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.

Disciplined writing

Here are a few hints about writing papers and grant applications.

First, just do it! Start writing. Worry about all the details on the second and third rewrite.

Second, review every single sentence. Is each sentence actually true? Any claim needs to be backed up with evidence (e.g., a reference). Never offer undefendable ground. In my view, far too many papers contain assertions that are debatable. For example, "this experimental data proves theory A is correct" should be replaced with "this experimental data is consistent with theory A".

Third, reconsider qualitative language and value judgements. You may write "the agreement between my theory and experiment is excellent" but others might think it is not particularly impressive. Perhaps it is better to be more circumspect and write "the theory and experiment are compared in Figure X" and let the reader make their own judgement.

Fourth, avoid vagueness. Be concrete, precise, specific, and informative. Instead of writing "there are lots of different theories of high-temperature superconductors" write something like "the diverse range of theories of the phase diagram of cuprate superconductors include theories that focus on resonating valence bonds, antiferromagnetic spin fluctuations, van-Hove singularities, quantum critical points,...." and give specific references.

Fifth, get feedback from others.

Sixth, keep at it. It is hard work, even for the experienced and gifted.

Signatures of an exotic Mott insulator

There is a really nice preprint Power-law dependence of the optical conductivity observed in the quantum spin-liquid compound κ-(BEDT-TTF)2Cu2(CN)3
by Sebastian Elsasser, Dan Wu, Martin Dressel, John A. Schlueter

The simplest possible picture of charge excitations in a Mott insulator is that it is similar to a band insulator but the origin of the gap is from correlations. Then one expects that

1. There is a non-zero charge gap.
2. The same energy gap occurs in the dc conductivity (activation energy) and the optical conductivity.
3. At finite temperature the subgap absorption in the optical conductivity arises due to thermal excitations across the energy gap. Hence, the subgap absorption decreases with decreasing temperature.

Indeed such behaviour is observed in the organic Mott insulator kappa-(BEDT-TTF)2Cu(NCN)2Cl which has an antiferromagnetic ground state [denoted kappa-Cl below].

However, this preprint reports qualitatively different behaviour in the title compound [denoted kappa-CN below]. Specifically,
A. There is activated behaviour in the dc conductivity.
B. Power law behaviour occurs in the optical conductivity, from about 20 to 1000 cm-1.
C. This "subgap" absorption increases in intensity with decreasing temperature.

Motivated by earlier, but less detailed, measurements showing similar results, Tai-Kai Ng and Patrick Lee, developed a theory which captures some of these properties. The key physics is the following. In the spin liquid state the elementary excitations are spinons which are charge neutral [and have a Fermi surface] and U(1) internal gauge fields associated with spin chirality fluctuations. An external electromagnetic field couples to the gapless spinon excitations indirectly via the internal gauge field. This produces the low frequency absorption.

A couple of cautionary caveats:

a. the experiment does require subtracting off a background signal associated with molecular vibrations, but that does appear robust to me.

b. the observed power law exponent is smaller than predicted by Ng and Lee.

However, doing reliable theory in this regime is so challenging, requiring uncontrolled approximations, I think it is unrealistic to expect this theory to get all the details right.

I thank Martin Dressel for bringing this work to my attention.

Understanding plagiarism by non-Western students

Undergraduate students from Western countries commit plagiarism. This has led to highly effective commercial software to detect it such as Turnitin.

Recently I have encountered several incidents of plagiarism from postgraduate students from non-Western countries. It is easy to be shocked and morally indignant about this, until one learns some of the cultural background and common educational practices in some countries. The following is helpful:

T. A. Abinandanan, an engineer at the Indian Institute of Science, says that the practice of duplicating information is deeply ingrained in Indian education. “Right from school our students are encouraged to take material from books and websites and use it in their charts and lab notebooks. In written answers, verbatim reproduction from textbooks is even rewarded with higher marks,” he told Nature. “Thus, when students start doing research, they have so much of this attitude to unlearn. This unfortunate case provides us yet another opportunity to redouble our efforts in training our students.”

This is taken from a News item that appeared in Nature earlier this year.

A key to stopping is this is education, central documentation of incidents, and strictly enforced penalties that increase substantially for repeat offenders.

Twisting fluorescent proteins

Seth Olsen and I just finished a paper A two-state model of twisted intramolecular charge transfer in monomethine dyes.
It presents a simple characterisation the ground and excited state potential energy surfaces associated with bond twisting of the chromophore for the green fluorescent protein. This is based on a 2 x 2 matrix Hamiltonian acting on diabatic states which have a simple valence bond interpretation.

In praise of publons

A publon is a "quantum" of publication. It is the least publishable unit of research.

I have found this to be a very helpful concept in doing and supervising research and deciding what and when to publish. The basic idea is that once you have a result that is publishable [a new equation or measurement or graph or numerical result] for a specific research project one should start to prepare a paper around it. Large projects can be broken down into publons. I first really took on board the concept after reading Peter Feibelman's A Ph.D is not enough. 

The advantages are

-research and publication can much more manageable

-it avoids procrastination, particularly by perfectionists who always want to get one more result...

-it placates the "bean counters" who expect quick results and large numbers of papers

-papers should have a single message and not several. publons prevent that single message being diluted or distracted from

-it can be helpful in building confidence and momentum in new situations: a new project, a new institution, a new position, a new student, or a new postdoc....

The disadvantages/dangers are

-results are published too quickly before they have been checked, compared to other methods, or for other systems.

-peoples publication list becomes inflated.

-too many short and superficial papers are published

-it is pandering to the "bean counters".

Talk in Bangalore on overdoped cuprates

Today I had a nice visit to the Physics Department at Indian Institute of Science. My host was Subroto Mukherjee. I gave a talk Interlayer magnetoresistance as a probe of Fermi surface anisotropies in overdoped cuprates. It was nice to get so many questions during the talk.

The more recent results are in a PRB written with Jure Kokalj and Nigel Hussey.

Talk on methine dyes at IISc

Tomorrow I am visiting the Solid State and Structural Chemistry Unit at the Indian Institute of Science in Bangalore. My host in Ramasesha. I will give a talk [slides] about effective Hamiltonians for the excited electronic states of methine dyes (including the chromophore for the Green Fluorescent Protein). All of this work was done largely by Seth Olsen.
A good introduction and overview is our recent J. Chem. Phys. paper.

The Higgs boson enters cocktail party chit-chat

It is amazing to me how much the Higgs boson has featured in the press and entered the public consciousness lately. I fear this is partly because CERN has such a large and effective press office. This was particularly brought home to me when my wife sent me the above cartoon which one of her friends had posted on Facebook.

On the one hand, I think it is good that physics is getting all this publicity. On the other hand I think the scientific significance of this discovery is being blown out of proportion, driven by the large budgets involved....

An annoying bug in Mac Mail

Over the past year I have encountered the following annoying and sometimes embarrassing problem with Mail on my Mac.
It seems that sometimes (apparently randomly to me) it strips attachments from incoming emails. A first I thought the sender had forgotten to attach the file, as sometimes happens. I asked them to resend it. Sometimes the attachment comes through on a second attempt. Other times it does not. The size does not seem to be relevant.
I thought the university system might be stripping the attachment and so I looked on the UQ Webmail and found that it was there in the original email there but not in Mail.

I welcome suggestions. I looked online and found other people with this problem, but no clear solutions.

Raman scattering from strongly correlated electron materials

At the cake meeting [weekly condensed matter theory group meeting] we took three weeks to go through the review article Inelastic light scattering from correlated electrons by Devereaux and Hackl.

I found it very helpful. Here are a few things I learnt.

Polarisation dependence
It is sensitive to anisotropy in Fermi surface properties via different light polarisations. This is because the relevant matrix elements depend on the polarisation.

In principle this can be used to detect unconventional superconductivity and even distinguish d-xy and d_x2-y2.

This polarisation dependence also shows signs of anisotropies in the metallic state of cuprates.

Background electronic continuum
In a simple metal the scattering off electron-hole pairs should cutoff at a wavevector of about v_F q where v_F is the Fermi velocity and q is the change in wavevector of the light. This is a relatively small energy. However, in strongly correlated materials such as the cuprates this extends to much higher energies. This is still not really understood.

Electronic Raman scattering played a key role in the cuprates in two regards. First, via two magnon scattering it provided the first experimental estimate of the antiferromagnetic exchange interaction J in the parent Mott insulator. [n.b. most people would agree that the large J is at the heart of the high-Tc]. Second, it showed the large "incoherent" background which is linear in omega, having a large influence on the development of the Marginal Fermi liquid phenomenology.

I was a little disappointed to see that interpreting the experimental results is not completely straightforward and that often there was a fair bit of noise in the data.

Overall it seems this is a promising technique which provides a complementary probe to others such as ARPES, optics, and transport measurements. No doubt in the next few decades there will be technological advances that will increase resolution and increase the signal to noise. It would be nice to see some measurements on organic superconductors.

Conserve your political capital

Everyone, and particularly junior people, need to accept the painful reality that within any department and university they only have a finite and limited amount of political capital and goodwill. So, conserve it like a precious commodity. Only cash it in for things you really need: e.g., getting tenure, adequate lab and office space, a strong letter of support for an important grant application, a reasonable teaching load, promotion, ....

Don't squander your capital and goodwill with minor gripes/requests/demands about lab space, small salary increments, not getting exactly the teaching you want, minor irritations about administrative procedures,...
Middle (and senior) management may quickly get tired of these minor concerns and will then be much less sympathetic and willing to address major concerns.

Clarifying the origins of mass and the Higgs boson

There is a nice article by Frank Close, Higgs boson: beginning of the end or end of the beginning.
It gives a succinct discussion of the background to the recent discovery and what it does and does not mean for elementary particle physics.

A couple of points I found helpful.
Reports in the media that "the Higgs boson is responsible for mass" are misleading.

First, almost all the mass in our bodies comes from neutrons and protons, and more than 99% of their mass comes from the binding energy of quarks (due to quantum chromodynamics) and not from the mass of isolated quarks.
The latter are hypothesized to get their mass from the Higgs field.
[Aside: the boson and the field are not the same thing. The boson is one specific excitation of the field].

Secondly, the CERN experiments do not establish that leptons and quarks obtain their mass from the Higgs field. Rather, only that the gauge bosons in electro-weak theory obtain their mass from the Higgs field.
Further experiments at CERN in the next few years may establish this though.

But, the existence of the Higgs field is relevant to chemistry in the following sense. Nuclei are so compact because of the large energy of pions, which is a reflection of the mass of quarks. Furthermore, the size of hydrogen atom is determined by the
mass of the electron.

Criteria for tenure?

Previously, I posted about Stanford's criteria for tenure in chemistry. It is striking that so little consideration is given to the metrics (grant dollars, number of publications, number of graduate students, number of citations, h-index,...) that lesser institutions use to judge staff.
In this light, it was particularly interesting to read the extract below of an interesting interview with Ken Wilson about his career.

PoS [Physics of Scale]

So by the time you come to Cornell, people don't really know what you're working on. I mean on the surface, there are many things that you're doing, but deep down, there's a commitment to try to explore how far can you go with quantum field theory.

KGW

The people at Cornell had more of an interest to know what I was doing than people at Harvard, because they were going to have to make a decision... Of course one of the things that happened was, as you may or may not be aware, is that they gave me tenure after only two years and with no publication record. In fact, there was one or two papers on the publications list when I was taken for tenure and Francis Low complained that I should have made sure there was none. Just to prove that it was possible.

History certainly vindicated Cornell's decision!
I wonder whether this really could happen today.
Quality science and the development of quality scientists takes time....

Deconstructing magnetoresistance in cuprates

In simple metals the temperature and magnetic field dependence of the magnetoresistance is encoded in some function of the product omega_c*tau.
omega_c is the cyclotron frequency which is proportional to the magnetic field B and independent of temperature.
tau is the scattering time, which is temperature dependent and field independent, and should have the same temperature dependence as 1/rho where rho(B=0) is the zero field resistivity.

These observations lead to Kohler's rule which is obeyed by simple metals.
A plot of the ratio of the rho(B)/rho(B=0) versus B/rho(B=0) should be independent of temperature. In 1995 Ong's group observed significant violations of Kohler's rule in the underdoped and overdoped cuprates.
In 1998 I pointed out that in one mysterious organic metal there were also significant violations. The paper also has an extensive discussion of reasons why Kohler's rule can fail.

Recently, Jure Kokalj, Nigel Hussey and I wrote a paper which showed that a wide range of transport properties of overdoped cuprates could be quantitatively described by an anisotropic marginal Fermi liquid picture. A referee suggested that we would not be able to describe the temperature dependence of the intra-layer magnetoresistance. It turns out that this also violates Kohler's rule but satisfies a "modified" form where omega_c*tau is related to the Hall angle. We found that with our model we could describe the available experimental data but the magneto-resistance was remarkably sensitive to the details of shape of the Fermi surface. The results are discussed in detail in our paper which just appeared in Physical Review B.