Wednesday, August 9, 2017

Subtle paths to effective Hamiltonians in complex materials

Many of the most interesting materials involve significant chemical and structural complexity. Indeed, it is not unusual for a unit cell for a crystal to contain the order of one hundred atoms.
Yet, for a given class of materials, one would like to find an effective Hamiltonian involving as few degrees of freedom and parameters as possible.

Following Kino and Fukuyama, twenty years ago I argued that the simplest possible effective Hamiltonian for a large class of superconducting organic charge transfer salts was a one-band Hubbard model on an anisotropic triangular lattice at half filling.
It seemed natural to then argue that the relevant model for the spin degrees of freedom in the Mott insulating phase is the corresponding frustrated Heisenberg model with spatial anisotropy determined by the anisotropy in the tight-binding model.

However, it turns out this is not the case.
There are some subtle quantum interference effects that I overlooked in the "derivation"  of these effective models, leading to a different spatial anisotropy.
This is shown by some of my colleagues in a nice recent paper, accepted for PRL.

Dynamical reduction of the dimensionality of exchange interactions and the "spin-liquid" phase of κ-(BEDT-TTF)2X 
B. J. Powell, E. P. Kenny, J. Merino


This raises questions about what the relevant effective Hamiltonian is for the metallic, superconducting, and (possibly) ferroelectric phases.

The paper's significance goes beyond organic charge transfer salts to the general problem of finding effective Hamiltonians in complex materials.

Similar interference effects have been found to arise in quite a different class of materials.

Heisenberg and Dzyaloshinskii-Moriya interactions controlled by molecular packing in trinuclear organometallic clusters 
B. J. Powell, J. Merino, A. L. Khosla, and A. C. Jacko

This also reminds me of subtleties (and debates) that occur in the Zhang-Rice "derivation" of the t-J model from a three-band Hubbard model.

Saturday, August 5, 2017

Who was the greatest theoretical chemist of the 19th century?

Dimitri Mendeleev, who proposed the periodic table of the elements, purely from phenomenology and without quantum mechanics!
He even successfully predicted the existence of new elements and their properties.

A friend who is a high school teacher [but not a scientist] asked me about how he should teach the periodic table to chemistry students. It is something that students often memorise, especially in rote-learning cultures, but have little idea about what it means and represents. It makes logical sense, even without quantum mechanics. This video nicely captures both how brilliant Mendeleev was and the logic behind the table.



A key idea is how each column contains elements with similar chemical and physical properties and that as one goes down the column there are systematic trends.
It is good for students to see this with their own eyes.
This video from the Royal Society of Chemistry shows in spectacular fashion how the alkali metals are all highly reactive and that as one goes down the column the reactivity increases.



The next amazing part of the story is how once quantum theory came along it all started to make sense!

Tuesday, August 1, 2017

The role of the Platonic ideal in solid state physics

In the book Who Got Einstein's Office?, about the Institute for Advanced Study at Princeton, the author Ed Regis, mocks it as the "One True Platonic Heaven" because he claims its members are Platonic idealists, who are interested in pure theory, and disdain such "impurities" as computers and applied mathematics.


This stimulated me to think about the limited but useful role of pure mathematics, Platonic idealism, and aesthetics in solid state theory. People seem particularly excited when topology and/or geometry plays a role.

The first example I could think of is the notion of a perfect crystal.

Then comes Bloch's theorem, which surely is the central idea of introductory solid state physics.

Beautiful examples where advanced pure maths plays are role are
Chern-Simons theory of edge states in the Quantum Hall Effect
and topological terms in the action for quantum spin chains, as elucidated by Haldane.

As I have said before I think topological insulators is a beautiful, fascinating, and important topic. However, I am concerned by the disproportionately large number of people working on the topic and the associated hype. I wonder if some of the appeal and infatuation is driven by Platonic idealism.

For a classic example of how Platonism leads to imperfect theory is Kepler's Platonic solid model of the Solar System from Mysterium Cosmographicum (1596).


Good theory finds a balance between beauty and the necessity of dirty details.

Can you think of other examples where Platonic idealism plays a positive role in condensed matter theory?

Saturday, July 22, 2017

Entering the strange world of Kurt Godel

The picture below is of Godel's rotating universe. It represents an exact solution to Einstein's gravitational field equations and has the strange property of closed timelike curves (i.e. one can travel into the past!). This mathematical solution was found by Kurt Godel while he was employed by the Institute for Advanced Study at Princeton.


I think I first encountered this picture in my final undergraduate year in the classic book, The Large Scale Structure of Space-Time by Hawking and Ellis, while working on a research project in general relativity.

Godel's universe is just one example of the fascinating science and stories recounted in the book
Who Got Einstein's Office? Eccentricity and Genius at the Institute for Advanced Study by Ed Regis, first published 30 years ago.

I only read the book this past week and loved it. It is a captivating blend of science, mathematics, personalities, history, philosophy, humorous anecdotes, gossip, eccentricities ...
I was so captivated that I read it during two situations I would not normally read something so "heavy": during a long flight [normally I watch reruns of The Big Bang Theory or Upper Middle Bogan [need to laugh!] or recently a Warren Buffett documentary... sorry better not mention that again...], and during "down time" in the evening after a busy day.

Regis nicely describes the continuum hypothesis, Einstein-Podolsky-Rosen (EPR) "paradox" in quantum theory, von Neumann machines, cellular automata, the Bourbaki seminar, parity violation, the solar neutrino problem, fractals, the stability of matter, ...

The personalities covered include Godel, Einstein, Herman Weyl, John von Neumann, J. Robert Oppenheimer, Freeman Dyson, T.D. Lee,  C.N. Yang, Andre Weil, John Bahcall, Stephen Wolfram, Ed Witten, .....

It is amazing how much Regis packs into less than 300 pages (in a paperback).

The tragic mental health problems of Godel are described in a sensitive manner.

One pathetic story concerns the endless quibbles of T.D. Lee and C.N. Yang.
(Aside: They actually did their Nobel Prize winning work on parity violation at the IAS. This is in contrast to the countless Nobel laureates who at one time have been affiliated with the IAS but did not do their prize work there.)
Lee and Yang (or is it Yang and Lee?) argued constantly about the order in which their names should be listed, not just as co-authors, and at the Nobel ceremony, but even in newspaper and magazine articles about them. Furthermore, it is crazy to read the wildly different and self-serving accounts of certain concrete events. Great scientists are all too human ......

Some people consider the book is a bit of a "hatchet" job and has a mocking tone that paints the IAS in a poor light and questions its value and existence. I would not agree. I think it does show that the IAS has produced a lot of important scholarship. Regis does raise some important questions I mention below. But, I did think that he did refer to the IAS as "the One True Platonic Heaven" too many times.

Regis is implicitly critical of the fact that there is very little interaction between different research groups and disciplines within IAS. However, there is one important story he missed: when Freeman Dyson and the number theorist Hugh Montgomery were introduced at tea at the IAS and they made a connection between random matrix theory (quantum physics) and zeros of the Riemann zeta function.

Some questions the book raises for me include:

Can you really "manage" genius?

How do you create an institutional environment that increases the likelihood of truly great discoveries and scholarship?

What is the best way to hire "great" people?

What is a good mix of young and old staff?

What is a good mix of permanent faculty, postdocs, and short term senior visitors?

When is the absence of students in a research institute good or bad?

When is the absence of experimentalists in an institution bad/good for theoretical physics?

How do you foster a healthy synergy between pure mathematics and theoretical physics?

How might you foster some constructive interaction between distinct disciplines: philosophy, mathematics, theoretical physics, economics, history, ....?

Here is Feynman's perspective (partly quoted in the book):
I don't believe I can really do without teaching. The reason is, I have to have something so that when I don't have any ideas and I'm not getting anywhere I can say to myself, "At least I'm living; at least I'm doing something; I am making some contribution" -- it's just psychological. 
When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don't get any ideas for a while: They have every opportunity to do something, and they are not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come. 
Nothing happens because there's not enough real activity and challenge: You're not in contact with the experimental guys. You don't have to think how to answer questions from the students. Nothing!
Governments have less and less interest in "research for its own sake" and "without constraints" [hallmarks of the IAS]. However, there is an increasing number of generous and wealthy philanthropic organisations who are very interested. These are important questions for them.

Although I lived in Princeton for four years around the time the book was being written I only recall going inside "the Brain Farm" [as a friend called it] once, and that was for a music concert. Nevertheless, I spent many pleasant hours walking, jogging, bird watching, and skiing in the beautiful woods located behind the IAS.

I thank Ben Powell for a conversation about the IAS, stimulating me to remember I had inherited a copy of the book from my parents.

I welcome thoughts on any of the questions and any good IAS stories...

Sunday, July 16, 2017

Lessons for universities from Warren Buffett

This is not about managing university endowments!

On a recent flight I watched the fascinating HBO documentary, Becoming Warren Buffett.
He may be one of the richest people in the world, and perhaps the most successful investor of all time. However, what is much more striking than his success is how he got there: in a completely counter-cultural (or iconoclastic) way.

Here a few lessons that I think are particularly relevant to universities as they struggle with their identity, purpose, and management.

Focus.
Several times Buffett and some of his admirers emphasised this. Good research of companies and understanding the market requires considerable focus. You can't be doing lots of different things or jumping into the latest fad. Universities need to focus on teaching and research. Faculty need to focus on just a few things they can do well.

The long view.
Buffett does not "play" the market. He finds companies that are undervalued or have enduring market share and keeps their stocks for decades.
Similarly, quality and innovative research requires large and long time investments. Rarely do things happen quickly.

Personal relationships are key.
Buffett has had a long and fruitful relationship with Charlie Munger. Furthermore, it seems within Berkshire Hathaway, the personal relationships between employees and with companies they invest in are key. If people can't get along or management is heavy handed or autocratic, long term productivity is unlikely.

Cut all the bureaucratic and management BS.
I might have misheard it but I got the impression that Berkshire Hathaway did not really have an HR or marketing department. Their reputation is their marketing. HR is worked out on a personal level.

It is all about the quality of the product.
Buffett looks for companies that have a quality product for which there is likely be long term demand. It does not matter how much slick marketing there is. In the long term, all that matters is the product. Similarly, universities need to focus on the quality of their "product": their graduates, and the content of the research they produce in papers and books.

Reputation is central.
Buffett preserves his and looks carefully at the companies he invests. Furthermore, a good reputation takes decades to establish but can be lost in days (e.g. through scandal). Universities with a good reputation should really think twice before they "cut corners"  to boost revenue [e.g. by offering expensive Masters degrees by coursework to international students but actually involve students taking undergraduate classes].

Integrity and leadership by example.
This is an important component of Buffett's reputation. Just one example is how he released his tax returns during last years US Presidential campaign. University managers who take/demand ridiculous salaries should think about how that undermines their ability to lead.

Jobs should be fun.
Buffett keeps working because he is having fun not because he wants to make more money. Universities should carefully consider whether they are creating an environment that employees enjoy.

I was also struck by Buffett's generous philanthropy, his concern about economic inequality, and that it would be hard to find a billionaire who was anymore the antithesis of Trump.

Thursday, July 6, 2017

Are theoretical physics and chemistry amenable to online collaboration?

Last week at UQ we had a very nice mathematics colloquium, "Crowdsourcing mathematics problems" by Timothy Gowers.
He first talked about the Polymath project, including its successes and marginal contributions.
He then talked about a specific example of a project currently underway on his own blog, concerning transitive dice. This was pretty accessible to the general audience.

This is where a well defined important problem is defined on a blog and then anyone is free to contribute to the discussion and solution. A strength of this approach is that it makes use of the complementary strengths, experience, and expertise of the participants. Specifically, solving problems includes:
  • selecting a problem that is important, interesting, and potentially ripe for solution
  • defining the problem clearly
  • breaking the problem down into smaller parts including conjectures
  • sketching a possible heuristic argument for the truth of the conjecture
  • giving a rigorous proof of the conjecture
  • finding possible counter-examples to the conjecture
  • connecting the problem to other areas of mathematics
This can be efficient because dead ends and mistakes are found quicker than someone working in isolation. 
People are more motivated and engaged because they are excited to be working on something bigger than themselves and what they might normally tackle. And they enjoy the community.
What about assigning credit in such group work? There is a clear public record of who has contributed what. Obviously, this does not work for bean counters looking at traditional metrics.
This approach mostly attracts senior people who are secure in themselves and their career stage and more interested in solving problems than getting individual credit.

The cultural differences of pure mathematics and physics was striking. The talk was given on whiteboards and blackboards without notes. No powerpoint! The choice of research problems was purely based on curiousity, not any potential practical value or the latest fashion. It is fascinating and encouraging that the pure mathematics community is still "old school" with the focus on quality not quantity.

Aside: Gowers is also well known for initiating a boycott of Elsevier journals.

Now, my question. 
What is stopping theoretical chemistry and physics from such a "crowd sourcing" approach? 
Is it that the problems are not amenable? 
Or is it largely that we are too driven by a system that is fixated on individual credit?

Monday, July 3, 2017

A molecular material and a model Hamiltonian with rich physics

Some of my UQ colleagues and Jaime Merino have written a series of nice papers inspired by an organometallic molecular material Mo3S7(dmit)3. They have considered possible model effective Hamiltonians to describe it and the different ground states that arise depending on the model parameters.
There is a rich interplay of strong correlations, Hund's rule coupling, spin frustration, spin-orbit coupling, flat bands, and Dirac cone physics.
Possible ground states include some sort of Mott insulator, a Haldane phase, semi-metal, ...

A good place to start is the following paper
Low-energy effective theories of the two-thirds filled Hubbard model on the triangular necklace lattice 
C. Janani, J. Merino, Ian P. McCulloch, and B. J. Powell

The figure below (taken from this paper) shows some of the molecular structure and some of the hopping integrals that are associated with an underlying decorated honeycomb lattice.


This model could be called kagomene, because it interpolates between the kagome lattice and the honeycomb lattice (graphene). The figure below is taken from this paper, which uses DFT and Wannier orbitals to estimate the tight-binding parameters and the spin-orbit coupling. Interaction driven topological insulator states are possible on this lattice.



There are a few things that are not "normal" about the physics, arising from the 4/3 band filling and the molecular orbitals that are delocalised over the triangles. Specifically, the orbital degeneracy does not arise from atomic orbital degeneracy (cf. d orbitals, or t2g and eg), but rather the E representation associated with C3 symmetry of the triangles.

Hund's rule coupling. 
This involves the E orbitals and arises purely from the Hubbard U on the non-degenerate orbital on a single lattice site.

Spin-orbital coupling.
This is Spin Molecular Orbital Coupling, where the electron spin couples to the angular momentum associated with motion around the triangle, not the angular momentum of degenerate atomic orbitals.

Haldane phase.
The associated spin-1's arise from the triplet ground state of four electrons on a triangle.
A DMRG study shows that this is the ground state of a three leg-ladder Hubbard model at 2/3 filling.

Many interesting and important open questions remain about the general phase diagram of the Hubbard model on the kagomene lattice. For example, the nature of the Mott insulator, different types of topological order, the possibility of superconductivity.....

Hopefully, these studies will stimulate new experimental studies and synthesis of new chemical compounds in this fascinating class of materials.