The 2016 Nobel Prize in Physics

’for theoretical discoveries of topological phase transitions and the topological phases of matter’

The three laureates were rewarded for pivotal contributions to our understanding of the topological properties of particular physical systems such as two-dimensional superfluids and one-dimensional spin chains. Their research is fundamental to a large and active field of research in condensed matter physics with connections to particle physics. Although the research that was recognised is basic and rather abstract, it could eventually lead to new types of electronics and computers.

The Kosterlitz-Thouless transition

Part of the prize was awarded for a description of the phase transition between superfluids or superconductors and the normal state in a two-dimensional layer of atoms (such as liquid helium) or electrons. This was work carried out in the early 70s when Kosterlitz was engaged in postdoctoral research with Thouless at the University of Birmingham.

There was seemingly a paradox in the theory of superfluids because in the simplest and generally accepted theory of phase transitions (Ginzburg-Landau), there was no critical temperature, Tc, over which a two-dimensional superfluid becomes normal — that is, when the superfluid properties, such as the absence of viscosity, disappear. This runs counter to the knowledge that the superfluid should be less robust in two dimensions than in three dimensions and that there is a critical temperature in 3D.

The solution to the paradox was that one had previously failed to take topological excitations into account. In a 2D superfluid the topological excitations rotate on the surface. The superfluid is described as a continuous complex wave function with amplitude and phase. If you make one rotation around the vortex, the phase must change with an integer multiple of 2π. (The phase is illustrated by the direction of a vector in Figure 1.)

Vortices can be generated thermally, but because that requires a lot of energy, they come in pairs that stay together and rotate in the opposite direction. Kosterlitz and Thouless showed that at a certain temperature (now called the KT temperature, TKT) entropy tops the energy required to pull apart the pairs and superfluidity is destroyed. Consequently, one also has a phase transition in two dimensions, but a special kind of topological phase transition. The Kosterlitz-Thouless transition is now a fundamental concept in theoretical physics that has been confirmed experimentally in superfluids and superconductors and also describes how two-dimensional solids melt.

Haldane’s hypothesis

Magnetism (such as in iron) is a complex phenomenon that has to do with the interplay between magnetic ions or electrons. Antiferromagnetism in particular, when nearby ions’ magnetic moments (spins) align in a counter direction, is an intrinsic quantum mechanic and many-body phenomenon. Just as in the case of superfluids, it is especially intricate in low (one or two) dimensions. The theoretical treatment of a one-dimensional chain of spins is a classic problem that largely was solved by Hans Bethe (who incidentally was Thouless’ PhD supervisor) back in 1931, using what is now called Bethe’s approach.

A bit like a metal, the chain has excitations with arbitrarily low energy. If you instead place two chains next to each other to form a ladder, it turns out that the system possesses an energy gap between the ground state and the lowest excited state. In the early 1980s Haldane was able to show that the difference between these two types of systems can be explained with the help of topology. In a semi-classical description of a spin as a unit vector, there is a topological term corresponding to the winding number of the vector over a sphere (space-time). Exactly as in the case of a vortex in a superfluid, this is an integer. For ladders with an even number of legs, the term yields nothing, and without the topological term, the semi-classical description has an energy gap, which now is called the Haldane gap.

 

Vortex (a), calculation of the winding number around a closed curve which contains the vortex (b), a vortex/anti-vortex pair that at a distance cancels each other out.

Vortex (a), calculation of the winding number around a closed curve which contains the vortex (b), a vortex/anti-vortex pair that at a distance cancels each other out.

 

For ladders with an odd number of legs, the topological term yields a varying symbol in the summation of different spin configurations, consistent with a state without an energy gap. This led Haldane to present the hypothesis that chains with an odd number of legs do not have a gap, while ladders with an even number of legs have a gap. This subsequently has been confirmed both theoretically and experimentally. The ground-breaking aspect of this was that the difference between the two systems could be understood via a topological argument.

 

Spin chain (a), two-legged spin ladder (b), the unit vector of a sphere that corresponds to winding number 1.

Spin chain (a), two-legged spin ladder (b), the unit vector of a sphere that corresponds to winding number 1.

Topological insulators, majorana states, spintronics and quantum computers

Recently, similar models experienced a real renaissance from the realisation that certain semiconductors, now called topological insulators, can be classified by a topological number. This is where topology finds concrete expression in that the material has what is known as a chiral state on the surface of the crystal, where the electron’s spin is strongly linked to its orientation. It is hoped that this quality can be used in a new type of electronics, known as spintronics, in which the electron’s spin instead of its charge conveys information.

The majorana state, in which an electron effectively divides itself into two parts, is a related area that currently is attracting a lot of attention, both theoretically and experimentally. There are hopes that these exotic states can be applied to quantum computers, where the robust nature of quantum bits are guaranteed by the fact that they are topological.

 

Mats Granath, Mats Granath, Docent,
Department of Physics, University of Gothenburg