Relating non-Fermi liquid transport properties to thermodynamics

On tuesday I had nice discussion with Raghu Mahajan, Maissam Barkeshli, and Sean Hartnoll about their recent preprint Non-Fermi liquids and the Wiedemann-Franz law.

Aside: I generally find that discussing a paper with the authors before/after I have read it greatly increases my understanding. Here are a few things that became clearer to me.

In this paper "almost conserved quantities" means quantities for which the relaxation time is very long. Thus in a Fermi liquid the quasi-particles have very long lifetimes and so one can think of the quasi-particle number for every wave-vector near the Fermi surface as being "almost conserved". This means there are many conserved quantities.

However, they consider a system in which there is a Drude peak in the frequency dependent conductivity but fermionic quasi-particles are poorly defined due to large scattering. Optimally doped cuprates might be an example of a real material with this property. I thought that one dimensional models that exhibit this are Luttinger liquids. They have a Drude peak due to a collective bosonic mode but no fermionic quasi-particles. However, they are close to integrability which corresponds to having an infinite number of conserved quantities.

Note, this is different from most of the bad metals I discuss on this blog: they have no Drude peak and no quasi-particles. Although Aristomenis Donos and Sean recently considered a model (based on the holographic correspondence) that does have this property.

A Drude peak but no quasi-particles means there is one dominant relaxation timescale, that for momentum relaxation. This is what they mean by only one almost conserved quantity. This is a bit like hydrodynamics.

Central to the paper is a "memory matrix formalism" for transport properties. Some justification (and an intuitive understanding) for that is given in this paper. Central to that is the real part of static correlation functions [thermodynamic quantities] between the total momentum P and the electrical current J and heat current Q.

A Wiedemann-Franz type ratio can given in terms of these thermodynamic functions. The actual Lorenz ratio is much less than one.

This is because of a cancellation of the two terms in 

where the first term obeys the modified ratio

This is the central result of the paper. The ratio of two transport quantities is determined by the ratio of two thermodynamic quantities.

It will be nice to see extensions of this approach to give the thermopower (alpha/sigma=Seebeck coefficient) and the Hall coefficient. Both these quantities are fairly independent of scattering time in a Fermi liquid.

I think that in the absence of thermal conductivity due to phonons (unrealistic) the thermoelectric figure of merit could be larger than one.

In some sense this work is similar in spirit to that of Shastry on the Hall coefficient and thermopower. He considered the high frequency limits of these quantities for strongly correlated electron models and showed they could be related to equal time expectation values of operators (thermodynamic quantities).
He also considered Kelvin's formula for the thermopower.

I have one minor quibble. They say that CeCoIn5 violates Wiedemann-Franz (WF) at low temperatures. However, in a PRL Michael Smith and I showed that the relevant experimental paper in Science involves a spurious extrapolation to low temperatures. At sufficiently low temperatures we claim WF will hold. I think this alternative point of view should be stated in the paper.
It does seem awfully hard to find violations of Wiedemann-Franz.

A marginal Fermi liquid talk at Stanford

Hopefully only the Fermi liquid, and not the talk content, is marginal!

On thursday I am giving a seminar for the Stanford Institute for Materials and Energy Sciences.

Here is the current version of the slides: Overdoped cuprates are anisotropic marginal Fermi liquids.

The most recent results in the talk are described in great detail in a PRB written with Jure Kokalj and Nigel Hussey.

Another distinct experimental signature of the chiral anomaly

Previously I posted how a negative classical magnetoresistance with an unusual angular dependence would provide a "smoking gun" for the chiral anomaly.

Another one is described

Probing the chiral anomaly with nonlocal transport in Weyl semimetals
Sid Parameswaran, Tarun Grover, D.A. Abanin, D.A. Pesin, and Ashvin Vishwanath

The figure below nicely describes the proposed experimental setup and signature.

I thank Sid for describing these results to me.

The most desirable citations

Not all citations are equal.
This highlights how they are an imperfect measure of scientific impact.
Broadly they might be classified as being of two types.

1. Token.
"There are many papers on this topic, including Jones et al."

2. Substantial.
"We use the results [method, equation, material, or concept] of Jones et al. to obtain new results".

There is a big difference. For token citations, the existence of the citing paper is really independent of the cited paper, i.e., it would not really matter whether or not the cited paper existed.
In contrast, for substantial citations,  parts [or even the whole] of the citing paper would not exist if the cited paper did not exist.

Unfortunately, too many of the citations I receive are token rather than substantial. I find this discouraging and embarrassing. Even worse, a few times I have been cited as "McKenzie has shown X to be true" when my paper actually showed X to be false!

Thus, it is very satisfying when someone actually uses your results in a constructive manner.
Recently, Tony Wright and I published a paper Quantum oscillations and Berry's phase in topological insulator surface states with broken particle-hole symmetry. One of our results was to propose a method to more robustly extract a signature of the Berry's phase from experimental data.

Tony pointed out to me a recent preprint that actually uses our method (see the Figure below).

The preprint is by a Korean Rugby League team [13 players]

Spin-Chiral Bulk Fermi Surfaces of BiTeI Proven by Quantum Oscillations
Joonbum Park, E. Kampert, Kyung-Hwan Jin, Man Jin Eom, Jongmok Ok, E.S. Choi, F. Wolff-Fabris, K.D. Lee, N. Hur, J.-S Rhyee, Y.J. Jo, Seung-Hoon Jhi, and Jun Sung Kim

A bad metal talk at Berkeley

Hopefully the talk isn't bad, just the metals!
On Friday I am giving a seminar in the Physics Department at Berkeley.

Here is the current version of the slides for my talk.

The main results in the talk are in a recent PRL, written with Jure Kokalj.

The organic charge transfer salts and the relevant Hubbard model are discussed extensively in a review, written with Ben Powell.

Defining non-trivial quantum effects in chemical dynamics

Bill Miller has a very nice Perspective: Quantum or Classical coherence? in the Journal of Chemical Physics. I thank him for explaining some of it to me today.

He clearly defines what he considers to be a truly quantum effect in chemical dynamics.
It is particularly interesting because by his definition Rabi oscillations are not quantum. They are just like two coupled classical harmonic oscillators.

He starts with a Feynman path integral representation of some time-dependent correlation function and considers the semi-classical (SC) limit. The correlation function can be written as an initial value representation (IVR). If one linearises the paths (LSC) one obtains classical Wigner functions and one cannot capture quantum interference effects [e.g. double slit interference which involves paths with more than infinitesimal separation].

Tao and Miller considered the semi-classical path-integral representation of the spin-boson model. They use the Meyer-Miller-Stock-Thoss representation to map the "spin" [two-level system] to a pair of harmonic oscillators [for condensed matter physicists these are Schwinger bosons]. This allows a semi-classical treatment of the two-level system. Then the Rabi oscillations are just like transfer of energy backwards and forwards between two coupled classical harmonic oscillators. This leads to the figure below.
The "Present model" refers to the LSC-IVR treatment, i..e there is no quantum interference.

Ishizaki-Fleming refers to a much more sophisticated "quantum" treatment working towards describing the much-hyped [and probably mistaken] "quantum coherence" of exciton transfer in photosynthetic systems.

The article also contains several other concrete examples: some with and some without quantum coherence. A Tully non-adiabatic problem is particularly interesting because the nuclei exhibit quantum coherence but the electrons don't.

Copper sulphate is a spin liquid

It is amazing since a common science project for school children is to make blue crystals of copper sulphate [CuSO4.5H2O]!
[Although I was surprised and disappointed when my son just told me he never did it].

Perhaps, one may not have to look so hard for quantum materials.

The first X-ray crystallography experiment [by von Laue] was also performed on copper sulphate pentahydrate.

It turns out that the Cu2+ ions (spin-1/2) form chains that are very weakly coupled to one another and so are effectively one-dimensional antiferromagnetic Heisenberg chains above the three-dimensional Neel ordering temperature of about 100 mK.
[Caveat: strictly speaking half of the Cu2+ ions form chains; the other half are essentially isolated and non-interacting].

Minor caveat: the relevant intrachain exchange interaction J ~ 0.25 meV and so one only sees the spinons for temperatures of order a Kelvin.

I first learned all this in the introduction of this Nature Physics paper.

I dislike "arbitrary units" on graphs

It is not unusual in papers to see graphs in which the vertical scale is given in "arbitrary units".  The most common occurrence of this may be experimental measurements of some spectrum, for example, a graph of the absorbance versus frequency (photon energy) of a solution of a specific molecule.
However, some theoretical papers do this too.

There are several reasons why authors may do this.

Laziness. It can be hard work and confusing to work out the actual physical units for some theoretical calculations.

Complexity.  For experiments it can be extremely difficult to normalise and calibrate some detectors.

Uncertainty and embarrassment. Parameters such as detector efficiency, sample thickness,
geometric corrections, solution concentration can involve large uncertainties so the horizontal scale may be unknown by as much as an order of magnitude.
But I think these uncertainties should be reported because they present a challenge for improvement.

I realise that the measured "units" may be detector dependent and not particularly useful to experimenters in different labs. For example, the units may be "number of clicks per second in homemade photodetector 3 in Professor Smith's lab".
But I still think those details should be reported.

I strongly dislike the use of "arbitrary units" for the following reasons.

1. It belies the fundamental fact that any physical quantity does have actual units.

2. For experiments it means that the reported measurements are not reproducible.
i.e. they cannot be checked. The shape of the spectrum can be checked but not the magnitude.

3. This practice limits the comparison of theory and experiment.
The absolute intensity [spectral weight] of a spectrum will be predicted by theory.  It is important to test this experimentally. Just because a theoretical calculation gives the correct spectrum does not mean it is correct.
Absolute units allow one to test sum rules.

[Aside: I have a prejudice/suspicion that matrix elements may be generally more theory sensitive than energies. For example, in quantum many-body theory it is possible to get a very accurate ground state energy with a variational wave function that has a small overlap with the true ground state.]

Another example is the temperature dependence of transport properties.
Sometimes resistivity is reported in arbitrary units or the resistance [rather than resistivity is reported].
Simple theories can sometimes get the temperature dependence of the properties such as resistivity and thermopower correct but the absolute magnitude can be off by orders of magnitude.

For a nice example of people working very hard to normalise spectra, test sum rules, and get physical insight, see the paper
Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain.

A less impressive example [that I was involved in] is in the paper Transition dipole strength of eumelanin.

So, do the hard yards. Don't use "arbitrary units".
If you referee a paper that does, request the authors to do better.

Justifying ourselves

Maybe I am reviewing too many CVs with inflated claims. This Dilbert struck a chord.

A distinct experimental signature of the chiral anomaly

There is a nice preprint
Chiral Anomaly and Classical Negative Magnetoresistance of Weyl Metals
by Dam  Son [a string theorist!] and Boris Spivak

This clearly shows a distinct experimental signature of the chiral anomaly associated with Weyl metals.
[Aside: I think that Weyl should not get his name on this for the same reasons discussed in this post.]

The key physics is summarised in the Figure below.

The Dirac cones [another misnomer?] associated with the chiral anomaly must come in pairs.
Consider the case where the magnetic field and electric field are parallel [and in the z-direction]. The magnetic field induces an anomalous charge current that destroys charge at one Dirac point and creates it at the other. This leads to an anomalous current that is proportional to the square of the magnetic field strength. Thus, the resistance decreases with increasing magnetic field, i.e. (classical) negative magnetoresistance. This is in distinct contrast to traditional classical orbital magnetoresistance which is normally positive.

Might this occur in a real material?
For the case of pyrochlore iridates [with 24 Dirac cones!] this magnetoresistance was discussed by Vivek Aji in a PRB.

I thank Boris Spivak for helpful discussions about this work.

Arthur Wightman (1922-2013): Doyen of Mathematical Physics and Gentleman Scientist

I just learned that Arthur Wightman died earlier this year. He is probably best known for axioms of quantum field theory, super-selection rules, and a famous book, PCT, Spin, Statistics and all that.
Arguably, Wightman's greatest legacy is being the advisor and mentor to a selection of Princeton Ph.D students who went on to distinguished careers, mostly in mathematical physics.
There are some nice testimonials on the Princeton Physics web site. Reading them it struck me that Wightman would have measured rather poorly on today's common metrics [grant money, numbers of publications, journal impact factors, numbers of Ph.D students, citations]; yet, he had an incredible scientific impact!

Wightman was extremely helpful and generous to me when I was a beginning graduate student at Princeton in the mid 1980s. In particular, I had a paper from my undergraduate thesis that I was trying to publish. He gave me great encouragement, some helpful feedback, and arranged for it to be published in the Journal of Mathematical Physics. To quote the beautiful testimonial of John Preskill, "Though I did not sufficiently appreciate it at the time, Arthur was incredibly generous with his time."

The following observations by Jurg Frohlich are particularly poignant:

As Arthur Jaffe said, the disappearance of Arthur Wightman marks the end of an era. I fear it may also mark the gradual disappearance of an attitude and style among scientists that I associate directly with people like Arthur Wightman and Res Jost: 

Focus on the central problems of your field – even if they may not be doable immediately  

– generously share your time, insights and ideas with others, especially with young colleagues, 

generously support the careers of young scientists, 

maintain unerring intellectual honesty and integrity – in short, try to be a gentleman scientist!

More than his scientific oeuvre, I view the latter qualities as Arthur Wightman’s central legacy for which he will be remembered, and which, in a time when they are endangered, we should cherish!

Whatever happened to course profiles?

When I was an undergraduate I don't think Course Profiles [detailed descriptions of course content, assessment, policies, ...] even existed. If I recall correctly there was a Course handbook which contained a paragraph about each course. Sometimes on the first day of class the lecturer might hand out a one page sheet with some more details about the course.
Times have certainly changed.

Now at University of Queensland [and I presume at most other universities] the course profile can be 15 plus pages. Here is an example from a course I have taught. It contains very detailed descriptions of not just course content, learning methods, and assessment but how these map onto "graduate attributes."
Profiles have to include details about university policies about plagiarism, student appeals, disabilities, library resources, ....

 Hence, it is not surprising that students often don't read the details, including the ones that really matter (e.g., what topics will be covered when, what they should be reading, the key concepts in the course,...).

Preparation of these documents increases the administrative overhead to faculty of teaching.

But, my biggest concern is that Course Profiles have become quasi-legal documents, like a will that might be contested by potential heirs. In particular, the language about assessment and marking [grading] have to be very carefully crafted so that they are not open to dispute. Increasingly, disgruntled students will claim that assessment criteria was ambiguous or not applied in a manner consistent with their "reading" of the course profile. One has to anticipate bizarre possibilities such as a student getting a dozen medical certificates [for every wednesday of the semester] so they never have to do any laboratory sessions but can pass the course!

Is my experience common? Should we care? Can anything be done about this?

Does spatial homogeneity break down in strongly correlated electron systems?

In considering the electronic properties of the metallic phase of solids one almost always assumes spatial homogeneity of the the underlying electron fluid.
This is convenient and powerful. But, that does not mean that it is always correct!
Occasionally one might entertain the possibility of some sort of charge order and symmetry breaking such as a charge density wave or stripes.

Over the past two decades it has been found that the two-dimensional electron gases (2DEGs) that occur at interfaces in semiconductor heterostructures exhibit a metal-insulator transition that has confounded definitive theoretical understanding. Furthermore, there has been considerable debate about the relative importance and interplay of disorder (due to impurities) and strong electronic correlations.

A fundamental challenge is to describe the dependence of the resistance on the temperature, density, and magnetic field (parallel to the 2DEG). Furthermore, double layer systems exhibit extremely large and unanticipated "drag" resistance: current is passed through one layer and the

A nice review of the experiments is in a 2010 Rev. Mod. Phys. Colloquium article by Boris Spivak, Sergei Kravchenko, Steve Kivelson, and X.P.A. Gao.

The authors discuss the shortcomings of different theories and make the novel proposal that the key relevant physics may be the existence of new phases ["microemulsion phases"] between the Fermi liquid metal and Wigner crystal. The relevant [speculative and schematic] phase diagram is shown below. The horizontal axis is density. The vertical axis is 1/d where d is the distance between the 2DEG and a metallic ground plane in a ideal MOSFET.

A key to understanding the temperature dependent properties is the Pomeranchuk effect [analogous to that in 3He] where the Wigner solid has a larger entropy than the Fermi liquid due to spin entropy.

Spivak and Kivelson have proposed that the large drag resistance arises from a phase consisting of mobile bubbles of Wigner crystal embedded in a Fermi liquid.

A key to testing these novel explanations experimentally is the development of new probes for the direct imaging of the spatial inhomogenities associated with the microemulsion phases.

A recent post Is hydrodynamics ever relevant in metals? considered a more recent theory of Andreev, Kivelson, and Spivak to describe transport properties above the Fermi liquid degeneracy temperature.

A previous post Deconstructing the metal-insulator transition in 2DEGs considered an alternative theory of the temperature dependence of the electrical resistance, based on a Dynamical Mean-Field Theory treatment of an extended Hubbard model.

Rich economics: with real data

In a previous post What is wrong with this textbook? I made the disturbing observation that a popular introductory economics textbook did not contain any real data.

My son [an economics major] and I recently read Poor Economics: a radical rethinking of the way to fight global poverty by two MIT economists, by Abhijit V. Banerjee and Esther Duflo.
The book is all based on real data. The book has a very impressive website. For each chapter it has the associated data and figures, and interactive tools to work with the data.
[n.b. how this transparency contrasts to the two Harvard economists who were reluctant to release the Excel spreadsheet they used in a controversial study about government debt].

The authors address questions such as

Why do the poor remain poor? Are they trapped?
What is the most effective way to help them?
Why do many well-intentioned aid programs fail?

The emphasis is on finding strategies that have actually been proven to work, rather than programs Western ideologues [both conservative and liberal] think should work. The authors are also very compassionate pointing out how many of the mistakes and bad choices that the poor make are not that different to those of us in the affluent West make. For example, we often make choices (e.g. lack of exercise and poor diet) that are not in our best long term interests, even though we know what is the best thing to do.

Pauling's last blackboard

Today I visited the Linus Pauling Archives at Oregon State University.
[I was actually on vacation in Corvallis visiting my sister-in-law but I just had to take a visit].

They have assembled a lot of fascinating material online, which is worth perusing. For example, how a funding agency convinced him to start working on proteins, and the details of correspondence about his (erroneous) ideas about quasi-crystals.

But, in the actual library there is a small display featuring Pauling's last blackboard, his desk, some molecular models, some calculators, his two Nobel Prize medals, and his signature beret.

Pauling is definitely one of my scientific heroes. He made multiple landmark contributions. Most of us would be happy to do just one of the things he is known for. He was truly the master of multi-disciplinarity. He brought quantum physics to chemistry, structural chemistry to biology, and molecular biology to medicine. But he had "clay feet", failing to see problems with his ideas about the triple helix for DNA, vitamin C and quasi-crystals.

Curious fact: I was allowed to touch the Nobel Prize medals but my brother-in-law asked if I could have my picture taken with Pauling's signature beret. The librarian said no!

The promise of physics in India

Physics World recently ran a special report about Physics in India. The government is investing heavily in science through a range of worthwhile initiatives. But, the challenges are substantial. Here are a few things that stood out to me.

The Tata Institute for Fundamental Research (TIFR) admits about 100 graduate students every year. More than 10,000 take the entrance exam which is held in about 20 cities!
Although India has hundreds of universities and research institutes it seems that only the IITs and a handful of universities give an undergraduate education that is adequate preparation for a Ph.D.

TIFR is opening a new campus in Hyderabad. This involves hiring 250 faculty members in the next 12 years!
This will be larger both in space and faculty numbers than Mumbai.

This is in addition to the International Center for Theoretical Sciences being established by TIFR in Bangalore.

Students who do decide to pursue science Ph.D's have often resisted considerable cultural and family pressure to pursue engineering careers instead.

The five new IISERs [Indian Institutes for Science Education and Research] are combating the rote learning culture by including a substantial component of research in the undergraduate curriculum.

The cultural obstacles for women in physics are considerable, as discussed in a frank interview with Shobhana Narasimhan.

A few trivia.
The Tata Institute was set up with money from Tata but now is solely dependent on government funding. [This is in spite of the fact that the Tata group now has a market capitalization of almost $100 billion!].
During the rise of Nazi Germany, the director of the Indian Institute of Science, Raman [i.e. who discovered the Raman effect] tried to hire Max Born but is was opposed by local faculty.
After Homi Bhabha, the founding director of the Tata Institute, died in a plane crash, his office was closed off for thirty years.

While on the subject of India, The Economist has a fascinating article, Out of the Gloom, about how solar electricity is having a significant positive effect on the economic and educational development of rural India.

Addendum. Here is a helpful summary of some statistics about science education and research in India.  Although, as can be expected from Nature it mindlessly makes some comparisons based on metrics.

The chiral anomaly and topological insulators

Yesterday we had another cake meeting where people "shared their ignorance". 
Mine concerned: what is the relationship between the chiral (parity) anomaly in quantum field theory, edge states, and topological insulators?
I learnt something in the week from this PRL [more string theorists writing about Fermi liquids!], which at least taught me what the relevant "anomaly" equation is for the "violation" of charge conservation.

Ben Powell pointed out that there is a relevant blog post Monopoles passing through flatland by John Preskill. His beautiful post certainly provides the background information I need, in a particularly lucid form. Having read it once I now need to digest it properly. Otherwise, I will still be "ignorant.".

Science outreach to young school kids

I recently went and did some science demonstrations to several grade 3 classes at the local primary school [elementary school in the USA]. I mostly did it because a friend from church, who is a teacher at the school, asked my wife if we could do it. The students were studying a unit on heat transfer.

Here are a few random observations from the experience.

The kids think scientists are like rock stars! You are so cool!
I was asked to wear the white coat and wild hair and so obliged.

The teachers really appreciate it. I was told I could not do just one or two classes because it would not be fair for some of the teachers and students to miss out! I had to do all!

These kids have had an incredible "diet" of computer games and special effects in movies. But, seeing something simple LIVE such as the baking soda rocket or coke can crush really wows them. They are not at all jaded, unlike the kids in this scene from Big Bang Theory!

The kids love it if they can help. Even, just measuring a temperature from a thermometer.

I really want to avoid the "magic show" dimension and try and communicate something about what science is really about, e.g. taking measurements, keeping records, making comparisons, developing concepts.
I also had a go at the "3 states of matter: gas, liquid, solid" myth by asking them about liquid crystals.
It was impressive how some got the point.

I feel we should all make more of an effort to do this sort of thing.
But, I really only felt could do it because my dear wife took care of all the logistics including all the materials for the demonstrations. My main time investment was showing up on the day. Hence, I am mindful that for many of us it may be unrealistic.

Here are some of the demonstrations
Baking soda rocket using a film canister
Tea bag rocket
Coke can crush

Thanks for the positive feedback

Recently I was at a chemistry workshop in the US and a physics one in Korea. It was really encouraging for me to hear from a range of people that they read this blog and find it stimulating. I particularly appreciated that my readers range in seniority and are from groups whose work I respect (and would like to influence).

Sometimes I have worried if the 500 hits per day are coming from non-scientists searching for random topics on Google. This must be the case for "Who is following who?" which is actually about orbital hybridisation, but has received 1000+ page views!
But, now I know I am reaching my target audience. So, thanks to those who took the initiative to talk to me.