Students should learn the method of steepest descent

Today I gave a lecture to a Solid State Physics class on "Magnetic quantum oscillations and mapping out the Fermi surface." I basically follow chapter 14 of Ashcroft and Mermin.

The central result is Onsager's 1952 equation that the period of the magnetic oscillations is related to extremal areas of the Fermi surface perpendicular to the magnetic field.

[Aside: This is an amazing result because it only involves fundamental constants and so the interpretation of the experiments is not "theory laden", a rare thing in condensed matter].

There is one point I struggle to explain: why extremal areas?
Ashcroft and Mermin have a figure to justify this. It is lost on me, no matter how many times I read it and stare at the pictures.

Does anyone know a clear and convincing way to demonstrate this?

The only way I know how to get this result of extremal areas is to do a very fancy calculation (Lifshitz-Kosevich) which evaluates the magnetisation (or thermodynamic potential or partition function), summing over all the Landau levels and integrating over the momentum direction parallel to field. One then evaluates the last integral using the method of steepest descent (saddle point approximation), which then picks out the extremal areas of the Fermi surface.
However, this is way beyond what one should be doing at this level (final year undergraduate).
Furthermore, the students said they have never encountered the method of steepest descents before.
That is reasonable since neither did I as an undergraduate in Australia.
I only learnt it in graduate school after learning how to evaluate Feynman path integrals by this method. Only later did I learn it also applied simple one-dimensional integrals!

Should undergraduate students learn the method of steepest descent?
In mathematics and/or physics?

When is a property emergent?

I write posts with titles such as "The Dirac cone in graphene is emergent" and "Holes are emergent quasi-particles?"
What do I mean?
To me emergent properties and phenomena have the following distinguishing characteristics.

1. They are collective phenomena resulting from the interactions between the constituent particles of the system and occur at different length/energy/time scales.
For example, superconductivity results from interactions between the electrons and ions in a solid and involves energy (temperature) scales much less than the underlying interaction energies.

2. They are qualitatively different from the properties of the constituent particles.
For example, individual gold atoms in a metallic crystal are not "shiny". One cannot speak about superfluidity of individual (or small groups of) atoms.

3. The property is difficult (or almost impossible) to anticipate or predict from a knowledge of the microscopic constituents and the associated laws. In particular, emergent properties and phenomena (especially new phases of matter) are almost always observed experimentally first before they are explained theoretically. They are often discovered by serendipity.

4. The property is weakly dependent on microscopic details and can occur in a chemically and structurally diverse range of systems.
For example, many different metals are "shiny". Adding impurities or changing the mass of the electron has little effect. One can observe superfluidity in liquid helium and in cold atomic gases.

5. Understanding and describing the property involves introducing new concepts and organising principles. For example, symmetry breaking and order parameters.

Some of these ideas are contained in an old post, "Illustrating emergence with an example from geometry," which generated some nice comments.
I thank Fei Zhan for asking me the question.

I welcome comments.
Do you think the characteristics above are reasonable criteria?
How might they be sharpened or modified?

When a Dean fakes data

The Sunday New York Times magazine has a fascinating and disturbing article The Mind of a Con Man about Diederik Stapel, former Dean of Behavioural and Social Sciences, at Tilburg University in the Netherlands. He had a stellar academic career which was based on fabricating experimental data.

The article is rather long but worth reading. Here are a few of the extracts I found particularly pertinent:

Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct sexy results that journals found attractive. 

In his early years of research — when he supposedly collected real experimental data — Stapel wrote papers laying out complicated and messy relationships between multiple variables. He soon realized that journal editors preferred simplicity. 

What the public didn’t realize, he said, was that academic science, too, was becoming a business. “There are scarce resources, you need grants, you need money, there is competition,” he said. “Normal people go to the edge to get that money. Science is of course about discovery, about digging to discover the truth. But it is also communication, persuasion, marketing. I am a salesman. I am on the road. People are on the road with their talk. With the same talk. It’s like a circus.”  

Stapel’s atypical practice of collecting data for his graduate students wasn’t questioned,  [How many Deans do that ?] 

[The official report from the University stated] The field of psychology was indicted, too, with a finding that Stapel’s fraud went undetected for so long because of “a general culture of careless, selective and uncritical handling of research and data.” If Stapel was solely to blame for making stuff up, the report stated, his peers, journal editors and reviewers of the field’s top journals were to blame for letting him get away with it. The committees identified several practices as “sloppy science” — misuse of statistics, ignoring of data that do not conform to a desired hypothesis and the pursuit of a compelling story no matter how scientifically unsupported it may be.

It may be tempting for physicists and chemists to look down our noses at the social scientists, but I think these issues are just as pertinent for us. Don't forget Hendrik Schon!

As Kauzmann said: we tend to believe what we want rather than what the data tells us we should believe. Often the data is messy and inconclusive.

A refreshing experience

Recently I was asked by a university to evaluate an individual for tenure and promotion. The process was fascinating and I found refreshing.

The university send me copies of a selection of the individuals papers and a copy of a short CV. I was asked to review the scientific merit of the papers and thus comment on the suitability of the individual for tenure. There was no discussion of grant money received, numbers of Ph.D students graduated, number of publications, citation metrics, university "service", public outreach, journal impact factors, speaking invitations, .....

I found this refreshing, since it was in striking contrast to the values and emphasis of most institutions, which are very concerned with these other criteria, that I consider are secondary.

Did Fritz London surpass Einstein and Bohr?

I learned today of two impressive endorsements of Fritz London as a great theoretical physicist.

First, after John Bardeen got his second Nobel Prize he used the money to endow the Fritz London lectures at Duke University.

Second, in 2005 Phil Anderson wrote an essay in Nature, Thinking big which lauded London for having the vision that quantum theory was correct on all length scales, including the macroscopic, as manifested in superconductivity and superfluidity. Furthermore, Anderson argues this allows a "common sense" understanding of the quantum measurement problem.

London's vision is contrasted with that of Bohr and Einstein, of whom the "thoughtful curmudgeon" says

In reading about these [Einstein-Bohr] debates I have the sensation of being a small boy who spots not one, but two undressed emperors. Niels Bohr’s ‘complementarity principle’ — that there are two incompatible but equally correct ways of looking at things — was merely a way of using his prestige to promulgate a dubious philosophical view that would keep physicists working with the wonderful apparatus of quantum theory. 

Albert Einstein comes off a little better because he at least saw that what Bohr had to say was philosophically nonsense. But Einstein’s greatest mistake was that he assumed that Bohr was right — that there is no alternative to complementarity and therefore that quantum mechanics must be wrong. This was a far greater mistake, as we now know, than the cosmological constant.

I first learnt of these two endorsements in a beautiful essay by David Pines, Emergent behavior in quantum matter.

Aside: London's theory of the van der Waals interaction may have been the first case of deriving an effective low-energy Hamiltonian by integrating out high energy states, as discussed in the last point of this post.

Holes are emergent quasi-particles

When I first taught Solid State Physics [following Ashcroft and Mermin] I would introduce holes as the absence of an electron. I then discuss the effective mass of "electrons" and holes near the bottom and top of bands, respectively. I would not introduce the concept of a quasi-particle until several weeks later when I discussed electron-electron interactions and Landau's Fermi liquid theory.

Now I do it differently.
I explain how holes are an example of a quasi-particle with a positive charge and an effective mass [which can be significantly larger or smaller than the free electron mass].
This is a nice "simple" example of emergence. When you put interacting particles ["non-interacting" electrons interacting with a nuclei in a periodic lattice] together new entities emerge which has properties that are qualitatively different from the constituent particles.
[Aside: it is interesting that the only "interactions" between the electrons themselves are those associated with Fermi statistics]

As an aside, I then try to get students to think about some of the philosophical questions asking them to vote on and discuss the following questions:

Do you believe electrons exist? Are they real? Why?

Do you believe holes exist? Are they real? Why?

5 Grand challenges for condensed matter science

In 2007 an advisory committee for the USA Department of Energy published a report Directing matter and energy: 5 challenges for science and the imagination.

They decided a "grand challenge" must

• be scientifically deep and demanding
• be clear and well defined
• be relevant to the broad portfolio of basic energy sciences,
• promise real dividends in devices or methods that can significantly improve
• the quality of life and help provide a secure energy future for the US.

Here are their five grand challenges:

1. Control material processes at the level of electrons

2. Design and perfect atom- and energy-efficient syntheses of new forms of matter with tailored properties.

3. Understand and control the remarkable properties of matter that emerge from complex correlations of atomic and electronic constituents.

4. Master energy and information on the nanoscale to create new technologies with capabilities rivaling those of living things

5. Characterize and control matter away—especially far away—from equilibrium.

A good introduction to the full report is the 2008 Physics Today article, co-authored by Graham Fleming and Mark Ratner, co-chairs of the DoE committee.

It is six years after the report was written, but the challenges remains the same.

To me these 5 challenges are actually broader than "basic energy sciences" that the DoE should fund. In fact, they define what should be the research agenda for the chemistry and physics of condensed phases of matter in any country.