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On the blackboard

The blackboard is the theorist's best friend. Sadly in many departments the blackboard has been superceded by the interactive whiteboard, so TCM has developed an interactive blackboard. Be the first to road test this new technology by exploring the blackboard below, where several TCM members have left their favorite equations.

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Josephson's hyperscaling relation

Brian Josephson (TCM)                              

Prof. Simon's lecture notes.

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.

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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.

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Cooper's flux lattice equation

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.

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Books from TCM

TCM members are interested not just in cutting edge research, but also the dissemination of new ideas to the next generation, policy makers, and the public. A selection of general interest books written by TCM members are listed below:

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Materials grading factor

Gareth Conduit (TCM) & Bryce Conduit

Patents EP2778990 and US2014236548.

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.