Random Geometry in Condensed Matter Theory

A central task for theoretical physics is explaining how the strange choreography of the microscopic world - the quantum-mechanical dance of electrons - produces the multifarious characteristics of macroscopic stuff: metallicity, magnetism, superconductivity, and so on. Equally crucially, theory should reveal new macroscopic phenomena that have not yet been seen because we did not know to look for them. These are hard tasks, since an atom and a magnet (say) are separated by a staggering jump in scale and complexity.

Fortunately, our understanding of quantum mechanics, as applied to assemblages of many interacting particles, is currently exploding. New types of quantum materials are emerging into the light which - unlike a simple magnet - have no analogue in classical physics. (They rely on the "spooky action at a distance" unique to quantum mechanics and ideas from topology - the mathematics of knots, etc.) Separate work has shown that the central assumptions of statistical mechanics fail radically in some strongly disordered (dirty) quantum systems. Simultaneously, we are discovering that phase transitions between different quantum states are far subtler than we thought.

The unifying theme for this proposal is a very general picture of physical systems in terms of fluctuating extended objects - for example vortex lines, or flux lines, or 'worldlines' in space- time. Such geometric descriptions are often more useful than descriptions in terms of electrons. For example, certain exotic states (quantum 'spin liquids' and related 'topological paramagnets') are best viewed as as Schrodinger's-cat-like mixtures of different configurations of loops, representing flux lines in a field which emerges miraculously from the dance of the electrons.

Using pictures like this, I will tackle such questions as: how do we describe the new types of quantum phase transition theoretically? What do they teach us about quantum field theory? How do we realise the theoretically predicted topological states? How robust are they to perturbations and disorder? These are crucial questions for theoretical physics, which we must answer in order to explain the diversity of material behaviours that emerge from the (deceptively simple) laws of quantum mechanics.

(1) Impact on quantum condensed matter physics. Ninety years after Schrodinger wrote down his famous equation, we are still only beginning to understand the ways in which large numbers of quantum-mechanical particles can organise themselves. We have repeatedly discovered, either through experiments (for example in cuprate superconductors) or through theory (as for topological insulators and their recent generalisations) that our language for describing 'types of stuff' is incomplete. The context for this project is our long struggle to extend this language to describe those materials in which interactions have the most dramatic effects. This theoretical understanding will underpin the revolutionary technological discoveries of the future.

Firstly, this project will help push our understanding of quantum criticality beyond the comfort zone of traditional theory. The new tools developed will impact the large community grappling with quantum criticality in insulators, metals and superconductors. Secondly, the project will provide new heuristics for thinking about topological phases of matter. This will impact researchers working both on how such states behave theoretically and on how to realise them in the laboratory.

The research programme, which pursues topical subjects from a fresh perspective, will also contribute to the vigour of the condensed matter community in the UK. The impact on the local community will include transfer of knowledge to students and other young researchers through personal interactions, seminars and collaborations.

(2) Applications. Ideas from the project will eventually have a downstream impact on applications. Theoretical understanding of strongly correlated electrons will be essential for turning the unconventional physics of critical or topological systems into predictable, controllable technologies. Both critical systems and topological states hold the promise of important applications. The key property of critical systems is their enhanced sensitivity to influence (e.g. pressure, temperature, applied fields, doping, etc.), which could one day be used to create easily 'switchable' devices. A fuller understanding of quantum criticality may also be key to reaching higher temperature superconductivity. Many applications have already been suggested for the best-understood topological states, ranging from spintronic devices to - at the most amibitious end of the scale - topologically protected quantum computing, in which quantum entanglement and topology are used to safeguard quantum information from environmental noise.

(3) Impact on other academic fields. The three parts of this project all make use of a conceptual picture in terms of extended, linelike degrees of freedom, coupled with field theory for performing quantitative calculations. The picture of extended degrees of freedom leads to close connections with other fields, as described in 'Academic Beneficiaries'. The project will help to invigorate these fields (all of which are strongly represented in the UK), and to generate dialogue between these fields and hard condensed matter.

(4) Social impact. Scientific discoveries contribute to our shared culture. The project includes plans for broader dissemination via articles for a general audience, which will communicate the excitement of recent topological ideas in physics through simple examples. Generating enthusiasm for science will help encourage young people to study it, and will lead to greater engagement with science policy.