Mathematical Physics

Mathematical physicsWhat is it and why do we need it?

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Front cover image

Supersymmetry, conceptual artwork. Superstring theory is an attempt to explain all of the particles and

fundamental forces of nature in one theory by modelling them as vibrations

of tiny supersymmetric strings (David Parker/Science Photo Library)

Quantum states, conceptual artwork. In physics, a quantum state is a set of mathematical variables that fully describes a quantum system, such as the state of an electron within an hydrogen atom at any given time (Richard Kail/Science Photo Library)

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Mathematical Physics What it is and why do we need it?

Mathematical physics. What is it and why do we need it?

The synergy between mathematics and physics

Mathematical physics – its purpose and applications

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Entanglement entropy – characterising quantum entanglement of many-body systems

Entanglement entropy

Quantum applications

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From gravity to fluids and back

Fluid/gravity correspondence

The nature of gravity

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Holography, black holes and superconductors

Superconducting materials

Black holes

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Random matrix theory – linking quantum chaos with the Riemann hypothesis

The Riemann zeta function

Random matrix theory

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Skyrmions

A mathematical model of atomic nuclei

The application of Skyrme’s model to larger nuclei

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Topological insulators – a new phase of matter

Topological insulators

Applications of a not-so-new material

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Contents

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This quote attributed to Albert Einstein rarely fails to raise a smile. It is reassuring for those who struggle with mathematics, but it is also surprising. Even a cursory brush with physics confirms that this science is highly mathematical. Whether it is describing the force of gravity, as was Einstein’s goal, understanding a simple light switch, or designing sophisticated GPS satellites, physicists need numbers to measure and to quantify, and they need mathematical equations to describe the relationships between physical objects and the forces that act on them. Mathematics is the indispensable language of physics — which is why Einstein’s admission strikes many as amusing.

Mathematical physics What is it and why do we need it?“Don’t worry about your difficulties in mathematics. I assure you mine are greater.” Albert Einstein

Gravitational lensing. Optical CCD (charge coupled device) image of a large "luminous arc" associated with Galaxy Cluster 2242-02. Several theories have been put forward to explain such arcs. Most popular of these is that the arc is part of an Einstein ring: the smeared-out image of a distant galaxy or quasar whose light has been "lensed" by the gravity of the cluster of galaxies. Einstein proposed the effect of gravitational lensing in 1936 (NOAO/Science Photo Library)

p Albert Einstein (1879-1955), German-born physicist. Famous for his theories of relativity, Einstein has become a cultural icon, his name synonymous with genius. In 1905, whilst working as a patent clerk, he wrote four papers, including one on special relativity. From this the idea of a space-time continuum followed and is represented here as space warped into the shape of an hourglass (Bill Sanderson/Science Photo Library)

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Mathematical physics What is it and why do we need it?

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If all of physics is mathematical then what is meant by “mathematical physics”? The boundaries are not clearly defined. Despite his misgivings, Einstein himself could be counted as a mathematical physicist. His general theory of relativity was not a result of extensive experimentation, but of theoretical and mathematical considerations. Because it differs from Newton’s classical theory of gravity only when high energies and massive bodies are involved, the theory was difficult to test at the time. But it has stood up admirably to the experimental tests that have been developed in the near century since its inception.

Einstein’s work provides one of many examples of the synergy between mathematics and physics. In the middle of the 19th century, the pure mathematician Bernhard Riemann developed a range of new geometrical concepts. His aim was to drive geometry to abstract perfection; to render it

The synergy between mathematics and physics

independent of how we happen to perceive the physical space around us. Decades later, Einstein found a very physical application for Riemann’s mathematics: it was exactly what he needed to describe the geometry of space and time. Riemann’s geometry has now become an essential prerequisite for anyone wishing to understand Einstein’s physics.

Mathematics does not just provide tools for physics. It can also drive physical insight. A striking example is Paul Dirac’s prediction of antimatter. Dirac was searching for an equation to describe the behaviour of electrons, tiny building blocks of matter, taking account of insights from special relativity and the other great success story of early twentieth-century physics, quantum mechanics. When he found a suitable mathematical expression he realised it contained twice as many pieces of information as necessary, leading him to suggest that each electron comes with an anti-particle,

Mathematics does not just provide the tools for physics – it can drive physical insight. When Paul Dirac found a suitable mathematical expression to describe electrons, he discovered the existence of positrons

the positron. Its existence, and that of many other anti-particles, was later confirmed in the laboratory.

Dirac’s efforts contributed vital pieces to a jigsaw puzzle physicists are still trying to complete today, aiming to describe all the fundamental forces and particles of nature in one theoretical framework, known as the Standard Model of particle physics. The language of this framework is mathematics, and mathematical considerations have led to the discovery of other particles. A recent addition is the Higgs boson, traces of which were glimpsed at CERN in July 2012. Its existence had been predicted back in the 1960s, resulting from a mathematical model devised to explain events in the very early universe that led to matter acquiring mass. Two physicists, Peter Higgs from the UK and François Englert from Belgium, were awarded the Nobel Prize in Physics 2013 for their role in developing the theory that led to the recent discovery.

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The greatest missing piece of the puzzle is the gravitational force, which cannot yet be accommodated in the Standard Model. The nature of the mathematical problems involved has spawned string theory, a major contender for a unified theory of everything. It appeals through its coherence and mathematical elegance, but it will be many years before experiments will be available to test whether it provides a correct model. Despite this, string theory is promising to shed light on the physics of real materials, such as superconductors, that find applications in a variety of contexts.

However, it is not just mathematics that fertilises physics. Physical considerations can pose difficult mathematical problems which then turn into active research areas in their own right. For instance, in 1834 the Scottish engineer John Scott Russell observed a curious solitary wave on the Union Canal near Edinburgh that could not be captured by the existing theory of hydrodynamics. It took 60 years for physicists to develop an equation describing the wave and nearly another century to find a general method for solving it. Their efforts have spawned a vibrant area of mathematics concerned with

p The Standard Model

NA48 experiment apparatus. Engineers working on the north area 48 (NA48) particle detector at CERN (the European particle physics laboratory) near Geneva, Switzerland. This apparatus was used for the NA48 experiment to study the direct effect of CP (charge parity) violation, which may be able to explain matter/antimatter imbalance (CERN/Science Photo Library)

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certain types of equations and their wave-like solutions known as solitons. The results feed back into physics, where solitons have become ubiquitous. The skyrmions explored in one of the case studies in this booklet are examples of solitons (page 24). What may appear surprising is that they describe the protons and neutrons found in the nuclei of atoms.

Particle physics has also influenced Riemann’s geometry, which provided Einstein’s tools. The equations used to describe fundamental particles have a geometrical interpretation and come with a particular algebraic structure, which captures the symmetries physicists believe govern the laws of nature. This has resulted in vigorous research efforts in an area of pure mathematics that combines algebra and geometry. Insights from quantum physics have also nudged mathematicians nearer to the solution of one of

the greatest unsolved problems in pure mathematics, the Riemann Hypothesis, first posed by Riemann in 1859. It concerns the prime numbers, considered the fundamental building blocks of number theory.

Considering this two-way interaction, the field of mathematical physics is best described as consisting of two parts: physical research that proceeds primarily through mathematical means and areas of mathematics that work to solve the problems posed by physics. As such, mathematical physics does not pertain to specific areas of either of the two disciplines. It is impossible to predict which mathematical methods will find applications in physics and what kind of mathematical problems will arise from physical research. Mathematical physics is a dynamic field full of surprises.

p Hydrostatics. A page of illustrations showing various aspects of hydrostatic theory. At top left is a diagram illustrating the calculation of forces in uneven water surfaces, at bottom left are two methods of using weights to raise a water column. The remainder are illustrations of how water finds a common level in joined vessels open to the air irrespective of their shape. This page was first published in Daniel Bernoulli's 'Hydrodynamica' of 1738 (Royal Institution of Great Britain/Science Photo Library)

Mathematical physics is best described as consisting of two parts: physical research that proceeds primarily through mathematical means and areas of mathematics that work to solve the problems posed by physics

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But do we really need it? Its main goals are to expand the boundaries of knowledge and provide a fundamental understanding of physical reality. This is a fascinating pursuit in its own right, but it is also a prerequisite for discoveries and applications that directly impact on our lives. GPS technology would not be possible without insights from general relativity, which in particular describes how the Earth’s gravity distorts space and time.

Quantum physics is paving the way towards superfast quantum computers, opening up a new era for information technology. The medical sciences will continue to benefit from progress in particle physics. For example, the development of high Tesla superconducting magnets for CERN’s Large Hadron Collider allows a better understanding of the superconducting materials used in MRI scanners. There are now a small number of 7-Tesla MRI scanners in clinical use worldwide, and a tiny number of scanners with extremely high field strengths have been developed for state-of-the-art

brain research. The discovery of topological insulators, one of the first truly new materials to be discovered in decades, was only possible due to the light shed by a mathematical understanding of materials.

A common feature of many such applications is that they could not have been predicted before the underlying theory was developed — they are the results of open-ended research conducted over

a sustained period of time. In the current economic climate this type of research is under threat as government and funding agencies hope for more immediate returns. This booklet presents some examples of recent successes in mathematical physics and their potential impacts, highlighting the UK’s role as a world leader in this research area. If we want to pursue fundamental research in physics and benefit from the economic impacts the resulting technological innovations will bring, we must ensure the UK maintains its position as a world leader in mathematical physics.

Mathematical physics – its purpose and applications

The discovery of topological insulators, one of the first truly new materials to be discovered in decades, was only possible due to the light shed by a mathematical understanding of materials

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Quantum entanglement is a very counterintuitive idea. It is possible to entangle two or more particles, such as electrons or photons, so that it seems that the particles are connected; for example, a spin measurement of a particle in an experiment instantaneously affects the others, whether they are in the next room or the next galaxy. When Albert Einstein first encountered the concept of quantum entanglement, he famously thought this “spooky action at a distance” was proof that the theory of quantum mechanics was wrong. Despite Einstein’s

Entanglement entropy –characterising quantum entanglement of many-body systems

p Quantum entanglement, computer artwork. One of the strangest consequences of the quantum theories is that some quantum events can become entangled. Two particles will appear to be linked across space and time, with changes to one of the particles affecting the other one (Harald Ritsch/Science Photo Library)

p Behaviour of the entanglement entropy near the transition of a one-dimensional quantum magnet. One can see a peak in the entanglement entropy (S) at the quantum critical point where the applied magnetic field, lambda, equals 1

Quantum entanglement is a key resource in quantum information processing and forms the bedrock of any design of quantum algorithms

concerns, this effect has been observed in experiments and quantum mechanics has gone on to be one of the most successful scientific theories ever proposed.

Quantum entanglement is a key resource in quantum information processing and forms the bedrock of any design of quantum algorithms. However, to make the most efficient use of this resource we need to be able to quantify it exactly and develop methods for measuring entanglement in real quantum systems.

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Entanglement entropyIn a simple quantum system of just two particles for example, it is easy to quantify how entangled the system is. But it is far harder to measure the amount of entanglement in a system of many particles. Over the last decade John Cardy (University of Oxford) and his colleagues, including Pasquale Calabrese (now at the University of Pisa), have contributed to the systematic study of exact predictions for entanglement entropy, a way of quantifying the degree of entanglement of such many-body systems. Their elegant approach uses all the tools of modern mathematical physics, including those developed in quantum field theory, integrable systems and string theory.

The spin of an electron can have one of two values: it can point either up or down. For a system of two electrons, there are 22= 4 possible states (shown in the figure, left). A quantum system can occupy all these states simultaneously.

Any simulation of this quantum system will need to allow for all these possibilities at the same time. For a system of N electrons, there are 2N possible states, a number that increases rapidly with N, and all of these must be encoded into any simulation of the system.

Now, if two particles are not entangled you can treat them independently, ignoring the one when making calculations for the other. But the more entangled things are, the harder the system is to simulate. You have to take all the entangled variables into account and solve the relevant equations for all these particles at the same time. One can think of the entanglement entropy of a system as a measure of how much information is needed to simulate the quantum system in a computer.

Numerical analysis had already revealed that certain large quantum systems were easier to simulate than others, though it was not known why. Entanglement entropy explains this, with the calculated value of entanglement entropy closely agreeing with the numerical results for the difficulty of simulating equivalent quantum systems.

p� The spin of an electron, a system of two electrons, there are 22= 4 possible states

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Quantum entanglement will help to characterise a deeper understanding of superconductors and quantum magnets, which will aid the development of novel high-tech devices

Entanglement entropy now informs the numerical simulations of these large quantum systems. Such simulations are traditionally very hard as you have to simultaneously solve Schrödinger’s equations describing their quantum state for many particles. Understanding the degree of entanglement of a large quantum system indicates how many particles need to be considered: the smaller the entanglement entropy the fewer the particles that are needed to simulate the system numerically.

There are hopes to test the predictions of entanglement entropy experimentally in the future. Proposals are being developed to measure the degree of quantum entanglement of a system, comparing it with the predicted value of entanglement entropy. Some of these experiments involve the design of quantum switches and other components that potentially could be used in the development of quantum computing. The measurement of entanglement

Quantum applicationsThis mathematical tool can also be used to characterise the quantum critical point of quantum systems, for example, the point at which a material becomes superconducting. As the material nears this point the electrons become more and more entangled, something described most clearly by the increase in the superconductor’s entanglement entropy. This has shed new light on the processes involved in these transitions. Another example is that of a quantum magnet, composed of many interacting spins. As the applied magnetic field approaches the critical point where the phase transition occurs, a sharp peak in the value of the entanglement entropy can be seen (illustrated in the figure, right). Measuring the entanglement entropy could provide a way to track the physical process. Deeper understanding of these materials, both superconductors and quantum magnets, will aid the development of novel high-tech devices.

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entropy in many-body systems will be vital in understanding how to scale up these components to eventually build functioning quantum computers.

Quantum computing will revolutionise how we store, process and use information. A full-scale quantum computer has the potential to perform certain calculations significantly faster than any silicon-based computer. While a normal computer can only work on one calculation at any time, a quantum computer can work on millions simultaneously. Such a technology could have a tremendous effect on financial asset movement, potentially leading to a more efficient financial market and, perhaps, a more predictable one.

Entanglement entropy is an example of how the abstract tools of mathematical physics, often developed for theoretical mathematics with no hint of their future application, are now integral to the research and development of essential future technologies.

p Quantum computer, conceptual artwork. Quantum computers, which are under development, are based on quantum mechanics and the principle of representing information using quantum properties. Quantum computing has the potential to massively increase computing power (Harald Ritsch/Science Photo Library)

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If, as Einstein postulated, gravity is the result of massive objects such as planets and stars warping space and time, then what can be said about the nature of this malleable spacetime? What can be said about those ultimate gravitational objects, black holes, which warp spacetime to the extreme?

Over decades, mathematical scrutiny of Einstein’s equations has led to an interesting idea: that aspects of spacetime might resemble a fluid. If this is true, then can they be captured by the central equations of fluid dynamics, developed nearly 200 years ago by the French engineer and physicist, Claude-Louis Navier, and the Cambridge-based mathematician

and physicist, George Gabriel Stokes? Conversely, can we address open questions about fluids using insights from gravity?

One of the most exciting recent achievements in mathematical physics has been to provide an explicit correspondence between Einstein’s equations and the Navier-Stokes equations, which works both ways. On the fluid side it gives a new tool for studying one of the last remaining questions of classical physics, that of how to describe turbulent fluid flow, and also for studying substances that seem to defy a description in terms of classical physics. On the gravity side it may give deep insights into the nature of spacetime.

s Black hole. Artwork of the spherical region where light is trapped around a black hole, with surrounding interstellar material (blue) being pulled inwards (Victor De Schwanberg/Science Photo Library)

From gravity to fluids and back

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The fluid/gravity correspondence was formulated in 2008 by Veronika Hubeny and Mukund Rangamani (both at Durham University), Shiraz Minwalla (Tata Institute of Fundamental Research in India) and Sayantani Bhattacharya (now at the Indian Institute of Technology in Kanpur).

The fluids to which it currently applies are not ordinary ones. They are nearly perfect fluids

which exhibit little viscosity and heat conduction. Examples are superfluids, such as liquid helium cooled to very low temperatures, which seems to defy the laws of physics, appearing to self-propel and climb up walls. Another example is the mysterious quark-gluon plasma, a particle soup thought to have last existed in nature right after the Big Bang 13.7 billion years ago, and which has recently been produced in tiny quantities in the laboratory.

Fluid/gravity correspondence

Superfluids and the quark-gluon plasma are also examples of so-called strongly coupled systems. These are made of particles interacting so strongly that they can no longer be considered individually. Such systems lack a coherent theoretical description, but they may find analogues on the gravity side of the correspondence, represented by mathematical descriptions of black holes.

The equations here can be more tractable and the hope is that the correspondence will illuminate how these systems work.

Over recent years, hundreds of research projects around the world have exploited the fluid/gravity correspondence. On the topic of fluid motion, researchers have used it to tackle a regime in which the traditional macroscopic approach breaks down. In this non-hydrodynamic regime, which includes turbulence, the velocity or temperature of the fluid shows large

Superfluids to which the fluid/gravity correspondence applies are not ordinary, some seeming to defy the laws of physics by climbing up walls

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variations on very small scales, so a microscopic description would, in principle, be needed. In 2012 Pau Figueras (University of Cambridge) and Toby Wiseman (Imperial College London) used solutions to Einstein’s equations, which describe a new type of theoretical black hole, to derive for the first time a description of both the hydrodynamic and part of the non-hydrodynamic regime for a certain class of fluids. In 2013, Paul Chesler, Hong Liu and Allan Adams (all at MIT) used gravity to derive, from first principles, a description of turbulence in a class of superfluids flowing on a surface. Surprisingly, their work predicts that in this

turbulent motion large structures tend to break down into smaller ones, just like a uniform stream of smoke breaks down into smaller eddies. That is the exact opposite of what happens to ordinary fluids flowing on surfaces. Here smaller structures tend to merge to form larger ones.

Gravitational insights may also give results in connection with the quark-gluon plasma. A physical characteristic of the viscosity of the plasma (the viscosity to entropy density ratio) can be described in terms of the number 1/(4pi), where pi is the famous mathematical constant relating the diameter of

a circle to its circumference. It has been suggested that 1/(4pi) is a universal bound for this characteristic in all fluids, although this idea has been under active debate. Using the fluid/gravity correspondence Bhattacharya, Hubeny, Minwalla and Rangamani have derived the same value in a simpler way than had previously been possible, providing a new angle on the problem. They hope that the correspondence will help to shed more light on this poorly understood quark-gluon plasma.

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The nature of gravityUnderstanding strongly coupled systems is a major aim researchers are hoping to achieve, but the fluid/gravity correspondence also raises questions about the nature of gravity.

How should we interpret concepts such as turbulence in gravitational systems? Physicists are now taking the first steps towards answering this question by identifying properties of black holes that correspond to turbulence in fluids. An example is work published in 2013 by Adams, Chesler and Liu, which uses the fluid/gravity correspondence to construct “turbulent” black holes. The identifying feature uncovered by the researchers was an approximately self-similar structure in the horizons of their black holes: they look the same at several scales of magnification.

The black holes that are involved in this kind of research are not real astrophysical objects. They are mathematical descriptions of black holes that exist in unusual mathematical spaces. But the hope

is that results like these will shed light on aspects of gravity in our own world that we do not yet understand. At the same time researchers are working to extend the fluid/gravity correspondence to capture less exotic fluids and spacetimes.

The overarching goal of this research is to understand the physics that governs our universe, from the smallest to the largest scales. What applications will eventually arise from it cannot be foretold. A better understanding of strange metals and superfluids may lead to materials that become superconducting at high temperatures, but it may also open up uses we cannot yet imagine. A theoretical description of turbulence may ultimately find applications in the full range of contexts in which this motion occurs, from aircraft design to meteorology.

The most fascinating question remains the fundamental one: the fluid/gravity correspondence hints towards a type of universality in nature we do not yet understand. Further research is needed to illuminate its true meaning.

s Turbulence is a common phenomenon in nature. Hurricane Katrina seen from NASA’s Terra satellite (NASA)

A better understanding of strange metals and superfluids may lead to materials that become superconducting at high temperatures, but it may also open up uses we cannot yet imagine

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Superconductors have intriguing properties: these materials exhibit zero electrical resistance when cooled below a critical temperature and they expel magnetic fields. The applications are broad; superconductors are used in MRI scanners, particle accelerators like those at CERN, and to levitate trains, to name just a few. However, the critical temperatures are very low and the cost of the required cooling limits their use. An important aim is to find a material that superconducts at room temperature. A step in this direction is to try to understand the quantum properties of a class of known, but still mysterious, "high temperature superconductors". This is where recent research using an interesting principle that emerged from the mathematical physics of string theory is showing promise.

t Demonstration of magnetic levitation of one of the new high-temperature superconductors – yttrium barium copper oxide. Discovered in 1986, the new superconducting ceramic materials are expected to lead to a technological revolution and are the subject of intensive worldwide research. The photograph shows a small, cylindrical magnet floating freely above a nitrogen-cooled, cylindrical specimen of a superconducting ceramic (made by IMI Ltd). The glowing vapour is from liquid nitrogen, which maintains the ceramic within its superconducting temperature range. Photographed at the University of Birmingham (David Parker/Science Photo Library)

Holography, black holes and superconductors

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Superconducting materials

fundamental forces of nature: the electromagnetic, the weak nuclear and the strong nuclear forces. While the elegance of string theory is highly appealing, technology has not yet advanced far enough to test it in experiments.

Nevertheless, in 1997 a remarkable observation within the mathematics of string theory was made by the physicist Juan Maldacena (now at the Institute for Advanced Study in Princeton), which led him to conjecture a far-reaching correspondence between two apparently very different theories. He developed a mathematical toy universe which, by very loose analogy, can be thought of as an inflated balloon. The physics on

the boundary of that universe, corresponding to the surface of the balloon, is described by a quantum theory with no gravity. The physics in the interior is described by a string theory, which does incorporate gravity. Remarkably, these two physical systems are equivalent: the quantum theory defined on the boundary is sufficient to describe all the physics in the interior and vice-versa. One direction of this two-way correspondence has given Maldacena’s discovery its name: it is known as the holographic principle because the boundary physics captures the interior physics just as a 2D hologram captures a 3D image.

Superconductivity was first discovered in 1911 by the Dutch physicist Heike Kamerlingh Onnes, who observed it in mercury. By the 1960s physicists had found many more examples of superconducting materials and had developed a comprehensive understanding of the sub-atomic quantum processes that made these “conventional” superconductors work. In 1986, however, new types of superconductors were discovered with much higher critical temperatures, which could not be described by the existing theory. Since then physicists have proposed a range of different approaches to explain the fundamental principles underlying these “high temperature superconductors”. However, despite an enormous amount of work, there is no consensus as to which approach, if any, is correct. Radically new paradigms may be needed, one of them potentially emerging from string theory, which provides a mathematical description of the world aiming at unifying gravity with the other three

p��The Japanese JR–Maglev MLX01-1 uses superconductors (Daylight9899)

An important aim in superconductor research is to find a material that superconducts at room temperature. A step in this direction is to try to understand the quantum properties of a class of known, but still mysterious, "high temperature superconductors"

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Black holesUniversity), Chris Herzog (now at Stony Brook University) and Gary Horowitz (University of California, Santa Barbara). Shortly afterwards, in 2009, Jerome Gauntlett and Toby Wiseman (both at Imperial College London), and their former colleague Julian Sonner, now at MIT, demonstrated that such black holes exist within the mathematical framework of string theory. These findings, and others, have set off a flurry of activity in the area, with encouraging results. Research has revealed theoretical black holes whose properties share similarities with properties seen in real materials. For example, Gauntlett and Aristomenis Donos (University of Cambridge) have recently found theoretical black holes with a halo of matter exhibiting a striped pattern. A similar striped phase, where the electric charge density has a periodic behaviour, has been observed in several materials including high temperature superconductors.

Further research is necessary to ascertain whether this approach will lead to the ultimate insight into the physics behind high temperature superconductors, but it may provide the basis for a new paradigm. What is more, many other types of exotic black holes are being discovered in string theory, and via the holographic principle these point to an untold variety of new materials with exotic properties yet to be discovered. This demonstrates the power of mathematical physics: its sophisticated mathematical techniques find applications in physical contexts far removed from the ones in which they were developed.

It is this principle that physicists are now using to try to understand superconductors. A beautiful feature is that the thermal properties of the boundary world can be described in terms of the mathematical properties of theoretical black holes in the interior of the model, as we have seen already in the fluid/gravity context. The results are intriguing: the phase transition that turns a material into a superconductor at the critical temperature, represented in the boundary of the model, corresponds to the existence of a new kind of theoretical black hole in the interior of the model, which comes with a halo of charged matter around it. It is the properties of this halo that characterise the black hole, via the holographic principle, as a superconducting phase.

The first breakthroughs regarding superconducting black holes were made in 2008 by Steven Gubser (Princeton University) and by Sean Hartnoll (Stanford

p Black hole, artwork. Technically, a black hole is a region of spacetime where, by nature of its great mass, gravity prevents anything from escaping; this includes light (Henning Dalhoff/Science Photo Library)

Research has revealed theoretical black holes whose properties share similarities with properties seen in real materials

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In 1859 the mathematician Bernhard Riemann posed a hypothesis that remains one of the most important unsolved problems in mathematics. Mathematicians have been studying prime numbers for millennia as they are the atoms of arithmetic – any integer can be written as a unique product of prime numbers. Much is known about primes and they have become a vital part of the economy due to their role in the encryption of digital information

Random matrix theory –linking quantum chaos with the Riemann hypothesis

used in e-commerce and secure communication. Yet despite our reliance on prime numbers there is much we still do not understand. In particular, a fundamental question remains open: how are prime numbers distributed along the number line? In the 1970s a surprising connection was discovered between this purest area of mathematics and quantum physics. This analogy has enabled mathematical physicists to give new insights into the Riemann hypothesis.

p Computer generated image of a mathematical saddle (Riemann surface). It is a way of representing a form of space where the conventional rules of geometry no longer apply (Alfred Pasieka/Science Photo Library)

Much is known about primes and they have become a vital part of the economy due to their role in the encryption of digital information used in e-commerce and secure communication


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The Riemann zeta functionRiemann considered a particular function, the Riemann zeta function, which describes the distribution of the prime numbers. The points at which this function is zero can be thought of as the sea level points in a landscape defined by the function. The Riemann hypothesis states that these points, rather than being scattered over the landscape, all lie in a straight line (called the critical line). If this is true, it gives important information about prime numbers; Riemann himself wrote a formula connecting the zeroes with the primes.

p A wave of the type occurring in quantum chaos, associated with a classically chaotic system with a downward force – quantum chaos of Galileo's falling body (Sir Michael Berry/University of Bristol)

Random matrix theory has crossed the boundary of mathematical physics and made a huge impact on a completely different field, connecting deep ideas between mathematics and physics giving novel insights in both areas

In the 1960s a similar formula was developed in an entirely different area. Dynamic physical systems (for example, electrons moving in atoms or molecules, or light moving in a laser) can be described in two ways: using the classical physics developed by Newton or using the newer theory of quantum mechanics. The formula linked the energy levels found in such systems when described by quantum mechanics to the periodic trajectories one finds when describing the systems using Newtonian physics.

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Random matrix theoryThe two formulae looked remarkably similar: the Riemann zeroes were like the quantum energy levels and the primes were like the classical trajectories. Furthermore, these formulae suggested the classical orbits in question should be chaotic. This connection deepened in the 1980s when Sir Michael Berry (University of Bristol) discovered that the distributions of the quantum energy levels of all chaotic systems were statistically the same. Remarkably, the Riemann zeroes had the same statistical distribution. This universality meant that all these systems look essentially the same and could be described using random matrix theory. This statistical universality applies at short range. Over longer ranges, the same 1980s theory predicted non-universal deviations from random matrix theory, also for the Riemann zeroes, where quantum chaos theory gave a detailed description, new to mathematicians, of their statistics.

In 2000 Jon Keating and Nina Snaith (both at the University of Bristol) used this connection to shed new light on the Riemann zeta function. Any function can be characterised by its moments – a sequence of numbers that describe its shape to greater and greater degrees of accuracy. The first

our understanding of the Riemann zeta function and the Riemann hypothesis.

This result has led to a new and stimulating collaboration between number theorists and mathematical physicists. After making these calculations of the moments, Snaith and Keating have gone on to answer more general questions in number theory, such as how quickly the Riemann zeta function grows in height, a question that is similar in spirit to the Riemann hypothesis.

New questions about the Riemann zeta function are now being studied as a result of known properties of the analogous quantum chaotic systems. Similarly, known results about the Riemann zeta function have prompted new directions in the study of quantum chaos. Random matrix theory has crossed the boundary of mathematical physics and made a huge impact on a completely different field, connecting deep ideas between mathematics and physics giving novel insights in both areas.

moment can be thought of as the mean, the average height of the function; the second moment the variance, the range of fluctuation in the height; and higher moments give more detailed information about the fluctuations themselves.

Number theorists had been struggling for nearly a hundred years to calculate the higher moments of the Riemann zeta function along the critical line. The first two moments had been calculated by the mid-1920s but there was then little progress until the 1990s. At this point number theorists had guessed the next two moments, giving the sequence: 1, 2, 42 and 24024. However, the methods they had used failed beyond this point and they could not generate any sensible answers for the next numbers in the sequence.

Using the analogy with quantum chaos, Keating and Snaith expressed this problem in terms of random matrices. Not only were they able to calculate the moments already known, they developed a general formula for the whole sequence. Their solution to this long-standing problem in number theory revealed that the sequence of moments had surprising mathematical properties, which have deepened

p Riemann zeta function in the critical strip, showing the zeroes where the colours meet (Sir Michael Berry/University of Bristol)

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What does the nucleus of a carbon atom look like? Given the success of nuclear physics, be it in power generation or medical applications, it may come as a surprise that we still do not have a detailed answer to this question. Atomic nuclei are too small to observe directly and the fundamental theory describing the smallest building blocks of matter, quantum chromodynamics

Skyrmions (QCD), does not allow concrete calculations of nuclear properties. It gives us the theory of matter, but the equations are so difficult to solve that even simple questions, such as how protons and neutrons (collectively called nucleons) are configured inside a nucleus, remain unanswered. To study nuclei directly from QCD would require immense computing resources that are

beyond current capabilities, and are likely to remain so for some time. Applications, such as nuclear power, hinge on approximate models that are disconnected from the fundamental theory. These work well, but much can be gained from a true understanding of the nature of nuclei and sub-atomic particles.

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A mathematical model of atomic nucleiSometimes the path to such an understanding winds in surprising ways. In the early 1960s the British physicist Tony Skyrme, then Senior Principal Scientific Officer in the Theoretical Physics Division of the Atomic Energy Research Establishment (AERE) at Harwell, devised a mathematical model of atomic nuclei, based on a novel idea that provided a unified description of matter and force. The nuclear force is described by wave-

like solutions of the theory, but Skyrme recognised that the theory also has particle-like solutions that describe nucleons. A useful analogy is to consider a piece of elastic band held at both ends. If one end is wiggled then waves carrying energy move along the elastic band: these waves are the

analogue of the nuclear force. The elastic band can also be twisted, which is the analogue of the particle-like solutions describing nucleons. Skyrme discovered the particle-like solution with a single twist (now known as a skyrmion) and identified the number of twists (the skyrmion number) with the number of nucleons. Studying nuclei requires finding solutions containing several skyrmions. This is a difficult problem that is of

mathematical interest in its own right, with fascinating geometrical aspects.

Despite its ingenuity, Skyrme’s model was quickly superseded by the newly-fledged theory of QCD, but in the 1980s it made an unexpected reappearance.

t�Simulation results showing a Skyrmion in the magnetisation vector field in a chiral ferromagnetic material at the nanoscale (Mark Vousden and Hans Fangohr/University of Southampton)

In an attempt to make explicit calculations more feasible, scientists tried to simplify QCD using novel approximations. Remarkably, Edward Witten (Institute for Advanced Study in Princeton) found that the Skyrme model emerged from these approximations, together with an understanding, directly from QCD, of Skyrme’s identification of the particle-like solution with the nucleon.

We still do not know what the nucleus of a carbon atom looks like. Applications such as nuclear power hinge on approximate models that are disconnected from the fundamental theory

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The applicationof Skyrme’s model to larger nucleiIn the 1990s researchers started applying the Skyrme model to larger nuclei, beyond the single proton and neutron. Among these researchers were the mathematical physicists Richard Battye (University of Manchester), Nicholas Manton (University of Cambridge) and Paul Sutcliffe (Durham University), and the pure mathematician Sir Michael Atiyah (University of Edinburgh). According to classical physics protons and neutrons can be thought of as rigid spheres that cluster together to form the nucleus of an atom. In Skyrme’s model the relevant question was how several skyrmions, representing protons and neutrons, would combine to form a larger nucleus.

Although Skyrme’s equations are simpler than those of QCD, significant computing power was still needed to answer this question. Battye and Sutcliffe set the Cosmos supercomputer at the University of Cambridge to work for weeks at a time. The pictures they saw emerging came as a pleasant surprise. Skyrmions merged to form beautiful symmetric structures, shapes related to the famous Platonic solids described over 2,000 years ago.

Why these particular shapes should form was initially a mystery. Each shape corresponds to a nucleus with a given number of basic skyrmions (representing protons and neutrons) but there was no obvious explanation for the relation between the number of skyrmions and the shape that forms. For example, when four skyrmions merge you might expect

Skyrmions with skyrmion numbers 1, 4, 12, 32,108 (Dankrad Feist and Chris Lau/University of Cambridge)

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the number four to correspond to a defining feature of the resulting shape. Yet the shape that formed was a cube, whose defining features, its faces, vertices and edges, come in the numbers six, eight and twelve, respectively, but not four. Battye and Sutcliffe would play a game trying to guess what shape was going to emerge after starting with a certain number of skyrmions. Often the result was a surprise and more interesting than they had imagined.

The explanation was found by revealing a hidden mathematical structure. The shapes correspond to the symmetries of a class of mathematical objects that lie concealed within Skyrme’s model. There is also a physical consequence of these symmetries:

they predict the possible spin states of the nucleus that is being modelled. These are nuclear properties that can be measured in the laboratory. It is gratifying that the sophisticated mathematics at the heart of Skyrme’s model, with the aid of geometry, can link up real nuclear data.

Skyrme’s model remains a simplification of QCD and we cannot be sure that real nuclei always have the predicted structure. But the agreement with experimental data is encouraging. Recent results provide new insights into some of the most abundant elements in nature, such as helium, lithium and carbon, and how these vital elements formed from the primordial soup of particles in the very early universe.

But this is not all. The description of nuclei in terms of skyrmions has recently been obtained within string theory, so different approaches are converging to provide a growing body

of evidence in support of Skyrme’s idea. Related skyrmions also appear in the completely different context of magnetic materials at the nanoscale. These have potential applications in data storage: the skyrmions’ twist can be used to represent the ‘0’ and ‘1’ bits that underlie information technology; the presence of a skyrmion in a medium denotes ‘1’ and the absence denotes ‘0’. Since a skyrmion’s twist cannot be undone, skyrmions may provide a robust alternative to conventional storage devices; the latter become unstable as devices get smaller and their capacity increases. Whether it will be possible to manipulate large numbers of skyrmions in the necessary manner is still unclear, but a team of researchers led by Roland Wiesendanger (University of Hamburg) have recently taken the first steps by “writing” and “deleting” single magnetic skyrmions on an ultrathin magnetic film.

p A system of 22 skyrmions in a 2D magnetic film with periodic boundary conditions (Mark Vousden, Marijan Beg, Hans Fangohr/University of Southampton)

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The discovery of a new type of material is a rare and precious thing. As well as deepening our understanding, new materials spur research, and innovative applications drive the development of future technologies. Mathematical physics has led the way in discovering a new class of materials with novel properties that promise to revolutionise the area of electronics and quantum computation.

For more than 80 years physicists believed they had a solid understanding of how insulators worked. The Swiss physicist Felix Bloch solved Schrödinger's equations for the wave functions of electrons in a solid in 1928, revealing that the wave functions of the electrons form discrete energy bands. The lower energy bands in an insulator are completely filled with electrons and there is a large gap to the next band: insulators do not conduct electricity as there is no way for the electrons to flow.

Topological insulators – a new phase of matter

t Quantum waves in topological insulators. Computer model showing interference patterns formed by quantum waves in a type of new material known as a topological insulator. This is an example of computational and quantum models being used to predict the properties of new materials (Ali Yazdani/Science Photo Library)

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Unlike the surfaces of ordinary metals that lose conductivity as they oxidise or corrode, the surface of a topological insulator will always conduct electricity, even in the presence of impurities and defects

Topological insulatorsThere is, however, a subtly different solution to Bloch's equations than this ordinary band insulator. In a flurry of papers appearing within just a few weeks in 2006, several groups of mathematical physicists independently predicted the existence of a new type of insulator – a topological insulator.Such insulators have surprising properties: while the interior looks like the ordinary band insulator, the surfaces conduct electricity. These surface states are also incredibly robust: it does not matter where or how the material is cut, the resulting surface will always be conductive. Moreover, unlike the surfaces of ordinary metals that lose conductivity as they oxidise or corrode, the surface of a topological insulator will always conduct electricity, even in the presence of impurities and defects.

This is because the conductive states at the surface of these materials are topologically protected. Topology is the area of mathematics that is concerned

with how things are connected: two shapes are topologically equivalent if you can bend or stretch one into the other, without cutting or tearing. The most famous example is that a coffee cup is topologically the same as a donut. Topological insulators, however, are not concerned with the shape of the physical material but instead the shapes of the mathematical forms that describe the states of the electrons. The mathematical description of the electron states in a band insulator can be transformed (in an exactly analogous way to bending and

p Topological insulators are a new state of matter, topologically distinct from ordinary insulators (Yulin Chen/University of Oxford)

stretching a coffee cup into a donut) to one that essentially ties each electron to a specific atom. However, this cannot be done for the mathematical descriptions of the electron states on the surface of a topological insulator. These electrons are fundamentally mobile as their mathematical forms are topologically different to the electrons in a band insulator.

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Applications of a not-so-new materialWhat is surprising is that these topological insulators turned out not to be new materials after all, although the properties inferred from revisiting Bloch’s equations were definitely novel. Indeed, a few years after the theoretical prediction of these properties came their experimental confirmation, and the materials found with these properties were available on the shelf of any chemistry laboratory and were commonly used in industrial processes. A deeper understanding of how these materials work will impact on their existing use in academic and industrial settings.

The novel properties of topological materials also promise to revolutionise electronics. Over the last decades electronic components have shrunk in size, enabling them to run at increasingly faster speeds. However, there is a limit to how far this miniaturisation can go: the electric current passing through these components generates heat and they must be of a certain size in order to dissipate that heat energy. However, spin currents, which transfer the spin state of an electron rather than its electric charge, do not produce heat. Spintronics, electronics based on

these spin currents, should lead to decreased chip size and, hence, faster computer processors.

Topological insulators are an ideal material to build such spintronic components. On the surface of these materials the spin of an electron is coupled to its electric charge: the direction of the spin of the electrons is always perpendicular to the direction of the electric current. This property allows the manipulation of spin currents and these materials could form part of the first functioning spintronic components.

p Artist's impression of an electron wave being transmitted through a step edge (Courtesy: Ali Yazdani/Princeton University)

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It is highly likely that topology plays an important part in many materials. Steve Simon (University of Oxford) and Rahul Roy (now at the University of California, Los Angeles) have been working on classifying topological materials using mathematical techniques first developed in string theory in the 1980s. Roy is one of the leading theorists in the field and received the 2010 McMillan Award for his prediction of 3D topological insulators. Simon is part of the UK wide collaboration, TOPNES

(Topological Protection and Non-Equilibrium States in Strongly Correlated Electron Systems) that aims to provide a theoretical and experimental foundation for quantum mechanical electronics. As well as providing promising avenues for producing smaller and faster computer components using spintronics, this collaboration is also investigating new materials and components for quantum computing.

One of the greatest challenges in quantum computing is isolating a system so that it can maintain a particular quantum state. Simon and Fiona Burnell (the latter now at the University of Minnesota) are examining if the topological properties of these materials can be harnessed to create a topological quantum computer. Such quantum systems promise to be far more stable due to the topological protection inherent in these materials.

The prediction and discovery of topological insulators has been one of the most important advances in condensed matter physics in the last 20 years and promises to transform this field. It is also an illustration of a growing trend, in which mathematical physics will lead the way to new materials, guiding the work of experimentalists in identifying the most promising areas for future discoveries.

p (a) The spin of electrons on the surface is correlated with their direction of motion (b) The lattice structure of bismuth telluride and the predicted relativistic "Dirac cone-like" electronic structure formed by the surface electrons (c) The electronic structure measured by angle-resolved photoemission that confirmed the theoretical prediction and the topological nature of bismuth telluride (Yulin Chen/University of Oxford)

a b c

Mathematical physics will lead the way to new materials, guiding the work of experimentalists in identifying the most promising areas for future discoveries

For further information contact:IOP Institute of Physics76 Portland Place, London W1B 1NTTel +44 (0)20 7470 4800E-mail: [email protected]

Charity registration number 293851Scottish Charity Register number SC040092

Published: September 2014

Writers: The Millennium Mathematics Project, University of CambridgeDesign: h2o creative

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