Nerd Christmas continued with the announcement of the Nobel Prize for Physics on October 4th, 2016. This year’s winners are Drs. David Thouless, F. Duncan Haldane, and J. Michael Kosterlitz. They won by using an esoteric branch of theoretical math, called topology, to explain the physical behaviors of very thin, very cold sheets of atoms. Their discoveries, which began in the 1970’s, are remarkable because they explain previously inexplicable behaviors of matter at extremely cold temperatures. They are further remarkable because at the time, these discoveries were completely theoretical, and have since been backed up by experimental proof.
Topology is not everyday math. To understand more about it, I enlisted my friendly neighborhood topologist, Dr. Michael Abel, who is a postdoctoral fellow at Duke University. The thrust of topology is answering questions about how space and matter deform without fundamentally changing. Topology and the more familiar field of geometry, which describes physical characteristics like length and volume, form “two sides of the same coin,” mathematically speaking, Dr. Abel says. Topology is more concerned with the “overall structure of… matter,” and those characteristics are not tied to the matter’s length or width.
Consider a solid rubber ball. Its smooth surface contains no holes, rendering it topologically very simple. By squishing and stretching the ball, you could form a plate or a cup. In doing so, the geometry of the ball has changed—it might be longer and have a larger volume. However, without drastically changing the rubber ball, say by drilling a hole in it, its topology does not change.
Thus, you cannot make a coffee cup or a pretzel out of the rubber ball without drilling a hole and changing the ball’s topological properties. In this case, the topological change in question is the number of holes in the rubber ball. The ball could have zero, one, two, or more holes, and each of these scenarios produces a topologically different rubber ball. Importantly, the ball cannot have 1.5 holes, or 2.837 holes- it can only have a whole number of holes. It is this part of topology, the investigation of structural changes by whole number values, that is important for understanding why the Nobel Prize was awarded to Drs. Thouless, Haldane, and Kosterlitz.
It turns out that topology is very useful for understanding strange states of matter- namely, very thin films of atoms chilled to extreme temperatures. These films are so thin, they are practically 2 dimensional- having length and width, but no thickness. Additionally, the cold is so extreme—in the neighborhood of absolute zero—that molecular movement almost completely ceases.
In such an environment, the extreme cold creates a high degree of order in the film. This order is so orderly that the film becomes superconductive or superfluid, which means that charges or liquids flow without friction- in the large-scale world, think of a top that never stops spinning. By heating the film, disorder arises, but only in multiples of a whole number. In these states of matter, disorder does not increase gradually, as it does when water is heated. It increases in a stepwise fashion, as when adding holes to the aforementioned rubber ball.
Studying matter in extreme conditions such as low temperature or high pressure helps us understand our world better. When fluids or charges flow, they normally encounter friction and lose momentum, which is why rocks erode in riverbeds and electricity cannot travel indefinitely through high-voltage wires. However, near the lower limit of temperature itself, materials abandon friction and flow freely. The research recognized by this Nobel Prize has implications for electronics, magnets, and many modern devices. However, its most valuable insight is the greater understanding that humans gained from learning about nature near the limits of the rules of our universe. It helps researchers fine-tune our physical theories of the universe, and adds to our understanding of matter itself.
The Inner Life of Life 