is a more accurate way of explaining the motions and behavior of matter. A scientist who works in that field is known as a physicist.

plasma     (in chemistry and physics) A gaseous state of matter in which electrons separate from the atom. A plasma includes both positively and negatively charged particles.

poles     (in physics and electrical engineering) The ends of a magnet.

quantum     (pl. quanta) A term that refers to the smallest amount of anything, especially of energy or subatomic mass.

quantum mechanics     A branch of physics dealing with the behavior of matter on the scale of atoms or subatomic particles.

resistance     (in physics) Something that keeps a physical material (such as a block of wood, flow of water or air) from moving freely, usually because it provides friction to impede its motion.

subatomic     Anything smaller than an atom, which is the smallest bit of matter that has all the properties of whatever chemical element it is (like hydrogen, iron or calcium).

superconductor     Materials that have no resistance to the flow of electricity, typically only when they are cooled below a certain temperature. Superconductors also repel all magnetic fields, which allows them to float in the air when they are placed inside a strong magnetic field.

theoretical     An adjective for an analysis or assessment of something that based on pre-existing knowledge of how things behave. It is not based on experimental trials. Theoretical research tends to use math — usually performed by computers — to predict how or what will occur for some specified series of conditions. Experimental testing or observations of natural systems will then be needed to confirm what had been predicted.

topology     (in math) The study of the properties of shapes and their relationships to each other. Shapes are related when they have similar properties even after deformation (such as bending, stretching, shrinking). They will not be similar if cut, torn or have had some pieces glued (or otherwise patched) onto it.

transition      The boundary where one thing (paragraphs, ecosystems, life stage, state of matter) changes into another. Some transitions are sharp or abrupt. Others slowly or gradually morph from one condition or environment to another.

vortex     (plural: vortices) A swirling whirlpool of some liquid or gas. Tornadoes are vortices, and so are the tornado-like swirls inside a glass of tea that’s been stirred with a spoon. Smoke rings are donut-shaped vortices.


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