Signatures of Majorana particles

How and when do Majorana fermions arise in a quantum many-body system?
What are their experimental signatures?

For some reason I find Kitaev's discussion somewhat ad-hoc. I do find the following more helpful and illuminating. Perhaps, it is just because it deals with things I am more familiar with.

Start with the transverse field Ising model in one dimension. It describes interacting localised spin-1/2 particles. J is the nearest-neighbour ferromagnetic interaction. h is the transverse magnetic field. At J=h it undergoes a quantum phase transition from a ferromagnetic phase to a paramagnetic phase.
One performs a Jordan-Wigner transformation which maps the spin-1/2 operators onto spinless fermion operators. This is a non-local transformation. The Hamiltonian then becomes quadratic in the fermion operators and so is analytically soluble via a Bogoliubov transformation. This means the "quasi-particles" are spinless fermions.
[For the details see this article which also includes the inhomogeneous case].

This nicely illustrates the profound fact that in a quantum many-body system the emergent quasi-particles  can have quantum numbers and statistics that are different from the underlying constituent particles (see here for more).

Now, it turns out that for an open chain that the spinless fermions defined at the end of the chain are actually Majorana particles. In a sense the end of the chain "splits the fermions in two". I like this because of some similarities to what happens in a Haldane spin-1 chain. The spins at the end are "split in two" into spin-1/2 excitations, as discussed in this earlier post, Edge states define the bulk.

What might be experimental signatures of these unusual edge states?
Brijesh Kumar and Somenath Jalal recently made an important observation based on calculations published by Pfeuty in 1970.

The long-range spin correlations in the bulk of the ferromagnetic phase it scales with p^2 where p is the order parameter.

In striking contrast, for an open chain the correlations between the end spins scales with p^8, a dramatically different dependence. They suggest this is a signature of the Majorana character of the edge excitations.

I thank Brijesh Kumar for explaining his preprint to me. His paper contains some other interesting results about how to experimentally realise this in a chain of cavity QED systems [based on Cooper pair boxes coupled to microwave cavities]. Hopefully, I will blog about that later.

Against jargon and memorisation

This is a classic scene from the Bollywood movie 3 Idiots.

What makes a good undergraduate research project?

First, I am no expert. I have probably supervised less than a dozen undergraduate projects in my whole career.

What should be the primary goals? Hopefully, the student will

• learn some science (including some combination of concepts, theory, and techniques)
• learn something about how research is done (searching and reading the literature, trying different things, asking good questions, making mistakes, brainstorming, ...)
• experience some of the joys and frustration of doing science (including feeling dumb).
• get to personally interact with a range of scientists (faculty, postdocs, grad students)

The dominant goals should not be:

• use the student as slave labour
• get a publication
• keep the student happy
• recruit the student to do a Ph.D in the same group

Projects I don't like include ones which

• are highly technical [the students learns a lot of jargon or advanced techniques but does not know the basics, or context]
• mostly use prepackaged software (e.g. for computational quantum chemistry) [knowing something about what software is out there and how easy it can be to use can be a good thing, but it becomes dangerous when the student does not learn its limitations or the underlying principles. or if they start to think running code is doing research].
• are just too hard or speculative for undergraduates and they get nowhere.
• are so straight-forward the supervisor knows the answer before one even starts. they just need a slave to turn the handle...

Projects I like

• are as simple as possible
• illustrate important concepts
• allow the student to actually understand what is going on
• connect theory and experiment 
• challenge the individual students preconceptions and prejudices [e.g. theoretical physics is just mathematics, theorists should not worry about experiment, I can't do units, I don't want to do any computational work ...]

In some sense, this post is largely meant for supervisors. My main advice to students is: choose the supervisor NOT the topic.

I welcome comments, both from supervisors and those who have experienced good and bad projects.

How do chemical subsitutions change the colour of a dye?

Last week I heard Seth Olsen give a nice talk about his recent paper
Why Bindschedler's Green is redder than Michler's Hydrol Blue

It addresses the important and subtle question of what happens in methine dye molecules when the central carbon atom is replaced by a nitrogen atom:

I was going to write a summary. But, the abstract of the paper is beautifully written, summarising the main results. And so, here it is.

We offer a new physical interpretation of the color shift between diarylmethane dyes and their azomethine analogues. We use an isolobal analogy between state-averaged complete active space self consistent field solutions for corresponding methines and azomethines to show that the shift contains a significant contribution from configuration interaction between a methine-like ππ* excitation and an nπ* excitation out of the azomethine lone pair. The latter does not exist in the corresponding methine systems. This picture is qualitatively inconsistent with traditional models of the shift based on molecular orbital perturbation theory of independent π-electron Hamiltonians. A key prediction is the existence of a dipole-allowed band in the blue/near-UV spectra of the azomethines, which has polarization parallel to the lowest energy band. This forces a revision of past assumptions about the nature of the low-energy spectra of the azomethines. We show that a band at the predicted energies was observed as far back as 1938, but its significance at the time appears to have been unrecognized.

Who coined the word photon? and when?

I would have thought it was Einstein, or some other physicist, around 1905.
However, it was actually the distinguished chemist G.N. Lewis, as late as 1926!

I learnt this in a nice article from "This Month in Physics History" in the APS News. It also discusses Lewis' possible suicide due to depression.

On a lighter side, this reminds me of a silly achievement on my own: getting the term "squashon" into the scientific literature (see this paper from my Ph.D).

Do grant applications ever get shorter?

I think when a grant application has a section F15.5 there is a problem!
My latest application is running at 76 pages. Only about 8 pages is actually about science. The rest is administrative details, publication lists, statistics, budgets, justifications, and "bragging" about how great all the Investigators and their institutions are.

Every year more information is required and the applications get longer.
The problem may be that every year or so a new administrator decides it would be "helpful" to request an additional piece of information. But, adding just 7 per cent per year doubles the application length every decade....

Is this really necessary? Not only does it take a lot of time to prepare, but it also takes a lot of time to review. Actually, the painful reality is that most reviewers (including me, sorry) don't read much of the "fluff" but just focus in on a few key pieces of information: the science proposed, what the Investigators have recently achieved/published, and whether the budget is reasonable.

My question is: are there any funding agencies that are actually trying to reduce the length and complexity of applications?

Different attitudes to Mathematica

The cartoon is from Saturday Morning Breakfast Cereal.
I thank my son Luke for bringing it to my attention.

This does raise an important issue. To what extent should students be encouraged or allowed to use Mathematica and Matlab?
It seems to me there needs to be a balance: between learning to use a powerful tool an understanding how it works.

For example, I think it is very important that students learn to sketch graphs of simple functions. This provides intuitive understanding and a way of checking that the computer is giving a reasonable answer.
Perhaps it is no different from pocket calculators.

Here is Ben Powell's comment on this post. It took me a while to get it!