Getting the same answer from complementary numerical methods

Developing reliable numerical methods that can give meaningful and useful results for lattice models [e.g. the Hubbard, Heisenberg, and t-J models] of strongly correlated electrons is a challenging and tedious task. An important outcome is when complementary methods (including analytical methods) give the same result!

There is a nice Physical Review E article by Marcos Rigol, Tyler Bryant, and Rajiv Singh which considers the application of a new numerical linked cluster algorithm (NLC) method to the t-J model. To put nicely things in context they state

In spite of its simplicity, understanding finite-temperature thermodynamic properties of the t-J model has proven to be a very challenging task. Quantum Monte Carlo simulations suffer from severe sign problems, which become a major difficulty at low temperatures. The two general approaches that have been commonly used to study this model are [exact diagonalisation] ED and [high temperature expansions] HTE. ED studies in which one fully diagonalizes the t-J Hamiltonian are difficult since they can only be done for very small systems, as a consequence of which finite size effects are very large. A more efficient approach to this problem is the finite-temperature Lanczos method (FTLM), which has been developed by Jaklič and Prelovšek (JP). Within this approach the full thermodynamic trace is reduced by randomly sampling the eigenstates of the Hamiltonian. This allows one to study larger systems sizes in an unbiased way, but still finite size effects become relevant as the temperature is lowered.

The outstanding question concerning high temperature expansion (HTE) methods is whether they can give reliable results at the "low" temperatures relevant to experiments. Here it should be stressed that the energy scales t and J are of the order of 1000 K. On the other hand HTE and NLC have the distinct advantage that they are valid for the infinite lattice and do not suffer from finite size effects (a problem for FTLM and ED).

The figure below shows the very encouraging result that the complementary methods NLC and FTLM are in agreement for a calculation of the temperature dependence of the entropy, down to temperatures as low at about 0.1t, and for a range of dopings. 

This consistency increases the confidence of the reliability of both methods to describe the metallic phase of strongly correlated electron models. Some of the key underlying physics may be that the correlations are short-ranged in this regime of dopings and temperatures.

Two earlier posts considered the significance of FTLM results for understanding the metallic phase of cuprates at optimal doping. 

Can scientists save the planet?

No. Businesses (small and large) are more likely saviours.

My January edition of Physics Today arrived in the snail mail this week!
Nevertheless, it was still worth reading.

The most interesting article were interviews with Steve Koonin, a former BP Chief Scientist and DOE Undersecretary for science and Ellen Williams, the current BP Chief Scientist. Two quotes from the Koonin interview particularly struck me

many of the people reading Physics Today are in the academic world, and if they want to really change energy, I would strongly recommend six months or a year out in the private sector, whether in a big company or a small startup. It really is a very different mindset than what a basic [academic] researcher has.... 

For some of our biggest problems, whether energy or other big problems in society, the technology is in many ways the easy part. The rate-limiting steps for many of our problems are societal: How people behave, what incentives there are, etc. I think the social sciences have a lot to bring to that discussion that has not really been exploited yet. That’s the direction I’m headed in; it’s still science, and it’s still in some ways goal-driven. But we’ve got to pay attention and better understand the human issues here: Policy, behavior, economics, perception, and how we fuse that with technology. 

This highlights something I (and some commenters on this blog) have said before. Perhaps there is too much emphasis (particularly in the chemistry community) on shifting basic research to trying to improve the efficiency of specific candidate materials and devices [e.g. Gratzel cells, bulk heterojunction organic solar cells, thermoelectric materials, hydrogen fuel cells, ...].

Picture is the 32 megawatt solar farm at Brookhaven National Lab.

Tailor your talk to the audience

The first key to preparing a good talk is taking into account the backgrounds and interests of your audience. This preparation begins with the title and abstract. You need to motivate people to come!

You should NOT give the same talk to experimentalists as to theorists, or the same talk at a specialised conference and a departmental colloquium. This may seem obvious but it is amazing how much it happens. It actually does require significant work, experience, and discipline to give relevant, appropriate, and enjoyable talks.

Next week I am giving the UQ Quantum science seminar. It is attended by physicists, mostly theorists, working in condensed matter, BECs, quantum information, and quantum optics. Below is the abstract I prepared.

A quantum physicist's view of hydrogen bonding

Hydrogen bonding plays a central and diverse role in chemistry and biology. It is key to the unique properties of water, the double helix structure of DNA, and the unique folding of proteins. Yet it is arguably the most poorly understood form of chemical bonding. Indeed the International Union of Pure and Applied Chemistry (IUPAC) recently gave a new definition of hydrogen bonding.
As a physicist I recently investigated hydrogen bonding with three goals:
i. to use the simplest possible model
ii. to describe a wide range of phenomena
iii. to elucidate the role of quantum physics in hydrogen bonding.

I consider a model which describes hydrogen bonding and proton transfer between two molecules due to the quantum mechanical interaction between the orbitals of the H-atom and of the donor (D) and acceptor (A) atoms in the molecules [1].
The model is based on a effective Hamiltonian which acts on two diabatic states and has a simple chemically motivated form for its matrix elements.
The model gives insight into the "H-bond puzzle" [2], describes different
classes of bonds (weak, low-barrier, and strong), and  gives a quantitative description of empirical correlations between the donor-acceptor distance and binding energies, D-H bond lengths, the softening (hardening) of D-H stretch (bend) vibrational frequencies.
A key testable prediction of the model is the UV photo-dissociation of symmetric
H-bonded complexes via an excited electronic state with an exalted vibrational frequency.

[1] R.H. McKenzie, Chemical Physics Letters 535, 196 (2012).
[2] G. Gilli and P. Gilli, The Nature of the Hydrogen Bond (Oxford UP, 2009).

Adapting teaching to student needs

In Bangalore recently it was nice to meet Shobhana Narasimhan. Following discussions about the challenge of teaching science in the developing world she sent me an interesting and helpful paper she wrote Training the Future Scientist: Making the Transition from 'Knowledge’ to 'Synthesis’.
After describing the context of Indian education [which utilises copious amounts of rote learning], she reviews Bloom's taxonomy of learning objectives [knowledge, comprehension, application, analysis, evaluation, synthesis], and then discusses specific initiatives she has taken when teaching Introductory and Advanced Condensed Matter Physics to graduate students.

Although geared to the Indian context many of the ideas are relevant and adaptable to other contexts. The paper showcases the importance of establishing where students are at, what their needs are, and adapting our teaching accordingly.

Introducing topological insulators

On his website Joel Moore has a copy of the slides from a colloquium  New topologically ordered phases of condensed matter that he gave in 2009.

I found this a helpful introduction/review. Sometimes talk slides are more helpful than review articles. A picture can be worth a thousand words...

Chemical hardness is the Hubbard U

Chemists are very good at coming up with new concepts and organising principles which can describe a wide range of chemical trends. Two examples are electronegativity and chemical hardness. The latter provides a basis to understand the principle of hard and soft acid and bases (HSAB): hard acids prefer to co-ordinate (bond) with hard bases while soft acids prefer to co-ordinate with soft acids.

Can one provide a quantitative measure of "chemical hardness"? A 1983 JACS paper by Robert Parr and Ralph Pearson suggested that for a specific atom or molecule it could be defined as the second derivative of the ground state energy with respect to the number of electrons N

or the discrete version 

where I is the ionisation energy and A the electron affinity. For comparison, the electronegativity is I+A, and the first derivative of E with respect to N is the chemical potential.
Aside: Physicists can relate eta to the charge compressibility.

At the end of their article they point out the fascinating fact (to me) that eta is equal to the Hubbard U. [I think this is just the case for an open shell system, i.e. where N is such that there is one electron in the HOMO.]

The argument that Parr and Pearson use to just the HSAB principle from their definition of hardness was not very clear to me. Perhaps it is clearer in an earlier paper of Klopman that they cite. I was hoping to see something like a simple proof [based on an asymmetric 2 site Hubbard model?] that the acid-base bonding is a maximum when the difference between the U on the two sites is minimal? But, maybe this is not possible...

A recent J Phys. Chem. article by James Reed on the subject considers some of the complexities of the subject.

Finding order in the cuprate confusion

In Science last week there was a Perspective Cuprates Get Orders to Charge by John Tranquada about a paper that finds long-range charge order in YBCO in the same doping region at which quantum oscillations are observed. The Perspective is worth reading because it gives a nice overview of the issues.

On the other hand, I find it disappointing that Tranquada takes a view, similar  to other experimentalists, that little progress has been made in theory:

Although there are many theoretical proposals to describe this unusual “normal” state, no new simplifying paradigm has yet appeared.

I disagree. I think "plain vanilla" RVB theory does provide the simplifying paradigm. I acknowledge that not everyone agrees and there are still outstanding questions. But not all the theories out there are equal. There have varying amounts of supporting evidence and internal consistency.

My somewhat ill-informed and prejudicial view is that stripes, charge ordering, and quantum oscillations are a "red herring" which may only occur in some families of materials. The essential physics is in the RVB = Gutwiller projected BCS state and that these other phenomena are just perturbations on top of it.

I welcome other views.