The order of things

I have now finished my first draft of chapter 4, of Condensed Matter Physics: A Very Short Introduction. The main purpose of the chapter is to introduce the idea and significance of the order parameter.

 I welcome comments and suggestions. However, bear in mind that my target audience is not the typical reader of this blog, but rather your non-physicist friends and family.

I think it still needs a lot of work. I may split this chapter into two.

The goal is for it to be interesting, accessible, and bring out the excitement and importance of condensed matter physics.

Completing the square

When studying quantum many-body theory, sometimes one gets lost in all the indices, functional integrals, Feynman diagrams, ...
Then one can lose sight of the fact that some techniques are really just the same as in simple mathematics. Examples include the method of steepest descent and cumulant expansions.

In basic algebra, a simple exercise is to complete the square in a quadratic equation, i.e. to make use of the following identity.

Suppose one has the following Hamiltonian. If describes a field q that couples linearly to a different field s, with a coupling constant s.

 Now if we complete the square and do a displacement of the field q we are left with the new Hamiltonian.

This now describes a free field q (i.e. non-interacting) and there is an attractive self-interaction of the field s with coupling constant a^2.

A related example is the Hubbard-Stratonovich_transformation. This allows one to introduce a new field that couples to the original field and then ``integrate out" the original field to leave a new interacting field theory. Two important and related examples are the following.

1. The Ising model is equivalent to a Landau theory for a scalar field (order parameter) and so they are in the same universality class. There is a nice treatment of this in Negele and Orland

2. Introduction of a superconducting order parameter to describe a fermion system with an attractive four-fermion interaction in the Cooper channel. There is a natural generalisation to superfluid 3He. I first encountered this approach in a book by Popov.

How do you maintain work-life balance?

This friday I am giving a 5-10 minute talk on work-life balance to a group of postdocs and young faculty from the School of Mathematics and Physics at UQ. I was asked because I previously gave a School Colloqium about mental health.

There will be three speakers and time for discussion.

Humorous aside: When I was asked in person to give this talk, I thought I heard that the invitation was from the ``Early career comedy". So I thought, ``I guess they are using comedy (such as skits) to cope with the stress of their work situation. But, I am surprised that they asked me because I am not really that funny."
After a few minutes of discussion, I realised that I had misheard. The invitation was from the ``committee" not for a ``comedy"!

What do you think I should say or not say?

Here are a few preliminary thoughts.

Start with empathy.
I remember this life stage as stressful and I did not always manage it well. Indeed, several times I have had mental health problems, which possibly could have been avoided if I had a better life-work balance.
I don't want to give a long list of do's and don'ts, but rather suggest some things to think about and discuss.

Know yourself.
Everyone is different. Don't compare yourself to others.
What are your values? What is most important to you?
What are the sources of stress, pleasure, satisfaction, and relaxation for you?
What are your expectations and presuppositions?
(e.g., if I work X hours a week, I will get a paper in a luxury journal, and then I will get a permanent job in academia.)

Know your environment.
Have a sober and realistic assessment of your chances of a permanent job in academia. It has more to do with chance than how many hours you work, the number of luxury papers, or grant $.
Universities don't necessarily want what is best for you, but rather what is best for senior management. Too often, their platitudes about work-life balance seem to be corporate well washing.
What aspects of your environment (phone, internet, boss, peers, family, ..) makes it hard for you to have a good work-life balance?

Be pro-active.
Set boundaries. Say no!
Do the basics (eat well, drink well, sleep, exercise, downtime).
Take breaks and vacations.

Small group discussion questions.
What do you do for ``downtime"?
Are you living consistently with your values?
How can you support one another to have a better work-life balance?
What boundaries do you need to set?

What do you think?

Don't be written off!

One of the most basic skills needed to succeed, or even survive, in professional life is to be able to write well. This is true whether you work in science, industry, business, or an NGO.
Of course, there are exceptions where an individual is incredibly gifted at the technical side of a job and can't even write a coherent paragraph. But, sorry, that individual is probably not you! Furthermore, even they need a collaborator or manager who is good at writing.

Most young scientists struggle to write a paper or a grant application, particularly when English is not their first language.

Here are a few suggestions on how to improve your writing skills over time.

First, accept that writing is hard work. Even John Grisham says that!

Accept that developing your writing skills is a project of a lifetime. This means starting early.

If you are an undergrad, take some humanities courses that require writing essays. Take writing lab reports seriously.
If you have to write a thesis, start writing it now.
Take a writing course. Take another.

Practise.
Write papers yourself. If you are the first author you really should write the first draft, including the introduction yourself. Don't let your boss (or someone more experienced) do it or expect them to. Your draft may be poor and get heavily edited or even discarded completely. But you will learn from the process and with time confidence and competence will follow.
Write a blog, even if no one reads it.

Learn by osmosis.
Read scientific authors known for the clarity and beauty of their writing. eg. David Mermin and Roald Hoffmann.
Read a lot and read broadly publications (newspapers and magazines) that are known for their excellent writing: The New York Times, The Economist, The New Yorker,...
Read famous novels and non-fiction books.

Read slowly and thoughtfully. Don't just skim everything.
I also suspect you may be better off reading hard copies.
Try to notice whether a piece of writing: makes sense, is hard to understand, is enjoyable to read?

Any other suggestions?

Why is condensed matter in flatland so interesting?

I am working on a chapter on condensed matter physics in dimensions different from three for Condensed Matter Physics: A Very Short Introduction. 
This is a rich subject since it is associated with high-Tc superconductors, quantum Hall effects, Haldane spin chains, Kosterlitz-Thouless transition, critical phenomena in 4 - epsilon dimensions, .....
Obviously, I cannot only give the flavour of things.

I would like to get your perspective on a few questions. For some, I have my own answers but want to hear others. Bear in mind the answers have to be accessible to a non-expert audience.

1. What is the central idea or concept?

2. What is an analogy to explain how dimensionality changes things?

3. What is an example of cross-fertilisation with another field of physics or science?

4. What is a significant technological application where low-dimensionality is central?
[High mobility MOSFETs are not an example because the devices are not really using a property that only occurs in two dimensions].

Simple model Hamiltonians can describe complexity

An important idea in condensed matter physics, both soft and hard, is that the rich phenomena seen in materials that are chemically and/or structurally complex can often be described by relatively simple model Hamiltonians that involve only a few parameters. This is particularly true when the model and system have competing interactions. This often leads to two inter-related phenomena, that I have previously described for strongly interacting quantum many-body systems.

• the emergence of new energy scales that are much smaller than the energy parameters in the Hamiltonian
• subtle competition between different ground states that involve distinct types of ordering (or not)

These phenomena also occur in classical systems. A nice example is described in this 1993 paper.

hcp Ising model in the cluster-variation approximation 
R. McCormack, M. Asta, D. de Fontaine, G. Garbulsky, and G. Ceder

The authors studied the Ising model on the hexagonal close-packed (hcp) lattice in a magnetic field. The authors are all from materials science departments and are motivated by the fact that the problem of binary alloys AxB1_x can be mapped onto an Ising model.
Rich phase diagrams result by varying the relative concentration of the atoms A and B (e.g. gold and silver), or equivalently the difference in the chemical potential between A and B, or the relative size of the interatomic interactions, or the temperature. The phase diagram can contain many competing phases with well-defined stoichiometry: A, B, AB, A2B, A3B, A2B3, A3B5, ...
Furthermore, even for a single stoichiometry, there can be multiple possible distinct orderings (and crystal structures).

The hcp lattice can be viewed as layers of two-dimensional hexagonal lattices where each layer is displaced relative to others. A unit cell is shown below on the left, where V1, V2, and V3, denote nearest-neighbour (nn), next-nearest neighbour (nnn), and nnnn interactions.
For the case of perfect packing of hard spheres V1=V2.
Note, that even when only nn interactions are present, and they are antiferromagnetic, that the system is frustrated, and for a single layer the Ising model does not order at finite temperature and has a massively degenerate ground state (i.e. non-zero entropy).
The figure on the right shows a way to represent this unit cell and the interactions in terms of two hexagonal lattices superimposed on top of each other.

 The authors show that there is a plethora (menagerie) of possible ground states and stoichiometric orderings.

We predict 32 physically realizable ground states with stoichiornetries A, AB, A2B, A3B, A, B, and A4B3. Of these structures, six are stabilized by NN pairs and eight by NNN pairs; the remaining 18 structures require multiplet interactions for their stability. 

This is a nice example of how a simple model can describe complex and rich behaviour. It is also a nice example of emergence in that many of the details don't matter such as the identity of the atoms or the form of the interaction between them.

The commercial applications gap

These days too many seminars, papers, and grant applications begin with great claims about the potential commercial applications of the research being discussed.
We should be skeptical about any hype concerning technological applications of basic research in materials science.
There is a big gap between a commercial device/material and what you can do in the lab with millions of dollars worth of equipment on a milligram of a material or a single electronic device.

It does not matter whether it is a photovoltaic cell, a catalyst, or a superconducting wire. All of the following demanding criteria must be met. Furthermore, it must be better than any existing technology and any competitor on most of these counts.

Cheap to manufacture.
Scaleable to mass production.
Durable. Often on the scale of years or decades.
Reliable and reproducible. Devices, whether batteries or computer memories, must work all the time.
Healthy. Not expose the user or manufacturer to toxic materials.
Use materials available in abundance (silicon, water, ...) rather than scarce ones, such as some rare earth elements.
Environmentally friendly.

Thanks to Tanglaw Roman for emphasizing these issues to me.

This post was partly stimulated by re-reading the front page of The New York Times from March 20, 1987, which features an article Discoveries bring a `Woodstock' for physics. The article describes the famous session on cuprate superconductors at the 1987 March meeting of the American Physical Society. It is worth reading to see how so little of what was promised then has not happened (unfortunately).

Can you think of other criteria that new technologies must meet to be commercially viable?