Speaking to the Nobel press conference via telephone, Haldane said, “I was very surprised and very gratified...It’s only now that there’s a lot of tremendous new discoveries based on this original work...It’s taught us that quantum mechanics can behave far more strangely than we could guess.” In describing his original work, he said “It was really just a toy model demonstration of something...Like most discoveries, you stumble onto them and you just have to realize there’s something interesting there.” Haldane also thanked Nobel prize winner Philip Anderson – his former tutor at the University of Cambridge – for inspiring his unorthodox approach to condensed-matter physics.

### British born

David Thouless was born in 1934 in Bearsden, Scotland. He completed his PhD in 1958 at Cornell University in the US. He was professor of mathematical physics at the University of Birmingham in the UK before joining the University of Washington in 1980 – where he is emeritus professor.

Duncan Haldane was born in London in 1951 and completed a PhD at the University of Cambridge in 1978. He is Eugene Higgins Professor of Physics at Princeton University.

Michael Kosterlitz was born in 1942 in Aberdeen, Scotland and studied physics at the University of Cambridge before gaining a PhD at the University of Oxford in 1969. He is Harrison E Farnsworth Professor of Physics at Brown University in Rhode Island. Kosterlitz has previously worked at the University of Birmingham, the Instituto di Fisica Teorica in Turin, Italy and Cornell University.

### Of doughnuts and coffee cups

The science behind this year's prize tied together three concepts in physics and mathematics, namely: topology; quantum phase transitions; and states of matter. The result is what the Nobel committee described as "beautiful mathematical and profound physical insights". Indeed, the laureates' research has laid the theoretical basis for a variety of condensed-matter staples including superconductors and thin magnetic films.

Common to the work of Haldane, Kosterlitz and Thouless is the concept of topology. This a branch of mathematics that describes properties that remain unchanged when an object is changed or deformed in a series of steps. An old but popular example of such topological changes is that a doughnut-like shape can be transformed into that of a coffee cup and vice-versa. So topologically speaking, both shapes are identical.

In a classical sense, all matter exists as either a solid a liquid or a gas. A phase transition occurs when matter changes from one form to another, such as liquid water turning to ice. Quantum effects do not normally play a role in these familiar phase transitions because they are washed out by thermal fluctuations. However, at very low temperatures near absolute zero, matter takes on strange new phases and quantum effects become very pronounced. A good example of this is that electrical resistance disappears at temperatures approaching absolute zero; or the spin of a vortex in a superfluid seems to flow forever without slowing down.

For a long time, it was believed that any ordered phases would be destroyed in flat 2D systems, even at absolute zero, due to thermal noise – this in turn meant that there could be no phase transitions. But in 1972 Kosterlitz and Thouless overturned that idea by identifying a completely new type of phase transition in such extremely thin layers, where topological defects play a crucial role. As a result, they were able to show that superconductivity or superfluidity can occur in 2D layers at low temperatures. The pair also calculated that the phase transition would occur at relatively high temperatures, above which superconductivity would disappear. According to the Nobel committee, the pair's work "resulted in an entirely new understanding of phase transitions, which is regarded as one of the 20th century's most important discoveries in the theory of condensed-matter physics".

### Wandering vortices

This topological change, now known as the KT (Kosterlitz–Thouless) transition, mainly occurs thanks to the configurations of tiny vortices of electronic spins on these 2D surfaces. At low temperatures, the spin vortices are tightly paired and as the temperature rises, the vortices suddenly separate from one another. This triggered a quantum phase transition from one state of matter to another. The KT transition has since been used to study superconductors and superfluids. It has also been applied to phase transitions that occur when a ferromagnetic thin film is cooled below the Curie temperature and the spins line up, giving rise to net magnetization.

Thanks to experimental advances, the early 1980s also saw the discovery of a number of new states of matter that defied explanation. A particular mystery was the 1980 discovery of the "quantum Hall effect" by German physicist Klaus von Klitzing, who won the 1985 Nobel Prize for Physics for that work. The classical Hall effect is based on the appearance of a measurable voltage across the two sides of a metallic sheet with a current passing along its length, which is placed in a strong magnetic field that is perpendicular to the sheet. The Hall voltage appears as electrons drift towards one edge of the sheet.

### Quantized steps

The quantum Hall effect is seen in 2D materials. Klitzing studied a 2D conducting layer sandwiched between two semiconductor layers, which was cooled to just above absolute zero and placed in a strong magnetic field. He found that the Hall voltage is quantized at very specific, discrete values. These values appeared to be independent of the material used and did not vary when experimental parameters such as the temperature, magnetic field or the amount of semiconductor impurities in the sample are changed. A large enough change in the magnetic field causes the conductance (which is also quantized) to change in fixed amounts – for example a reduction in the magnetic-field strength initially makes the conductance double, then triple and so on. A comparison of the current in the conductor and the Hall voltage showed that the resulting Hall resistance is *h*/*Ne*^{2}, with *N* being an integer, but why these integer steps took place was unknown.

Thouless found an appropriate solution by proving that these integers were topological in their nature. Indeed, he showed that understanding the collective behaviour of the electrons in the conducting thin-film layer was crucial and that the material could be thought of as a topological quantum fluid. In such a fluid, the conductance is described via the electrons' collective motion, and that their topology means that phase-transitions would occur at fixed steps.

### Mind the gap

Around the same time, Haldane was studying the properties of chains of magnetic atoms and how symmetry comes into play. Haldane claimed that magnetic chains would have fundamentally different properties depending on whether the magnetic atom was even or odd – i.e. has an integer or half-integer spin. He showed that even (integer) chains are topologically ordered (and inversion-symmetry remains unbroken), while odd (half-integer) chains are not topological (and inversion symmetry is broken).

Indeed, in 1988 Haldane worked out that there is a spin gap in the excitation spectrum for integer spin-chains, whereas half-integer spin-chains have a gapless excitation spectrum. At the time, Haldane's reasoning was questioned, but it has since been experimentally verified. The work has also helped to forge links between statistical mechanics, quantum many-body physics and high-energy physics – fields that now boast a large shared toolkit of theoretical techniques.

Today, condensed-matter physics regularly studies a variety of topological phases in 2D and 3D materials, as well as topological insulators, superconductors and metals. Indeed, these materials are thought to be at the frontline for potential uses in the next generation of electronic devices.

- This article first appeared on
*physicsworld.com*