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!

One of my research values

I deem it of more value to find out some truth about however light a matter than to engage in long disputes about the greatest questions without achieving any truth.

Galileo

I encountered this quotation in a Physics Today obituary for an Italian particle physicist Massimilla Baldo Ceolin.
[I often read the obituaries, even of people I have never heard of before. I usually learn some interesting physics and something about what it takes to be an influential scientist].

Are elemental metals quantum critical?

I doubt it.

There is an interesting paper
Similarity of Scattering Rates in Metals Showing T-Linear Resistivity
J.A.N. Bruin, H. Sakai, R.S. Perry, A.P. Mackenzie

The central result is the figure below.

The graph shows the magnitude of the estimated scattering rate per Kelvin versus the Fermi velocity for a wide range of materials.
The line alpha=1 corresponds to a value of k_B/hbar, comparable to what one gets from a simple dimensional analysis or the "minimum viscosity limit" of "quantum hydrodynamic fluids" described by some theoretical models connected to string theory.

What worries me about this graph?
It is that elemental metals [copper, silver, aluminum, paladium, ...] lie on the same curve. As far as I am aware, they are not strongly correlated. They are nowhere near a quantum critical point. The resistivity is due to electron-phonon scattering. So given that they "accidentally" lie on this "universal" curve suggests to me that the significance of other materials lying close to it may not be of much significance.

On the other hand, the authors claim this "universality" arises because the electron-phonon scattering is "highly efficient" involving "high momentum scattering". They suggest similar scattering occurs in quantum critical metals.

A couple of earlier posts discussed my skepticism/confusion about similar claims about the significance of the magnitude of the linear resistivity.

A key piece of experimental evidence is needed to rigorously justify claims of quantum criticality: measurement of a correlation length which diverges at the quantum critical point.

But, perhaps I am missing something....

Something more to worry about ...

"Table top" science is great.
But how about "table top" nuclear technology?

On the Back Page of the APS News there is an important debate about The Benefits and Risks of Laser Isotope Separation. The science is really interesting, but the prospect of being able to make enriched uranium with a "table top" facility is scary.

I felt Mark Raizen's view that "this will only work for light atoms so we don't have to worry about complex atoms like uranium" is a little naive given the history of science.

Who should be a co-author of your paper?

Only people who have made a "significant scientific contribution to the content of the paper".
In particular, getting a grant, employing someone, or being a lab director does not justify co-authorship.
I have observed that the issue cuts both ways with regard to seniority. It is not just senior people demanding to be co-authors. Sometimes it is junior people putting a senior person as co-author to try and "curry favour" with them or in the hope that it will increase the likelihood of publication.

Here are two frank and helpful articles in Nature Materials which discuss some of the relevant issues.
Authorship without authorization (2004)
Authorship matters (2008)

Like a lot of things, problems may be avoided if there are open discussions before employment or a collaboration begins.

Sometimes co-authorship may be a "grey issue". What constitutes a "significant contribution" can be highly subjective.  But I fear there are too many cases of unjustified co-authorship.

I welcome comments and stories.

Quantum nuclear motion in proton sponges

There is a nice paper Hydrogen Motion in Proton Sponge Cations: A Theoretical Study
by Yevhen Horbatenko and Sergei Vyboishchikov

Proton sponge is a trade name for a particular compound which is a strong base, i.e. it likes to bond to protons.

The proton sponge compounds are of particular interest to me because they work by the proton forming a strong hydrogen bond between two nitrogen atoms.

In this paper the authors first use quantum chemistry to calculate the adiabatic potential energy surface for the ground state as a function of the proton position. They then calculate the proton vibrational wavefunction and energy.

The three potential energy curves above correspond to the three distinct cases of hydrogen bonding: i) strong hydrogen bond, ii) low barrier hydrogen bond, iii) weak hydrogen bond. [They arise naturally in my simple model of H-bonds.]

I found two results of the authors particularly interesting.

First, the shape of the potential energy curve [and specifically the presence or absence of a barrier] depends on the level of quantum chemistry or density functional used in the calculation.

Second, the paper has a nice physical insight for strong hydrogen bonds that I have not seen before. The vibrational energy levels have spacing similar to that for a square well potential of with width comparable to the donor-acceptor distance. Specifically, the energy of the n-th level is proportional to n^2, whereas for a harmonic potential it is proportional to (n-1). This reflects how anharmonic the potential is.

Diluting your accomplishments

In applications for grants, jobs, tenure, and promotion one is asked to list a range of accomplishments: research, collaborations, teaching, community service, ...

There is real danger here that you produce a long list of activities and this can really dilute the impact of your actual significant accomplishments on the reader/reviewer. Hence, I think it is best to not list everything but highlight a few accomplishments and give some specific details of why they are significant.

On a related matter I think that universities are putting increasing pressure on faculty to be involved in a diverse range of activities so they can produce such lists. I noticed this particularly in a couple of tenure/promotion cases I recently reviewed. I was really impressed by how much the applicants had done but I wondered if they had focussed more on just a few of the activities whether everyone would have been better off.

Seminar at JNU

Today I am giving a seminar in the School of Physical Sciences at Jawaharlal Nehru University in Delhi.

"Spin frustration in organic Mott insulators: from quantum spin liquids to superconductors." Slides are here.

The talk material is covered in great detail in a review article, written with Ben Powell.

My host is Brijesh Kumar.

Aside: JNU is the best university for the humanities in India. It is a hot-bed of student and faculty political activism. Yesterday there was a strike. One of the student demands was that Ph.D scholarships be extended from 5 to 7 years! 

Being on campus reminded me a bit of an Australian campus when I was an undergraduate.

One of my Indian friends told me many faculty [and consequently students] believe that science and rationality are imperialistic Western social constructs, just like their counterparts in the West!

I think the West is guilty of many imperialistic atrocities; but science and rationality are not among them.

Enzymes are not different just better

Before I gave my talk in Bangalore about proton transfer in enzymes I reviewed some of the recent literature since I have not worked on the problem for a few years.

I found a very nice paper in Nature Chemistry
Taking Ockham's razor to enzyme dynamics and catalysis
by David R. Glowacki, Jeremy N. Harvey, and Adrian J. Mulholland

They consider a simple transition state model for the anomalous kinetic isotope effects that have been observed in several enzymes. These anomalies have previously been claimed to be evidence for quantum tunneling, breakdown of transition state theory, and require a new paradigm for enzyme catalysis.
The key feature of their model is the assumption that there are two transition states, not one, being associated with two possible conformations of the enzyme-substrate complex.

They end the paper quoting a 1991 Nature paper by Jeremy Knowles
Enzyme catalysis: Not different, just better.

Another nice reference removing the almost mystical interpretation of proteins [and containing some nice thermodynamics] is
Protein heat capacity: An anomaly that maybe never was by Alan Cooper.
I need to read it.

There is an important lesson here, particularly for proponents of "quantum biology". Extra-ordinary explanations require first ruling out simpler less glamorous explanations.

Should I do a postdoc in the same topic as my Ph.D?

No.
Breadth of experience is important; both for your development as a scientist and to demonstrate your versatility to potential employers. It should also be fun and interesting to work on something different.

However, a complete change of research field is not a good idea because the learning curve is so great meaning it is unlikely you will produce your first postdoc paper in a timely manner. (I recommend 6 months; others one 100 days).

It is a good if you can use some of the expertise, experience, and/or techniques you have developed in your Ph.D during your postdoc.

So on balance, here is my suggestion. Most projects involve applying a specific technique (experimental, computational, or analytical) to a specific system  (e.g., a class of materials or model Hamiltonians). A great situation is if you either:
use your Ph.D technique on a new system
or
apply a new technique to your Ph.D system.

For example, if you did a Ph.D using neutron scattering to study transition metal oxides do a postdoc using inelastic X-ray scattering on the same materials.
Or if you used an electronic structure technique (e.g. DMFT+LDA) to study iron pnictide superconductors to do a postdoc using this technique to study organic charge transfer salts.

Unfortuantely, I get postdoc applications which say in effect "I worked on obscure topic X in my Ph.D and I want to keep working on it for the rest of my life. I don't care what you are working on. But I am sure you will want to hire me..."

Everyone is different and every situation is different. There will always be exceptions to the above considerations, especially for the brilliant [not me or you!].
I welcome comments.

Memorisation or understanding?

Friends in India recommended my family watch 3 Idiots. It is the most commercially successful Bollywood movie ever. And it is about university education in India!
The movie tackles important issues such as

• rote learning vs. understanding
• learning for passion vs. career success
• parental pressure
• mental health and suicide
• abuse of authority by faculty 

Some of it is funny. Some of it is very sad.

Definitive evidence for a topological insulator

Last Friday I had a nice meeting at IISc Bangalore with Subroto Mukerjee.
One thing he emphasized to me is that if you see evidence of surface states with a Dirac cone (e.g. in ARPES or quantum oscillations) it is not unambiguous evidence that you have a topological insulator. That requires seeing an odd number of Dirac cones.