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2016 The Year of Topology in Condensed Matter Physics

Dr Zlatko Papic

Dr Zlatko Papic, former postdoc and current collaborator with 2016 Physics Nobel Prize winner Prof Duncan Haldane, explains the science behind the award and how it relates to the research currently being undertaken in the School of Physics and Astronomy at Leeds.

This year two of the most important physics prizes have been awarded to pioneers of topology in condensed matter physics. The 2016 Nobel Prize for Physics was divided between David Thouless, Duncan Haldane and Michael Kosterlitz for “Theoretical Discoveries of Topological Phase Transitions and Topological Phases of Matter”, while the Oliver Buckley Condensed Matter Physics Prize went to Alexei Kitaev and Xiao-Gang Wen for “Theories of Topological Order and its Consequences in a Broad Range of Physical Systems”

Geometry versus Topology

Geometry of an object is determined by its shape and can easily change by small deformations of the object. On the other hand, objects also possess certain properties that do not change that easily. One such property is the number of holes in an object; this property will remain the same unless the deformation is so strong that it creates new holes in an object. The branch of mathematics concerned with understanding and classifying such properties is called topology. A famous link between geometry and topology was discovered in the 19th century: the Gauss-Bonnet theorem, which links the total curvature of a surface to the number of its holes. By this theorem, objects such as the sphere and torus are topologically distinct because a sphere has no holes, while a torus has one, thus they have different “connectivity”. Topology beautifully captures our intuition about geometrical objects, but as physicists, why should we care about it? 

Topology in Condensed Matter Physics

It turns out that topology pops up in physics in many beautiful and unexpected ways. It was recently discovered that topology could characterise the quantum properties of condensed matter systems (i.e., crystals with regular lattice structures). The most important property of a crystal is its spatial periodicity: a lattice repeats regularly after every few atoms. This innocuous property actually makes it very complicated to describe crystals in our ordinary “real” space. Instead, it is mathematically far more natural to describe crystals in the so-called “reciprocal” space. Reciprocal space is obtained by taking the well-known Fourier transformation that converts spatial coordinates into momenta. This reciprocal space is the arena where our physicists’ version of the Gauss-Bonnet theorem can be applied, and this lead to a host of new “topological” phenomena. 

If we want to “see” new topological effects in solid state materials, we need to look for materials whose band structures have one or more “holes” in them. Unfortunately, materials with such band structures are not easy to come by. In the past 30 years or so, physicists have discovered ingenious ways of designing systems whose reciprocal-space topology is non-trivial. One experimental system where this can now be routinely achieved is the “quantum Hall effect”. This is a setup where a gas of electrons, confined to a plane, is placed in a perpendicular magnetic field. Although electron gas can be created by layering semiconductors such as GaAs and AlGaAs on top of one another, one can also get it “for free” in newer materials such as graphene. In fact, as Duncan Haldane showed in 1988, the quantum Hall effect in graphene can even be obtained in the absence of any external magnetic field. One can mimic the magnetic field by carefully tuning the amplitudes for electrons in graphene to hop from one atomic site to its neighbours. 

Topological Phases and Transitions

How does one detect a “topological” characteristic of a given experimental system? The “quantum Hall effect” means that transversal “Hall” conductance of the electron gas is locked around a specific value given only by Planck’s constant and electron charge. This is obviously a quantum phenomenon because in a classical gas of moving charges, the Hall conductance would vary with applied magnetic field. Instead, in the quantum Hall effect the conductance is robust and largely independent of external conditions or sample details. As shown by David Thouless and his collaborators, this is because the conductance is a topological invariant – its value, like the number of holes, cannot change under small perturbations. It could only change if we change the magnetic field by a very large amount. But this immediately prompts another question: if there are “topological phases” that matter can form (for example, the quantum Hall effect), are there also transitions between phases that involve topology? Indeed, as demonstrated by Michael Kosterlitz and David Thouless, again in two-dimensional systems, there could be new kinds of phase transitions. Unlike ordinary transitions (say, between water and ice), topological phase transitions are characterised by binding/unbinding of vortices – the so-called topological defects, which also play an important role in superconductors.

Topological Physics at Leeds

At the University of Leeds a number of groups are currently investigating topological phases of matter and associated phenomena.

In the Theoretical Physics Group, Dr Zlatko Papic focuses on topological phases with strong interactions between particles, such as the fractional quantum Hall effect, which gives rise to the phenomenon of “topological order” (the term coined by Xiao-Gang Wen). The group of Dr Jiannis Pachos studies potential applications of topological matter. In particular, due to its robustness, topological matter is a promising building block of a new types of quantum computers. In collaboration with the School of Mathematics Dr Pachos recently received £800K from EPSRC to study the effects of topology in physics and in quantum computation. 

Experimental activity in the Condensed Matter Group revolves around semiconductor nanonwires (Dr Satoshi Sasaki) and the use of topological insulators for spintronics (Dr Oscar Cespedes).

For more information 

Nobel Prize announcement:

Buckley Prize announcement: 

Jiannis Pachos, Cambridge University Press, 2012: Introduction to Topological Quantum Computation