A very strange metal

The linear chain compound Li0.9Mo6O17 exhibits a subtle competition between superconductivity, a "bad" metal, and a strange "insulating" phase. Recently large deviations from the Weidemann-Franz law were reported by Nigel Hussey's group.

The graph below shows the temperature dependence of the electrical resistance for current parallel to the chain direction. It has a "metallic" temperature dependence above about 30 K, and an "insulating" temperature dependence between the superconducting transition temperature around 1 K and 30 K. This is rather unusual and puzzling since one normally sees a direct transition from a metallic phase to a superconducting phase. Although there are other cases such as reported in this PRB [see Fig. 2 inset] for an organic charge transfer salt where a superconducting state occurs close to a charge ordered insulator [see also the Table in this PRL].

The data is taken from a Europhys. Lett. by Chen et al. which also reports a rather strange angular dependent magnetoresistance.

How much money does a "World class" university cost/need/want?

A lot!
This is a fact that I feel politicians who fund public universities just do not appreciate.
Here is a statistic that I find mind boggling.
Princeton University now has an endowment of $17.1 billion dollars!

Aside: the graph above shows how the endowment is now above pre-GFC levels.

What does that mean? Well, the university only has 5,000 undergraduates and 2,500 grad students. That means the average endowment per student is more than $2 million!
[This is the highest per student endowment in the world].
The university aims to spend the endowment at a rate of 4-5.75% on the annual operating budget. That means about $100 K per year is being contributed (indirectly) towards each students education. For reference annual tuition is about $36 K. Room and board are a further $12K per year.

How does the Mott insulating phase differ from the metallic phase near it?

For organic superconductors there is a first-order phase transition from a Mott insulator to  a superconductor with increasing pressure. This post concerns the relevant Hubbard model, that on an anisotropic triangular lattice at half filling, as discussed in this review.

With increasing U/t there is a first-order transition from a metal to an insulator.

This leads to a discontinuity in the double occupancy at the transition, illustrated in the sketch above.

The double occupancy D is shown below versus U/t for t'=0.8t. For reference D=0.25 for a half-filled system at U=0.

The figure is taken from a 2008 PRL by Ohashi et al.

D is calculated with Cluster Dynamical Mean-Field Theory (DMFT)

1.  A rough estimate of the  magnitude of D can be found from the Hellman-Feynman theorem D= dE_0/dU where E_0 is the ground state energy.
In the Mott phase this is dominated by the antiferromagnetic Heisenberg exchange J ~ 4t^2/U. Hence, D ~ (t/U)^2

2. The discontinuity in D at the metal-insulator transition  is relatively small, being about a 15 per cent change for T=0.1t, and less at higher temperatures. To me this suggests that in some sense the character of the metallic and insulating phases near the transition are not that different, just like a liquid and gas are hard to distinguish near the critical point.

3. These results are in contrast to Brinkmann-Rice theory [which ignores J] which gives D=0 in the Mott phase.

Covalent character of hydrogen bonds III

Following up on an earlier post about how indirect spin couplings in NMR (Nuclear Magnetic Resonance) may be a signature of the covalent character of hydrogen bonds I have been reading a range of papers on the subject. The Figure below shows how the calculated O-O nuclear coupling J correlates with the donor-acceptor distance [another example of an empirical correlation I have been highlighting].

The figure is taken from a 2000 JACS by Del Bene, Perera, and Bartlett.

One thing that is frustrating about reading most of these chemistry NMR papers is that they never explain the basic physics involved.

The Oxford Chemistry primer on NMR by Peter Hore has a useful section on Indirect coupling. He gives a nice simple argument explaining how [from 2nd order perturbation theory] the H-H coupling in the hydrogen molecule is roughly J ~ A^2/E  where A is the proton hyperfine interaction and E is the energy gap between the ground state and the lowest triplet state. This estimate gives J ~ 300 Hz, which is comparable to the actual value. Basically, when one flips one proton spin the A flips the electron spin, converting the ground state spin singlet into the excited triplet state.

The figure is taken by a nice webpage by Hans Reich

Deconstructing vertex corrections

Ultimately much of quantum many-body theory concerns calculating correlation functions which are measurable. For example, the conductivity can written as a current-current correlation function [Kubo formula]. The simplest approximation neglects vertex corrections and just calculates the "bubble" diagram consisting of the product of Green's functions.

What are vertex corrections? When do they matter? What sort of robust or general results are available about them?

Many people, including myself, often just ignore them. I fear this is partly motivated by difficulty rather good scientific criteria.

Below are a few things I am slowly learning, re-learning, and digesting.

Migdal showed that for the electron-phonon interaction the vertex corrections are small due to the smallness of the ratio of the electronic mass to the nuclear mass [alternatively the ratio of the speed of sound to the Fermi velocity].
But, Migdal's argument breaks down for an electron-magnon interaction.

Neglecting vertex corrections is equivalent to making the relaxation time approximation (RTA) when solving the Boltzmann equation. Then the quasi-particle lifetime equals the transport lifetime because one ignores dependence of the scattering rate on momentum transfer. Below is some helpful text from a review by Kontani:

 ....we have to take the Current Vertex Correction [CVC] into account correctly, which is totally dropped in the Relaxation Time Approximation [RTA]. In interacting electron systems, an excited electron induces other particle– hole excitations by collisions. The CVC represents the induced current due to these particle–hole excitations. The CVC is closely related to the momentum conservation law, which is mathematically described using the Ward identity [28–31]. In fact, Landau proved the existence of the CVC, which is called backflow in the phenomenological Fermi liquid theory, as a natural consequence of the conservation law [28]. The CVC can be significant in strongly correlated Fermi liquids owing to strong electron–electron scattering.

For specific types of interactions Ward identities allow one to relate the vertex function to derivatives of the self energy. Mahan's book (Section 8.1.3) discusses this in detail.

In the limit of infinite dimensions [in which Dynamical Mean-Field Theory (DMFT)] becomes exact, vertex corrections can be neglected.

In a recent PRB, Bergeron, Hankevych, Kyung, and Tremblay calculated the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. They found that at "high" temperatures (T > 0.2t) vertex corrections did not matter much, but were significant at lower temperatures near a quantum critical point. 

Its all in the title?

An earlier post, Entitled to a reading, pointed out the value of carefully choosing engaging titles for your papers. Contrast the titles of the following two papers. The subject and conclusions of the papers are similar.
Benzene forms hydrogen bonds with water published in Science.

Low-J rotational spectra, internal rotation, and structures of several benzene-water dimers published in Journal of Chemical Physics.

Arunan brought this contrast to my attention.

Should you follow a textbook?

Yes. Closely.

This is the conclusion I have slowly come to over the years. Furthermore, the more junior the class the more closely you should follow a text.
Often I have struggled to find a text I thought suitable or have drawn on material from several books. This has meant giving out lecture notes.

It seems closely following a book is most effective if you can actually get students to read it! This appears to be a major goal of people who use methods such as Peer Instruction.

Having said all that you can expect student complaints. "You are just telling me what it is in the book". "There is too much reading". "Why are we paying you?" "Don't you have any ideas of your own!"

I welcome your thoughts. I would be curious to learn of systematic studies which showed whether student learning (rather than satisfaction and comfort) was actually enhanced by closely following a text.

Postdoc in theoretical chemical physics available at UQ

Seth Olsen and I have just advertised for a postdoc to work with us at UQ. The flavour of our interests and approach can be seen in posts on this blog under labels such as organic photonics, quantum chemistry, conical intersections, and Born-Oppenheimer approximation.

Competing phases are generic to strongly correlated electron systems

One might think that the Hubbard model on the square lattice with infinite U would be relatively boring. For example, a "simple" theory like Brinkman-Rice [or equivalently slave bosons] would predict that the ground state is a metal, except at half filling where it is a Mott insulator.

There is an interesting preprint Phases of the infinite U Hubbard model by Liu, Yao, Berg, and Kivelson. Here are a just a couple of the results concerning the ground state on a ladder, that I found interesting.

For 3/8 filling [3 electrons per 4 sites] they find the ground state is an insulator with a charge gap (0.24t) and plaquette bond order. The spin degrees of freedom are equivalent to those of a spin-3/2 antiferromagnetic Heisenberg model. I think this can be "understood" this by starting from the limit of weakly coupled placquettes with 3 electrons per plaquette.

For 1/4 filling the ground state is an insulator with a charge gap (~0.1t) and a small spin gap and "dimerisation". 

The authors argue these phases will also be stable for the actual isotropic square lattice.

It is fascinating to me that one can produce such rich and diverse broken symmetry states starting from such a relatively simple model which has no competing interactions in the Hamiltonian.

An earlier PRB by Yao, Tsai, and Kivelson, Myriad Phases of the checkerboard Hubbard model suggests competing phases are a generic feature of strongly interacting systems. 

Writing your first grant proposal

For most people this is one of the hardest and most tedious things to do. Some helpful thoughts are Writing the first proposal by John Wilkins [who mentored me through my first proposal!]. There is also a useful chapter in A Ph.D is not Enough, by Peter Feibelman.

Three basic questions you need to make sure your clearly and convincingly answer:

• Is the proposed project important?
• Is it feasible?
• Are you the best person to do the work?

Should cats be herded?

Everyone agrees that you need to herd cattle and sheep. But what about cats?
Cats are best enjoyed and fulfil their purpose if they are left alone and allowed to be what they are.

What is the relevance of this? It is sometimes claimed that academic researchers are like cats. They are fiercely independent and groups of them are very difficult to manage. Hence, books such as Herding Cats: Being advice to aspiring academic and research leaders by Geoff Garrett and Graeme Davies.

I became aware of the existence of the book because Geoff Garrett, who is now Queensland's Chief Scientist, gave the UQ Physics Colloquium on friday. I have not read the book. Afterwards a colleague expressed reservations about the ideas presented, saying, "this is relevant to engineers, not physicists!"

One idea that was presented was the importance of having a "Big Hairy Audacious Goal" which creates team spirit. Although laudable on some level, I am hard pressed to think of examples in science that have been fruitful or that I personally find inspiring. Maybe I am jaded but ones such as "nuclear fusion in our lifetime", "build a quantum computer", "discover a room temperature superconductor", "cheap organic solar cells to save the planet", or "lets make our university number one" just don't seem that achievable via highly managed research teams. 

Furthermore, it seems that most Nobel Prize discoveries did not result from such programs, but rather from curiousity driven research by "independent" research groups. One obvious exception are Nobel Prizes for discoveries in elementary particle physics.

What do you think?

Two persistent misconceptions about hydrogen bonding?

Misconception 1?
         "H-bonding interactions are dominated by electrostatics"
This is stated in Assessment of the Performance of DFT and DFT-D Methods for Describing Distance Dependence of Hydrogen-Bonded Interactions, one of the most downloaded recent papers in Journal of Chemical Theory and Computation.
Earlier posts discuss the importance of covalent interactions, particularly in shorter (stronger) Hydrogen bonds.


Misconception 2?

“One of the most fundamental, yet enigmatic, of all chemical processes is the transfer of protons in liquid water, which occurs via ultrafast quantum tunneling in the hydrogen bonded network”.

This is stated in a 2003 Science article and challenged by Dominic Marx in Proton Transfer 200 Years after von Grotthuss: Insights from Ab Initio Simulations

proton transfer has often readily been explained by taking recourse to the appealing concept of “proton tunneling”,27–29 which was begotten shortly after the birth of quantum mechanics. It is essentially based on a static view derived from symmetrical one-dimensional double-well potentials V(δ), such as the one shown on the left in Figure 3 [see below], thus completely omitting the possibility of a barrierless scenario, as sketched in the central panel of this figure. 

For a discussion of the origin [in terms of covalent interactions!] of these three different potentials, see my preprint.

Deconstructing the chemical potential of the cuprate superconductors

I have been reading through the nice review Finite temperature properties of doped antiferromagnets by Jaklic and Prelovsek from 2000. They summarise their studies of the t-J model by the Finite temperature Lanczos method.

At first sight the graph below of the temperature and doping dependence of the chemical potential does not look particularly interesting [at least to me]. However, they highlight its significance.

Here are a few points.

• In a simple Fermi liquid the chemical potential has a positive, quadratic and weak temperature dependence. This is only seen for doping c_h=x=0.3
• For a wide doping range [0.05 < c_h < 0.3] the temperature dependence is approximately linear. The slope changes sign for approximately optimal doping (c_h ~ 0.15).
• The weak temperature dependence for c_h ~ 0.15 means that optimal doping corresponds to maximum entropy!  [This can be deduced via the Maxwell relation below. Don't you love thermodynamics!]

• This relation is also related [approximately] to the thermopower via a relationship [equation 8.6], which is essentially a restatement of the Kelvin formula [discussed  by Peterson and Shastry].
• The latter means the thermopower should change sign around optimal doping, as is indeed observed [more on that later].
• The large entropy near optimal doping emerges from the interplay of the localised spins [from the remnants of the Mott insulator] and frustration of the antiferromagnetic spin interactions via doping.

I would be interested to see a similar calculation for the Hubbard model on the anisotropic triangular lattice at half filling to see how the chemical potential varies as a function of U/t as the Mott insulator is approached from within the metallic phase.

When the data is "too good to be true"

Remember Hendrik Schon! A decade ago he published a string of very impressive Nature and Science papers that eventually turned out to be "too good to be true". It seems a similar thing has been happening in the field of social psychology. The AP reports

 three graduate students grew suspicious of the data Stapel had supplied them without allowing them to participate in the actual research. When they ran statistical tests on it themselves they found it too perfect to be true and went to the university's dean with their suspicions.
In the future, the university plans to require raw data from studies to be preserved and made available to other researchers on request - a practice already common in most disciplines.

Nature News reports

The commission found that co-authors of Stapel's papers seem to have been unaware of the fraud, naively trusting in Stapel's reputation and fooled by elaborate preparations for tests that were never actually carried out..... Stapel and a colleague or student came up with a hypothesis, and then designed an experiment to test it. Stapel took responsibility for collecting data through what he said was a network of contacts at other institutions, and several weeks later produced a fictitious data file for his colleague to write up into a paper. On other occasions, Stapel received co-authorship after producing data he claimed to have collected previously that exactly matched the needs of a colleague working on a particular study.....
The data were also suspicious, the report says: effects were large; missing data and outliers were rare; and hypotheses were rarely refuted. Journals publishing Stapel's papers did not question the omission of details about where the data came from. 

This is part of a Nature News piece which has the misleading title "Report finds massive fraud at Dutch universities". A more responsible and accurate title would be "Report finds massive fraud by one Dutch professor of social psychology". In  the comments section several Dutch researchers rightly object to the title.

Chemistry driven by conical intersections

Nonadiabatic Quantum Chemistry is a nice Chemical Reviews article by David Yarkony.

It is quite succinct but covers a significant number of specific chemical systems where non-adiabatic effects [including conical intersections] are important and have been treated theoretically.

Here I just mention one example for which theory has failed so far, the vibrationally mediated photodissociation of NH3 (ammonia) to NH2 + H.
Experiments find that if the excited state contains a symmetric (asymmetric) N-H  stretch the dominant decay channel is to the NH2 ground state (excited state). Yarkony says that calculations [e.g. this one from Truhlar's group, which contains the figure below] have not yet captured this vibrational selectivity.

I thank Seth Olsen for bringing the article to my attention.

Should university be fun, fun, fun!

There is an interesting (and somewhat depressing) article in the New York Times Why Science Majors Change Their Minds (It's just so darn hard). It discusses how in the US there is a big push to have more STEM (Science, Technology, Engineering, and Mathematics) graduates but even if many start these degrees they do not finish.
One contributing factor is that these courses are graded harder than humanities courses.
The article also discusses initiatives, particularly in engineering courses, to make the courses more "fun" and "relevant", especially via projects.
I think this is all commendable and valuable. However, I have a sneaking discomfort that people [students, faculty, and administrators] just don't want to face the painful reality that engineering and science education does involve a certain amount of tedious hard work and that ultimately a lot of jobs (in any field) just aren't that exciting or satisfying.
Or am I just a grumpy old man?

I thank my wife for bringing the article to my attention.

A sign of something important

The Hall coefficient is a fundamental property of metals. In simple Fermi liquid metals it is temperature independent and inverse proportional to the charge carrier density. It has the same sign as the charge carriers (electrons or holes). A major triumph of the Bloch model of metals is that it could explain the sign of the Hall coefficient for simple metals in terms of their Fermi surface.

In contrast, the Hall coefficient of cuprate superconductors has a complex temperature and doping dependence which defies a simple description. Basic questions about the Hall coefficient are:

• What determines its sign?
• What is the origin of its temperature dependence?
• What is the relationship between it and the structure (or absence) of the Fermi surface?  

A 2006 PRB by Tsukada and Ono describes measurements of the Hall coefficient in the cuprate LSCO. The graph below shows the temperature dependence of the Hall coefficient for a range of dopings x of La2-xSrxCuO4 in the overdoped region. For reference, optimal doping is around x ~ 0.2, and for x larger than 0.3 there is no superconductivity. Note the sign change with increasing x.

The authors emphasize how this is a tricky measurement because one has to be careful that the current paths that are measured [to get both sigma_xx and sigma_xy needed for the Hall coefficient] really do lie in the plane of the layers and do not contain spurious contributions (see this earlier post about the challenge of electronic transport measurements in highly anisotropic materials).

The sign change may be an important signature of strong electronic correlations. I find it interesting (and surprising) that the observed sign change at x=0.3 is obtained in a high temperature series expansion of the high frequency Hall coefficient for the t-J model [in this 1994 PRL by Shastry, Shraiman, and Singh (SSS!)]. [An earlier post discusses Shastry's approach]. [Note: this calculation does not have a t' hopping term, which may be relevant. For example, it has a significant effect on the shape and  curvature of the Fermi surface and the proximity to van-Hove singularities. See below].

An alternative explanation of the sign change in terms of Mott physics was given by Stanescu and Phillips.

There may be a more mundane explanation in terms of changes in the Fermi surface associated with the proximity of the van Hove singularity in LSCO. Indeed ARPES experiments do find an electron-like Fermi surface for x~0.3. Furthermore, experiments on Tl2201 [which does not have a close van Hove singularity] do not see any hint of a decreasing Hall coefficient [or sign change] as one increases the doping on the overdoped side towards samples with Tc=0. [Higher dopings seem problematic for Tl2201].
Furthermore, one can quantitatively describe the temperature dependence of data for x=0.3  [including the sign change with temperature] if one uses a realistic Fermi surface and assumes that the impurity scattering rate is anisotropic over the Fermi surface. See this PRB; I thank Nigel Hussey for bringing it to my attention.

I thank Jure Kokalj for some helpful discussions.

Converging to the "right" answer

This week I had an interesting experience. I was doing a calculation and comparing my result to experiment. The comparison was poor, with a discrepancy of a factor of about two. This was disappointing, but then I decided that the theory was just too simple and one should not experiment anything better than qualitative agreement... I just had to accept this.
But then I found a mistake in my Mathematica code. I realised I had to check everything more carefully. .. One of my variables I had defined incorrectly... I redid the plot. The agreement of theory and experiment was excellent.

But, now there is a real danger. I could stop checking for errors. Afterall, given I already found a couple there may be another one which will lead to new discrepancies.
I will let you know if I find any. But, I have to confess the motivation to find errors is less than it was..

I wonder how often this happens in science.  I think I recall that there are some famous historical examples, e.g. that over years the value of the speed of light and the charge on the electron have drifted, but at any particular time peoples values have always been within a standard deviation of the latest measurements.

Just remember Feynman's warning: "The easiest person to fool is yourself."

The challenge of a simple measurement

Just because an experimentalist claims to have measured a specific physical quantity does not mean they actually have measured the desired quantity. Theorists need to be particularly wary at uncritically accepting data.

To most people, especially theorists, measuring the electrical resistivity of a metal sounds like an almost trivial measurement! Surely, you just stick a sample of the metal between the leads of an ohm-meter and read off the resistance!
The temperature dependence of the resistance can provide significant information about scattering of quasi-particles in the metal and any decent theory should be able to describe it. A famous case it the "linear in T" resistivity of optimally doped cuprate superconductors, a signature of non- Fermi liquid behaviour.

Most of the interesting strongly correlated metals (cuprates, organic charge transfer salts, iron pnictides, ....) have layered crystal structures leading to anisotropic electronic properties. These are sometimes referred to as quasi-two-dimensional metals.
Accurately, measuring the resistivity (and its temperature dependence) in the three different directions though is a highly non-trivial exercise. Basically, this is because you have to be sure that the current is going through the sample in the direction you think it is.

This is highlighted in a recent Nature Communications article from Nigel Hussey's group. They state:

 In a quasi-1D conductor, it is especially problematic to measure the smallest of the resistivity tensor components, because even a small admixture of either of the two larger orthogonal components can give rise to erroneous values and distort the intrinsic temperature dependence of the in-chain resistivity. In Li_0.9Mo₆O₁₇, reported room-temperature values for the in-chain (b axis) resistivity range from 400 μΩ cm²³ to more than 10 mΩ cm^34, 35.

Reported values for the ratio of the a to b axis resistivity vary from about 2 to 100!
This is a very large discrepancy!

I wrote this post because I thought I had come up with a fancy theoretical explanation of why in one paper the resistivity anisotropy ratio was only ~4, whereas band structure predicts a much larger value. However, when I surveyed the literature I discovered the result I was so proud of explaining is probably an artefact!