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.