Nobel Prize in Physics 2016: Topology and Strange Phases of Matter

This note covers the Nobel Prize in Physics 2016: who won it, how David J. Thouless, F. Duncan M. Haldane and J.

Michael Kosterlitz used a branch of mathematics called topology to explain strange states of matter in very thin, very cold materials, how their discoveries unfolded across the 1970s and 1980s, why the work matters for future electronics and quantum computers, and the key facts for quick revision.

What was the Nobel Prize in Physics 2016 awarded for?

The Nobel Prize in Physics 2016 was given "for theoretical discoveries of topological phase transitions and topological phases of matter". This is the exact wording used by the Royal Swedish Academy of Sciences, which decides the physics prize every year.

In plain words, the three laureates showed that matter can exist in states that ordinary ideas about order and symmetry cannot explain.

They borrowed topology, a branch of mathematics that studies properties which only change in whole-number jumps, and used it as a new language to describe how thin layers of superconductors, superfluids and magnets behave, especially at temperatures close to absolute zero.

Half of the prize money went to David J. Thouless, and the other half was shared between F. Duncan M. Haldane and J. Michael Kosterlitz. The announcement was made on 4 October 2016, and the total prize amount that year was 8,000,000 Swedish kronor.

The Royal Swedish Academy of Sciences said in its press release that the laureates had "opened the door on an unknown world where matter can assume strange states", using advanced mathematics to study phases such as superconductors, superfluids and thin magnetic films.

Who are the laureates?

All three laureates were born in the United Kingdom and held their award-time affiliations in the United States, though some of the prize-winning work, such as the Kosterlitz-Thouless transition, was carried out in the United Kingdom.

Who was David J. Thouless?

David J. Thouless was born on 21 September 1934 in Bearsden, United Kingdom, and died on 6 April 2019 in Cambridge, United Kingdom. At the time of the award he was an emeritus professor at the University of Washington, Seattle, WA, USA. He received one half of the prize.

Thouless earned his PhD from Cornell University in 1958, supervised by the future Nobel laureate Hans Bethe.

He became a professor at the University of Birmingham in 1965, later moved to Yale University, and joined the University of Washington in 1980.

With Michael Kosterlitz he explained phase transitions in thin, cold layers in the early 1970s, and in the 1980s he showed that the precise integer steps seen in the quantum Hall effect were topological in nature.

Who was F. Duncan M. Haldane?

F. Duncan M. Haldane was born on 14 September 1951 in London, United Kingdom. At the time of the award he was the Eugene Higgins Professor of Physics at Princeton University, Princeton, NJ, USA. He received one quarter of the prize.

Haldane studied at Cambridge University, gaining a PhD in 1978 under the future Nobel laureate Philip Anderson. He worked in Grenoble, Los Angeles and at Bell Laboratories before joining Princeton in 1990.

In 1983 he showed that chains of magnetic atoms with different quantum properties behave in fundamentally different ways, and in 1988 he predicted that a topological quantum fluid, like that seen in the quantum Hall effect, could exist even with no magnetic field at all.

Who was J. Michael Kosterlitz?

J. Michael Kosterlitz was born on 22 June 1943 in Aberdeen, United Kingdom. At the time of the award he was the Harrison E. Farnsworth Professor of Physics at Brown University, Providence, RI, USA. He received one quarter of the prize.

Kosterlitz studied at Cambridge University and completed his PhD at Oxford University in 1969. He carried out much of his prize-winning work with David Thouless at the University of Birmingham, before becoming a professor at Brown University in 1982.

Together with Thouless he showed, in the early 1970s, that superconductivity and superfluidity could occur even in extremely thin, effectively two-dimensional layers.

What problem were the laureates trying to solve?

Physicists had long classified phases of matter, such as solids, liquids and gases, using the idea of symmetry breaking.

A liquid is uniform in every direction, but when it freezes into a crystal, that uniform symmetry is broken into a more limited, patterned arrangement. This way of thinking, developed by Lev Landau, worked well for ordinary materials.

But researchers believed that in a flat, two-dimensional world, thermal jostling of atoms would always destroy any ordered pattern, even at absolute zero. If there is no order, there seemed to be no possibility of a phase transition at all in thin films, superconducting layers or superfluid sheets.

A second, separate puzzle appeared in 1980, when the German physicist Klaus von Klitzing discovered the integer quantum Hall effect: in a very cold, very thin conducting layer placed in a strong magnetic field, electrical conductance jumped only in exact integer steps, with extraordinary precision, regardless of impurities or small changes in the field.

Nothing in existing band theory explained why nature should prefer such exact whole numbers.

Both puzzles needed a new mathematical idea that could describe order and precision without relying on ordinary symmetry breaking. That idea was topology, the branch of mathematics that classifies shapes by properties, such as the number of holes, that can only change in whole-number steps.

How did Kosterlitz and Thouless explain phase transitions in flat worlds?

In the early 1970s, Michael Kosterlitz and David Thouless met at the University of Birmingham and challenged the belief that thin, two-dimensional layers could never show ordered phases or phase transitions.

Their explanation centred on tiny whirlpool-like patterns called vortices, which can form in a thin magnetic film, superfluid or superconducting layer.

The logic of what is now called the Kosterlitz-Thouless (KT) transition can be set out as a sequence of steps:

1. At very low temperature, vortices and their mirror-image "antivortices" appear only as tightly bound pairs, because a single lone vortex costs too much energy to exist on its own.
2. As the temperature rises, the system gains more ways (more entropy) to arrange free vortices than the energy cost of separating a pair.
3. At a precise critical temperature, the balance tips: bound vortex pairs suddenly break apart, or "unbind", into independently moving vortices.
4. Above this critical temperature, the free-roaming vortices destroy the ordered, resistance-free behaviour, so superconductivity or superfluidity in the thin layer disappears.

The popular information page noted that this was "one of the twentieth century's most important discoveries in the theory of condensed matter physics". A related idea had earlier been proposed by the Soviet physicist Vadim Berezinskii, so the transition is sometimes also called the BKT transition.

Draw a flat square representing a thin cold layer.

On the left half, sketch two small circular arrows close together, one rotating one way (a vortex) and one rotating the opposite way (an antivortex), joined by a short line to show they are bound as a pair.

On the right half, at a higher temperature, sketch the same two arrows now far apart and moving independently, to show the pair has unbound.

How did Thouless and Haldane uncover topological phases of matter?

A second strand of the prize concerns the 1980s, when both David Thouless and Duncan Haldane found that some phases of matter cannot be described by symmetry breaking at all, but need topology instead.

In 1983, Thouless and his collaborators showed mathematically that the integer steps seen in the quantum Hall effect were an example of a topological invariant, a quantity that, like the number of holes in a doughnut, can only take whole-number values and cannot change under small, smooth disturbances such as impurities or a slightly altered magnetic field. This explained why the measured conductance steps were so exact.

At around the same time, Haldane studied chains of tiny atomic magnets and discovered a sharp difference depending on their underlying quantum property, called spin.

He showed that a chain built from "even" magnetic units behaves as a topological phase, with a gap in its energy levels, while a chain of "odd" units does not.

Later, in 1988, Haldane predicted that a topological quantum fluid like the one in the quantum Hall effect could appear even with no external magnetic field, in what is now called a Chern insulator.

The popular science text records that this 1988 idea was validated experimentally only as recently as 2014, using atoms cooled to near absolute zero, showing how far ahead of experiment the theoretical prediction had been.

What new topological phases followed from this work?

Once topology was shown to matter, physicists began finding many more examples beyond thin films and atomic chains. The press release notes that topological states now turn up "not only in thin layers and threads, but also in ordinary three-dimensional materials".

New names entered the field: topological insulators, materials that conduct electricity only along their surface while remaining insulating inside; topological superconductors; and topological metals.

Each of these is defined not by the ordinary pattern of a crystal but by an invariant number, rather like the fixed number of holes in a doughnut that cannot change without tearing the object apart.

The underlying mathematics behind these ideas, called a Chern number, measures how electron wave functions twist as they move through a material.

Because it must be a whole number, it cannot shift gradually when a material is slightly disturbed, which is exactly why the quantum Hall conductance steps are so precisely exact and resistant to impurities.

The press release said that research in this area "has boosted frontline research in condensed matter physics" over the decade before the prize, partly because of hopes that topological materials could be used in future electronics, superconductors or quantum computers. These remain active, open areas of research rather than finished technologies.

How did the discovery unfold?

Why does it matter?

The laureates' work changed how physicists think about phases of matter. Before their discoveries, phases were classified almost entirely by symmetry breaking, in the style developed by Lev Landau. The 2016 prize showed that this classification was incomplete, because topological phases exist that have no ordinary broken symmetry at all.

Practically, the ideas have fed directly into the search for new materials. The press release pointed to "the hope that topological materials could be used in new generations of electronics and superconductors, or in future quantum computers".

Because topological properties resist small disturbances such as impurities, they are attractive for building more stable electronic devices and more error-resistant quantum bits.

The work also opened an entire research field. The award ceremony speech noted that research into topological phases "has virtually exploded" in the years before the prize, with topological insulators, superconductors and other exotic states now studied across condensed matter physics.

Many of these states, and their possible uses, are still being actively explored rather than fully understood.

At a broader level, the prize is a reminder that new mathematics, in this case topology, can unlock physical understanding that older tools could not reach, a point the award ceremony speech described as mathematics combining "beauty" with experimentally tested "truth".

Quick facts for exams

The Nobel Prize in Physics 2016 was awarded to David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz "for theoretical discoveries of topological phase transitions and topological phases of matter".

It was announced on 4 October 2016 by the Royal Swedish Academy of Sciences. Thouless, who worked at the University of Washington, received half the prize. Haldane, at Princeton University, and Kosterlitz, at Brown University, shared the other half equally.

All three were born in the United Kingdom. The prize used the mathematics of topology to explain phase transitions in thin, cold layers and the precise steps seen in the quantum Hall effect. The total prize amount that year was 8,000,000 Swedish kronor.

Note: Source. The prize facts in this note are from the Nobel Prize's official site, nobelprize.org.

Glossary

• Topology — the branch of mathematics that studies properties of shapes, such as the number of holes, that only change in whole-number steps.
• Phase transition — a change from one state of matter, such as liquid, to another, such as solid or an ordered magnetic state.
• Superconductor — a material in which electrical current flows with no resistance, usually at very low temperature.
• Superfluid — a liquid, such as very cold helium, that flows with no viscosity or resistance at all.
• Vortex — a small, swirling, whirlpool-like pattern that can form in a thin magnetic, superconducting or superfluid layer.
• Kosterlitz-Thouless (KT) transition — the phase transition in two-dimensional materials caused by bound vortex pairs suddenly breaking apart as temperature rises.
• Quantum Hall effect — the phenomenon in which electrical conductance in a thin, cold, magnetised layer changes only in precise integer steps.
• Topological invariant — a quantity, such as the number of holes in an object, that can only take whole-number values and resists small changes.
• Chern number — a whole-number topological invariant used to explain the exact steps in the quantum Hall conductance.
• Spin chain — a line of atoms each carrying a small magnetic property called spin, studied by Haldane for its topological behaviour.
• Topological insulator — a material that insulates inside its bulk but conducts electricity along its surface, because of its topological properties.
• Chern insulator — a topological phase, predicted by Haldane, that shows a quantized Hall effect without needing any external magnetic field.
• Condensed matter physics — the branch of physics that studies the properties of solids and liquids, including unusual phases such as superconductors.
• Absolute zero — the lowest possible temperature, about minus 273 degrees Celsius, at which unusual quantum phases of matter can appear.

Common errors and misconceptions

• Misconception: The three laureates shared the prize equally. Correct: Thouless received one half, while Haldane and Kosterlitz each received one quarter.
• Misconception: The prize was for an experimental discovery. Correct: The citation specifically honours theoretical discoveries, though some of the predictions were later confirmed by experiment.
• Misconception: Topology in this prize means the general mathematics of shapes used everywhere in physics. Correct: It refers specifically to properties, such as the number of holes, that change only in whole-number steps.
• Misconception: The Kosterlitz-Thouless transition behaves like an ordinary phase transition, such as ice melting. Correct: The scientific background document states it does not break any symmetry, unlike ordinary transitions.
• Misconception: Klaus von Klitzing shared this 2016 prize for discovering the quantum Hall effect. Correct: Von Klitzing received his own Nobel Prize in 1985; the 2016 prize honoured the theoretical explanation of his discovery.
• Misconception: Haldane's 1988 prediction was confirmed immediately. Correct: The source states the model was validated experimentally only as recently as 2014, decades later.
• Misconception: All the laureates' prize-winning work was done in the United Kingdom, where they were born. Correct: The Kosterlitz-Thouless transition was worked out in the United Kingdom, but Thouless's and Haldane's later topological work on the quantum Hall effect and spin chains was done after they had moved to the United States.

Exam-style questions with model answers

Q1. State the official citation for the Nobel Prize in Physics 2016. [2 marks]

1. The prize was awarded "for theoretical discoveries of topological phase transitions and topological phases of matter".

Q2. Name the three laureates of the Nobel Prize in Physics 2016 and state how the prize money was shared among them. [2 marks]

1. David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz shared the prize, with Thouless receiving one half and Haldane and Kosterlitz each receiving one quarter.

Q3. Explain what a vortex is and how vortex pairs explain the Kosterlitz-Thouless transition. [4 marks]

1. A vortex is a small, swirling pattern that can form in a thin magnetic, superconducting or superfluid layer, together with its opposite partner called an antivortex.
2. At very low temperature, vortices and antivortices stay bound together in tight pairs, because a single free vortex would cost too much energy to exist alone.
3. As temperature rises, the system gains more possible arrangements (entropy) for free vortices than the energy cost of separating a pair.
4. At a precise critical temperature, bound pairs suddenly unbind into free vortices, destroying the ordered, resistance-free state and marking the Kosterlitz-Thouless phase transition.

Q4. What puzzle did the integer quantum Hall effect present, and how did topology help solve it? [4 marks]

1. In 1980, Klaus von Klitzing found that electrical conductance in a thin, cold, magnetised layer changed only in extremely precise integer steps, regardless of impurities or small field changes.
2. No existing theory of electron bands explained why nature preferred such exact whole numbers for the conductance.
3. In 1983, David Thouless showed mathematically that the integer steps were an example of a topological invariant, a quantity that can only take whole-number values.
4. Because this invariant cannot change under small disturbances, the conductance steps stayed exact even when the material had impurities, explaining the puzzling precision.

Q5. Discuss how the 2016 Nobel Prize in Physics changed the way physicists classify phases of matter, with reference to the contributions of all three laureates. [6 marks]

1. Before this work, phases of matter were classified mainly using the idea of symmetry breaking developed by Lev Landau, where an ordered phase, such as a crystal or magnet, breaks a symmetry that existed in its disordered form.
2. Kosterlitz and Thouless showed in the early 1970s that thin, two-dimensional layers could still undergo a phase transition driven by vortex pairs unbinding, even though this transition does not break any symmetry, which was a completely new type of transition.
3. In 1983, Thouless used topology to explain why the quantum Hall effect's conductance changed only in exact integer steps, introducing the idea of a topological invariant that is insensitive to small disturbances.
4. Also in the early 1980s, Haldane showed that chains of magnetic atoms with even and odd quantum spins behave in fundamentally different, topologically distinct ways, and in 1988 he predicted a topological fluid that needs no magnetic field at all.
5. Together, these discoveries showed that Landau's classification of phases was incomplete, because topological phases exist with no broken symmetry, and this insight has since led to the study of topological insulators, superconductors and metals across condensed matter physics.

Q6. Where was each laureate born, and where was each affiliated at the time of the award? [3 marks]

1. David J. Thouless was born in Bearsden, United Kingdom, and was affiliated with the University of Washington, Seattle, USA.
2. F. Duncan M. Haldane was born in London, United Kingdom, and was affiliated with Princeton University, New Jersey, USA.
3. J. Michael Kosterlitz was born in Aberdeen, United Kingdom, and was affiliated with Brown University, Providence, USA.

Key takeaways

• The Nobel Prize in Physics 2016 honoured theoretical discoveries of topological phase transitions and topological phases of matter.
• David J. Thouless received half the prize; Duncan Haldane and Michael Kosterlitz shared the other half equally.
• Kosterlitz and Thouless showed that thin, two-dimensional layers can undergo a phase transition driven by vortex pairs breaking apart.
• This Kosterlitz-Thouless transition was unusual because, unlike ordinary phase transitions, it does not break any symmetry.
• Thouless later explained the precise integer steps of the quantum Hall effect using a topological invariant.
• Haldane showed that magnetic atom chains with even and odd spins are topologically different, and predicted a topological fluid needing no magnetic field.
• The ideas of topology have since led physicists to study topological insulators, superconductors and metals.
• Researchers hope topological materials may one day be useful in electronics, superconductors and quantum computers.

Test yourself

What mathematical branch did all three laureates rely on?

They relied on topology, the branch of mathematics that studies properties, such as the number of holes, that change only in whole-number steps.

Who announced the Nobel Prize in Physics 2016, and when?

The Royal Swedish Academy of Sciences announced the Nobel Prize in Physics 2016 on 4 October 2016.

What happens to vortex pairs at the Kosterlitz-Thouless critical temperature?

Bound vortex-antivortex pairs suddenly break apart, or unbind, into free vortices, which destroys the ordered, resistance-free state.

What did Klaus von Klitzing discover, and how does it connect to this prize?

Von Klitzing discovered the integer quantum Hall effect in 1980; David Thouless later explained its precise steps using topology.

What was unusual about Haldane's 1988 prediction of a topological fluid without a magnetic field?

Haldane predicted it theoretically in 1988, but it was confirmed experimentally only as recently as 2014, using atoms cooled near absolute zero.

Name one material class, besides atomic chains and thin layers, now studied using topological ideas.

Topological insulators are now studied; they insulate inside their bulk but conduct electricity along their surface because of their topological properties.

Where did Michael Kosterlitz carry out much of his prize-winning work, and with whom?

J. Michael Kosterlitz did much of this work at the University of Birmingham together with David Thouless in the early 1970s.