Whoever knew that probing into the two-dimensional physical world could open a pandora’s box? David Thouless, Duncan Haldane and Michael Kosterlitz sure did!

The eminent trio were recently awarded the Nobel Prize for Physics 2016 for their work on explaining strange phenomena in the states of matter in a flat world – a world that can be considered to be laminar and two-dimensional. The realm of the single atomic particle, of electrons and protons, of quarks, is that of the quantum world. Many of you may have heard of the idiosyncratic characteristics of this realm such as the probabilistic outcome of measurement of properties in this realm. However, when such entities come together in the two-dimensional world, they are found to display some very interesting collective phenomena. Most importantly, it was found by the Haldane, Thouless and Kosterlitz that topology plays a major role in the physical ‘flatlands’, so to say.

**Figure**: Two-Dimensional physical systems, the ‘flatlands’, have interesting properties such as the presence of vortices in the spin-ordering that undergo phase-transitions as we vary the temperature of the system.

Topology is a branch of mathematics that describes properties of a space that change step-wise under *continuous deformations*, such as stretching and bending and twisting, but not tearing or gluing various sections of the space. With topology as the primary tool, this year’s Laureates presented surprising results, which have opened up new avenues of research.

Phases of matter, such as the solid, liquid and gaseous states, transition between each other when the temperature changes. It is also found that close to absolute zero, at zero Kelvin, matter assumes strange unheard-of phases and begins to behave in unexpected ways, with quantum physics that otherwise operates in the small-scale world suddenly becoming conspicuous! At very low temperatures, other strange properties emerge such as the cessation of the resistance otherwise encountered by all moving particles, such as the one experienced by electric current in the form of resistance.

**Kosterlitz-Thouless Phase Transition**

It was a long-held view that thermal fluctuations destroy any and all order in matter, even at absolute zero. Since the existence of ordered phases is a prerequisite for there to be transitions between them, the hypothized absence of order pre-empted the possibility of such phase transitions altogether. However in the 1970s, David Thouless and Michael Kosterlitz challenged this concept. They began looking into the problem of phase transitions in the two-dimensional world. This eventually led them to what is considered to be one of the biggest discoveries in the theory of condensed matter physics: the KT transition (Kosterlitz-Thouless transition). This involved what is known as *topological phase transitions.*

In the matter ‘flatlands’, the topological phase transition is quite different from an ordinary phase transition, such as that between water and water-vapour (via evaporation and condensation). The primary fundamental unit in such transitions are small vortices of matter. Vorticity can be described as the local rotatory motion of a continuous physical entity near a point. At very low temperatures the vortices of matter in the two-dimensional world form pairs. When the temperature rises, a phase transition takes place which makes the vortices suddenly move away from each other and sail off in the material themselves.

For a brief mathematical understanding of the transition, we will look into what is known as a two-dimensional XY model that consists of planar rotors of unit length arranged on a two dimensional square lattice. Given the convenience of writing mathematical formulas in LateX, I will include the derivation for the KT transition in a LateX-pdf snapshot below:

(c): Mrittunjoy Guha Majumdar

We can calculate the circulation integral for the field around a vortex.

(c): Mrittunjoy Guha Majumdar

Thus the distortion of the phase field from the vortex pair is able to cancel out at distances far from the centre of the two vortices large (compared to the separation R) between the vortex and the anti-vortex unlike the case of the single vortex which has a log-divergence even for large distances from the vortex location in the lattice.

It is the logarithmic dependence on system size of the energy of the vortex combined with the logarithmic dependence of the entropy that produces the subtleties of what can be called *vortex activation and unbinding transitions*. As the temperature is increased, vortex pairs become thermally activated. At certain temperatures, the elements of vortex pairs start to seperate and move away from each other. This is known as the Kosterlitz-Thouless (KT) Phase Transition.

Over the years, the KT transition has been found to be universal and proven experimentally, besides being a key concept in various branches of physics, be it atomic physics or condensed matter physics.

**Quantum Hall Effect**

After around a decade of the formulation of the KT transition theory, in 1983, David Thouless proved that in the presence of strong magnetic fields and at low temperatures, a topology was important in describing certain properties of matter, particularly in the mysterious phenomenon that came to be known as the quantum Hall effect (was discovered in 1980 by Klaus von Klitzing).

**Figure**: The Hall resistance varies stepwise with changes in magnetic field B. Step height is given by the physical constant h/e^{2} (value approximately 25 kilo-ohm) divided by an integer i. The figure shows steps for i =2, 3, 4, 5, 6, 8 and 10. The lower peaked curve represents the Ohmic resistance, which disappears at each step. (Kosmos 1986, published subsequently by nobelprize.org)

Duncan Haldane also arrived at a similar result, independently, at around the same time while analysing magnetic atomic chains. The quantum hall effect is the phenomenon of the electrical conductance in a two-dimensional layer assuming only particular integer values, a finding that could not be explained by the physics of that period. The answer was eventually found in topology. Topologically, any form with one, two, three, four,… holes in it belong to different categories, with objects in each category being capable of being converted to other objects in that category. So, for example, topological a sphere and a bowl belong to the same category and a spherical lump of clay can be transformed into a bowl, while a doughnut with a hole in the middle and a coffee cup with a hole in the handle belong to another (same) category and can also be remodelled to form each other’s shapes.

In the quantum Hall effect, electrons move freely in the two-dimensional layer between semi-conductors and form what is known as a *topological quantum fluid*. It is not possible to determine whether electrons have formed a topological quantum fluid if one only observes what is happening to some of them but when seen as a collective entity, a paradigmatic shift in the theory is introduced where topology becomes important. Since the electric conductance of a material describes the electrons’ collective motion and as topological categories vary in integral steps (associated with the number of holes in a form), the topological variation introduces variation in collective properties of electrons such as conductance leading to quantised (integral) values of electric conductance.

**The Future**

Building upon the inspirational work by this years’ Physics Nobel laureates, various kinds of matter such as topological insulators and topological superconductors have become areas of interest and research in recent years, primarily with the hope that topological materials will be useful for new generations of electronics and superconductors, or even possibly for building quantum computers.

Not to forget, the trio have been closely associated with the University of Cambridge. Duncan Haldane studied at Christ’s College, where he was awarded a Bachelor of Arts degree followed by a PhD in 1978 for research supervised by Philip Warren Anderson. David Thouless earned a Bachelor of Arts degree at Trinity Hall, while John Kosterlitz received his B.A. degree, subsequently promoted to an M.A. degree, at Gonville and Caius College.

Given the motivational achievement of the trio, I would like to conclude this articles with the hope that I share regarding the plethora of possibilities of research in the world of physical ‘flatlands’ and the need for greater focus on applications of such material to electronics and computing.