Interactive blackboard

The singular behavior in the vicinity of a second-order critical point is characterized by a set of critical exponents (e.g. alpha, beta, gamma, etc.). These power law dependencies of thermodynamic quantities are a symptom of scaling behavior. Mean-field estimates of the critical exponents are found to be unreliable due to fluctuations. However, since the various thermodynamic quantities are related, these exponents can not be completely independent of each other. Hence, using scaling ideas and theoretical principals such as the homogeneity assumption, a number of exponent identities may be obtained (e.g. Rushbrooke's identity, Widom's identity, etc.).

However, in order to obtain an identity involving the exponent nu, describing the divergence of the correlation length, we need to replace the homogeneity assumption with a generalized homogeneity assumption. This yields exponent identities, known as hyperscaling relations, which involve the space dimension, d. The most important of these is known as Josephson's identity. Josephson's identity does not agree with the mean-field exponents for spatial dimensions less than or equal to three and so, this must be taken into account when developing a valid theory of critical behavior.

In condensed matter physics we categorize matter into phases such as solids, liquids, and gases. Rubber is a rather unusual phase of matter characterized by being an unusually stretchy solid. A liquid crystal is another exotic phase, being a liquid of rod shaped molecules in which all the rods point in the same direction so, despite flowing like a regular fluid, it looks different from different directions.

A liquid-crystal elastomer is an even more exotic phase, pioneered by Mark Warner, consisting of a liquid crystal phase inside a rubber. This combination leads to a soft stretchy material that will double or triple in length upon cooling by a few degrees (illumination or application of electric field) paving the way for real-life applications in artificial muscles, soft robotics, and touchscreens that change shape on demand.

The Hall effect, first discovered by Edwin Hall in 1879, is related to the conductance of electrons across a 'two-dimensional' sheet of metal with a strong and uniform magnetic field applied perpendicular to it. The perpendicular magnetic field causes the electrons to deflect to one side of the sheet as they travel along. However, as these electrons begin to accumulate, the electric repulsion will eventually balance the magnetic deflection and the electrons will flow as before. This is know as the Hall effect and the induced voltage at which this balance occurs is known as the Hall voltage.

In 1980, Klitzing et. al. discovered that at strong magnetic fields and extremely cold temperatures, the graph of Hall voltage against magnetic field had distinct plateu regions. From the value of the Hall voltage at the plateau regions, they deduced that the Hall conductance must be quantized in multiples of the square fundamental charge divided by the Planck constant (e^2/h). This is known as the quantum Hall effect. A couple of years later, Tsui et. al. carried out the same experiment at even colder temperatures and stronger magnetic fields. Surprisingly, they found even more Hall plateaus, which showed that the Hall conductance may also be quantized in fractional multiples of e^2/h. This is known as the fractional quantum Hall effect.

However, the way in which Tsui et. al. achieved the first fractional quantum Hall state in the laboratory has many limitations, not least of which the fact that it was only applied to a two-dimensional gas of fermions. In 2003, Nigel Cooper and Jean Dalibard developed a robust way of achieving bosonic fractional quantum hall states for ultracold gases, using optical flux lattices. Ultracold atomic gases are ideal systems for this, as they allow studies of strong correlation phenomena for both fermions and bosons, and fractional quantum Hall physics can be approached for homogeneous fluids, as well as for atoms confined in optical lattices. New methods for achieving fractional quantum Hall states are always desired. However, a method which utilizes ultracold gases is particularly significant, for both theorists and experimentalists alike.

• Mark Warner and Eugene Terentjev. Liquid crystal elastomers. Vol. 120. OUP Oxford (2003).
• Alexander Altland and Ben Simons. Condensed matter field theory. Cambridge University Press (2010).
• Neil Ashcroft and David Mermin. Solid State Physics. Saunders, Philadelphia (1976).
• Mark Warner and Anson Cheung. A Cavendish quantum mechanics primer, Periphyseos Press (2013).
• Thomas Fink. The Man's Book. W&N (2014).
• Thomas Fink. The 85 Ways To Tie A Tie. Fourth Estate (1999).
• John Ziman. Principles of the Theory of Solids. Principles of the Theory of Solids (1972).
• David MacKay. Information Theory, Inference and Learning Algorithms. CUP (2003).
• David MacKay. Sustainable Energy - without the hot air. UIT (2008).

The standard approach to design new materials is an experimentally driven process of trial and improvement. The formula combines the individual likelihoods P of a materials fulfilling the multiproperty design criteria to give an overall grading factor G that can be maximized to propose the material most likely to exceed the target specification given the data currently available.

The computational approach has been used to design two nickel alloys for gas turbines, and two molybdenum alloys for forging hammers. Each alloy has thirteen individual physical properties that are predicted to match or exceed commercially available alternatives. For each alloy the properties have been experimentally verified, and then commercially exploited.

Density functional theory is a computational, quantum-mechanical modeling method used to investigate the electronic structure of many-body systems. At present, it is the most successful approach to compute the electronic structure of matter and has made an impact in various disciplines, ranging from microelectronics to geology. The appeal of density functional theory is in its efficiency. It can provide a sufficiently accurate description of electron correlation, at a computational cost which scales favorably with particle number (as the cube of the system size) compared to other methods (which scale with at least the fifth power of the system size).

The Order-N Electronic Total Energy Package (ONETEP) is a linear-scaling code for quantum-mechanical calculations based on density-functional theory. In certain situations, the scaling of density functional theory may be improved even further and reduced to a linear relation with controllable accuracy. This enables the exploration of an even broader range of systems, which previously would have taken too long to calculate.

A superconductor is a material that has no resistance to an electrical current flowing through it. A superconductor is described by not just the strength of the superconductivity, but also by an angle, much like the direction that a compass points in. By the 1950's the strength if a superconductor was understood and had been verified by experiment, however the angle had never been observed.

At this time Brian Josephson was a PhD student in TCM. During a graduate lecture he hit on an a way observe the angle of the superconductor in an experiment. The proposal was soon verified, becoming the first experimental observation of the angle of superconductor. Following the experimental verification, Josephson was awarded the Nobel prize in 1973. This became known as the "Josephson Junction", and has now been widely exploited in digital communications, microscopes, and medicine.

The Josephson effect is the phenomenon of supercurrent across a Josephson junction. This is a current which flows for an indefinite period of time, across the boundary of two superconductors coupled by a weak link. This "weak link" could be a thin insulator, a non-superconducting material, or indeed anything which weakens the superconductivity across the boundary. This effect was discovered by Brian Josephson in 1962.

This supercurrent can either be direct (DC) or alternating (AC). The DC Josephson effect is the presence of a constant supercurrent, without any applied voltage across the device. This current may be explained using the concept of quantum tunneling. Furthermore, the AC Josephson effect is the presence of an alternating supercurrent, due a constant voltage applied across the device. This alternating supercurrent oscillates with the Josephson frequency.

In 1973 Brian Josephson was awarded the Nobel prize "for his theoretical predictions of the properties of a supercurrent through a tunnel barrier" and, in particular, for the "phenomena which are generally known as the Josephson effects".

The line between theoretical and computational physics is blurred. Most theoreticians find themselves doing at least some computational work, sometimes using "computer experiments" to verify theory. Similarly computational physicists often have to develop new theories in order to improve the speed and accuracy of their codes.

Under the leadership of Prof. Heine and Prof. Payne, TCM has been a pioneer of using the power of computers to deliver new insights into condensed matter systems, and the history of computing in TCM goes back to the 1950s. TCM has been using parallel computers since the 1980's, and used Crays in the 1990's. Today, TCM members have access to supercomputers all over the world, with TCM driving the creation of the Cambridge High Performance Computing Facility, now the Cambridge Service for Data-Driven Discovery, and once ranked as the 20th most powerful computer in the world.

The largest calculations need regional and national scale facilities, but smaller calculations, and code development, are more conveniently and economically performed on local facilities. TCM hosts a network of Linux desktops, along with over a dozen powerful dual socket machines dedicated to running larger-scale calculations. A uniform environment helps users scale their codes on this hardware whose performance and memory capacity varies by about a factor of a hundred. This environment is tailored for computational physics, currently with just two different web browsers, but over half a dozen different Fortran compilers (and many other languages supported, such as C, C++, python and R, along with packages such as Matlab and Mathematica). Specialist IT support for research computing is provided, for TCM has had its own dedicated Computer Officer since the mid 1970s.

More often than not in condensed matter theory, the systems which we try to model, and hence the equations which we try to solve, are too complicated to be treated in an exact way. Consequently, theorists arm themselves with a set of tricks, approximations and short-cuts to speed up their calculations.

One such trick is known as the replica trick, and was pioneered in the study of spin glasses by Sam Edwards. At first glance, the formula appears to be a standard result. Simply applying L'Hopital's rule on the left-hand side with respect to n, and then taking the limit as n goes to zero, does indeed yield the natural logarithm of Z. However, the remarkable trick is not the identity itself, but rather the situations in which it may be applied.

The identity holds for all real n. In this case, computing the left-hand side does not have any advantage over computing the right-hand side. However, if we promote n to be an integer, then Z to the power of n usually becomes much easier to compute. Hence, we replicate the system n times, compute everything as a function of n, and then cross our fingers in taking the limit as n goes to zero. Bar a few mathematical subtleties, this works surprisingly well.

Sam Edwards used the replica trick to work out the ground state and properties of spin glasses. As Heine expresses it, "A whole industry on spin glasses and then neural networks developed".

One of the important distinctions in solid state physics is that between crystals and amorphous solids, such as glass. A crystal is defined as a homogenous solid, which has a regular and uniform pattern of atomic bonds. By contrast, the bond structure in glass is highly irregular. A spin glass is a disordered magnet, which has an irregular orientation distribution of magnetic spins, analogous to the irregular position distribution of bonds in glass.

At face value, these materials do not seem useful. They are not particularly good magnets, conductors or insulators. However, spin glass serves an ideal playground for theorists, which has far-reaching applications in statistical mechanics, complexity, as well as various branches of mathematics.

The theory of spin glass was introduced by Sam Edwards and Philip Anderson in 1975. By studying the theory of these materials, they showed that no overall ferromagnetic or antiferromagnetic state is possible in the material. However, a ground state does exist, where the spins are aligned in definite directions, even if these directions appear to be at random. Philip Anderson was awarded the Nobel prize in 1977 for "fundamental theoretical investigations of the electronic structure of magnetic and disordered systems".

An electron gas with contact repulsive interactions is a deceptively simple interacting system, yet it displays a remarkably rich range of phenomena. At mean-field level the electron gas was predicted by Stoner to undergo a mean-field transition into an itinerant ferromagnet. Moreover, quantum fluctuations mean that a variety of other inhomogeneous magnetic states are energetically favorable. The free energy expansion predicts that a magnetic spin spiral state should emerge, which was first observed in CeFePO in 2012 and PrPtAl in 2015.

A long-term goal of TCM is to be able to describe the properties of a material with a computer. This would allow materials to be designed computationally, rather than by an expensive process of experimental trial and improvement.

Gold is one of the best electrical conductors in nature. Each gold atom has 79 electrons orbiting it, however only one of these electrons is free to move over the metal, with the other 78 are tightly bound to the gold atom. To understand, model, and calculate the electrial conductivity of gold, it was realized that we can replace the gold nucleus and the 78 electrons with a new pseudo gold atom, and simply calculate the quantum mechanical motion of the final electron. Removing 78 of the 79 electrons gives a significant computational saving.

The pseudopotential approximation when applied to all of the atoms in the periodic table has made the computational study of the properties of materials practical and is now a multi-billion dollar industry.

The most well-known states of matter are solids, liquids and gases. These are ubiquitous because they arise in everyday conditions. However, there are other states of matter which arise under more extreme conditions. If we widen our scope beyond planet Earth, we notice that most observable matter is actually ionised gas (plasma), which is the fourth fundamental state of matter. Furthermore, many other low-temperature states of matter also exist. Perhaps the most significant of these is the Bose-Einstein condensate. As predicted by Bose & Einstein in 1924, when matter is cooled to near-zero temperature, the wavelengths of the individual atoms become so broad that they start to overlap, and the whole system can be descibed by a single, uniform wavefunction. This was first experimentally verified by Cornell & Wieman at Boulder, Colorado in 1995. Nowadays Bose-Einstein condensates are relatively easy to produce in the laboratory, and they are of great interest to both theorists and experimentalists because they provide a rare opportunity to observe quantum phenomena on the macroscopic scale.

It is important to note that matter does not have to be dense to condense. In fact, it is common to study the Bose-Einstein condensation of weakly interacting gases. In 2001, Cooper et al. investigated the ground states of weakly interacting bosons in a rotating trap. Due to its quantum single-wavefunction nature, when a Bose-Einstein condensate is rotated, quantised vortices form in the cloud. These vortex states can either consist of a tangle of vortices (a vortex liquid), or a regular repeating arrangement (a vortex lattice). Cooper et al. investigated the ratio of the number of bosons to the number of vortices in the cloud, as the condensate was rotated. They found that this ratio, known as the filling factor, determines the nature of these vortex states. At near-zero filling factors, the condensate forms a vortex liquid, as one might expect. From numerical calculations, Cooper et al. subsequently showed that the filling factor at which a vortex liquid becomes a vortex lattice is approximately six.

Vortices or "eddy currents" are of great interest to condensed matter theorists, particularly on the really small scale, as they may appear as 'defects' in particular types of materials. As opposed to regular vortices, quantum vortices only have a discrete set of allowed angular momenta, as well as other unusual properties. Under certain conditions, quantum vortices may arise in materials which are superconducting (i.e. have zero resistivity) or superfluid (i.e. have zero viscosity).

On a practical level, it is important to understand the physics of these vortices as they may, for example, interfere with the construction of electronic components, or change the macroscopic behaviour of a system. David Thouless, along with John Kosterlitz and Vadim L'vovich Berezinskii, studied the physics of how these vortices behave.

Analogous to point electrical charges, vortices may also be thought of as having a charge. Vortices which are spinning in the same direction are said to have the same charge, and so repel each other. Conversely, vortices which are spinning in opposite directions (known as a vortices and antivortices) attract. By deriving the vortex energy equation, Thouless showed that vortices and antivortices have a logarithmic attraction, which is exactly analogous to point electrical charges in two dimensions.

When an elastic structure is pushed or pulled it responds by changing shape. For example, if you pull on the two ends of an elastic band it gets longer. For small forces the shape changes are typically simple and small, but if you apply enough force the structure sometimes responds by changing shape in dramatic and unexpected ways. We call these shape changes elastic instabilities. The simplest example of an elastic instability occurs when you push on a thin elastic rod. For small forces the rod simply gets shorter, but at a critical degree of compression the rod will buckle into a bent shape.

In a linear system doubling the force applied simply doubles the system's response. In elastic instabilities adding more force produces a completely different response, so they must be driven by non-linearities in the theory of elasticity. These non-linearities divide into two types, material non-linearities that stem from the constitutive law of a given material and geometric non-linearities that stem from the fact that large deformations do not commute. The former are specific to a given material but can occur at small strains while the latter are universal but only occur when the shape changes are geometrically large, so they tend to be restricted to soft solids.

John Biggins worked on a new example of such a geometric elastic instability, called elastic fingering. When a joint between relatively rigid bodies is stressed, even though perfect adhesion is maintained, the exposed free meniscus in the thin elastic layer becomes unstable, leading to the formation of spatially periodic digits of air that invade the elastic layer. In practice, these rigid bodies may function as sealants, thermal, electrical, or mechanical insulators, bearings, or adhesives.

A principal goal of the TCM group is to develop tools to understand the materials that surround us in our everyday lives. Analytics provide generic insights, whereas computational calculations will deliver the properties of specific materials without time consuming and expensive experiments.

Under the leadership of Mike Payne, TCM worked to make the practical calculation of the properties of materials an everyday reality. This lead to the widespread adoption of the Density Functional Theory technique through the development of the CASTEP code. CASTEP is commercialized through Accelrys, and in 2015 passed $30 million in sales to industry.

The prediction of even very simple structures of materials has presented a great challenge to theorists, due to the large number of possible atomic arrangements. Ab initio random structure searching (AIRSS) is a simple, yet powerful, computational technique developed by Chris Pickard and Richard Needs used to overcome this problem.

AIRSS uses random numbers to suggest different structural configurations of atoms within a material. It then relaxes these structures to their corresponding energy minima. Using knowledge of potential energy surfaces, the method is then able to funnel these guesses to obtain a good estimate of the global energy minimum, and hence a new stable material.

The AIRSS technique was first applied to high-pressure phases of silane in 2006. However, the technique has since found more widespread applicability from discovering point defects, surfaces and clusters to completely new materials. The Morris group is currently using this method to improve the design of rechargeable batteries.

The 85 Ways to Tie a Tie refers to both the number of knots that can be tied in a conventional necktie, and a popular book on the history and science of ties and tie knots.

The book grew out of two journal articles by Thomas Fink and Yong Mao, in which they introduced a mathematical representation of tie knots and proved that, with a conventional necktie, there are exactly 85 possible ways to tie it. Of these, just over a dozen are sufficiently handsome or different from each other to be worn. These include the four traditional knots - the four-in-hand, the Nicky (a derivative of the Pratt knot), the half-Windsor and the Windsor - and a number of previously unknown knots.

The book itself studies the history of the knotted neckcloth, introduces some elementary notions from topology, and gives an overview of the 85 knots and the history of those previously discovered.

One of the most fundamental characterizations of a material is whether it is an electrical conductor or insulator, that is whether it allows electrons to easily flow through it. Until the 1950's the traditional theory of materials considered each single electron as interacting with the background crystal lattice. This predicted that the electrons would easily move together through the crystal, and so the crystal would conduct.

Mott realized that electron-electron interactions are important, which in fact resists electrons from flowing together through the crystal, making it an electrical insulator. This insight won the 1977 Nobel prize. Such Mott insulators are now the basis of solid state electrical switches and the transistors found in computers.

Condensed matter physicists are interested in behaviors that emerge when you have lots of particles grouped together. Most commonly this means lots of atoms grouped into a material, but it can also mean lots of cells grouped to make a biological tissue. Ben Simons has studied tissues such as skin, which are continually replaced even in adults. New skin cells are continually made by skin stem cells. Ben showed that these stem cells behave randomly, usually dividing to make a skin cell and a replacement stem cell, but also sometimes dividing to make two stem cells and sometimes dividing into two skin cells. More recently Ben has applied similar condensed-matter techniques to understand cancer and biological development.

Magnetoresistance is defined as the property of a material to change the value of its electrical resistance when an external magnetic field is applied. The effect is the subject of intense scientific research and numerous applications have been found, ranging from the magnetic field sensors in satellites, to the hard disk drives in computers.

In 2003, Parish and Littlewood presented a surprising finding on the magnetoresistance of semiconductors (i.e. materials which can only conduct electricity under certain conditions). In homogenous semiconductors, one would usually expect the resistance to increase quadratically with magnetic field, at low fields, and then saturate when the field becomes too strong. Parish and Littlewood discovered that in heavily disordered semiconductors, the resistance increases linearly with the magnetic field for all measured fields, and does not saturate.

This discovery has lead to the improvement of magnetic field sensors, which now can offer a controllable, non-saturating, and linear response to external fields. This allows the sensors to work for a larger range of magnetic fields, and also enables more precise measurements to be taken.

The "chain fountain" refers to the phenomenon observed when one end of a chain of beads is thrown out of a beaker and allowed to fall freely under gravity. Due to the structure of the chain, the momentum of one end falling downwards causes an arc of chain to rise up out of the beaker, reminiscent of a fountain. This phenomenon was first demonstrated and popularized by British science and television presenter Steve Mould. In his original video, Mould demonstrated the phenomenon using 50m of standard roller blind chain, to produce a fountain approximately one foot in height.

In 2014, John Biggins and Mark Warner published a paper, in Proceedings of the Royal Society of London, explaining the physics behind the phenomenon. Biggins & Warner argued that the formation of a fountain requires that the beads come into motion not only by being pulled upwards by the part of the chain immediately above the pile, but also by being pushed upwards by an anomalous reaction force from the pile of stationary chain. Furthermore, they showed that the height of the fountain is directly proportional to the height of the drop.

Coffee is the lifeblood of TCM. Indeed, we are so proud of our coffee machine, it is often said that TCM stands for "Theory of the Coffee Machine".

Our current machine, La Pavoni, is a three spout, double boiler espresso machine with two wands for steaming milk. We use freshly ground and Illy beans selected by a LASER beam, of which we consume a 3kg canister every week, and fresh milk, of which we've been known to get through 16 pints a day. Members of TCM can often be found practicing their latte art and fine-tuning the espresso grind according to local temperature, humidity, and air pressure. We believe that we serve the best coffee in Cambridge.

Before La Pavoli we had a jet-black Gaggia E90, a similar three-spout espresso machine donated by Mike Payne in 2000, which replaced a baby espresso machine purchased in September 1999. Before that we used a kettle.

Tea drinking is also supported in TCM, via a hot-water spout on La Pavoni, but is generally regarded as a bit of a health and safety hazard. Tap water is also available.