Why are quantum gases called degenerate?

In my recent tutorial on bad metals at IISER Pune a student asked me a basic question that I could not answer:
"Why is the degenerate Fermi gas called "degenerate"?
Is it anything to do with degenerate energy levels?"

So, I went in search for answers.

The Wikipedia entry on Degenerate matter is a bit rambling and I found it unhelpful.

I then went to the library and looked at a few textbooks and found a range of answers. Some books use the term "degenerate" without any elaboration.

In the discussion below it seems looking at the Oxford dictionary is helpful:

Having lost the physical, mental, or moral qualities considered normal and desirable; showing evidence of decline. 

technical: Lacking some usual or expected property or quality, in particular.

Here are a few entries

degenerate. (This use of the word is completely unrelated to its other use to describe a set of quantum states that have the same energy).

Daniel V. Schroeder, An Introduction to Thermal Physics, page 272.
[my favourite undergraduate text on statistical mechanics]

... a bit of terminology. At low temperature, quantum ideal gases behave very differently from the way a classical ideal gas behaves. ..... The quantum gases are said to be degenerate at low temperatures. This is not a moral judgement. Rather, the word "degenerate" is used in the sense of departing markedly from the properties of an "ordinary" classical gas.

Ralph Baierlein, Thermal Physics, page 192.

Gas degeneration proper. The quantitative study of the deviations from the classical gas laws when xi is not very small ...

Erwin Schrodinger, Statistical Thermodynamics (1936)

Here xi is the product of the particle density and the thermal deBroglie wave length cubed.

The definition of a quantum gas is one where xi becomes larger than one. (This occurs for "high" densities and "low" temperatures).

But then there is an interpretation in terms of degeneracy of energy levels because one can consider the case where each energy level has degeneracy g and condition for a non-degenerate gas (i.e. Maxwell-Boltzmann statistics to apply) is
g >> n_i ~ exp ((mu-Ei)/kB T) = number of particles in level i

I welcome comments.

Deconstructing noise in organic charge transfer salts

There are several things that I used to find very puzzling about electrical noise measurements on the metallic phase of organic charge transfer salts.

The  measured noise spectrum is close to (but not exactly) 1/f.

The disparity of time/energy scales.
What is the relationship (if any) between the noise (which is sometimes measured on time scales as long as one thousand seconds (mHz)) and microscopics (which one might calculate with quantum chemistry and/or Hubbard models, but typically involves energies larger than meV or frequencies that can be ten orders of magnitude larger)?

Obscure trends.
If one looks at the actual exponent alpha of the noise, 1/f^alpha. It varies in a non-monotonic way as the temperature T varies. This looks rather "random" to me (i.e. I found it hard to believe there was any systematics involved).

However, Jens Muller and collaborators have used a model due to Dutta, Dimon, and Horn (DDH) to nicely elucidate what is going on in a series of papers such as this one.

Origin of the glass-like dynamics in molecular metals κ-(BEDT-TTF)2X: implications from fluctuation spectroscopy and ab initio calculations 
Jens Müller, Benedikt Hartmann, Robert Rommel, Jens Brandenburg, Stephen M Winter, and John A Schlueter

Here are the basic ideas of the DDH model.
There is a distribution of relaxation times tau, which arise because there are a distribution of activation energies E for relaxation.

tau0 is a typical "attempt frequency"/molecular vibration frequency for something like a conformational change of a molecule.

One assumes that for a specific tau that the noise is simply Lorentzian. But one then averages over D(E), the distribution of activation energies.

One can then show that at a given temperature the noise has a 1/f^alpha form with an exponent given by,

A specific consistency test of the model is to then compare the measured alpha to that calculated from the above expression using the observed temperature dependence of the noise spectrum. This comparison is shown in the figure above. 

One can also invert the equation above to extract D(E), giving the result in the figure below.

These two points give a better understanding of where the temperature dependence of alpha comes from; it has a reasonable explanation in terms of the distribution of activation energies.

Furthermore, the origin of the low frequency noise is the relatively large value of the activation energies. This leads to conformational transitions being extremely rare. In particular, I find it amazing that the noise at the Hz scale is detecting the fact that in the macroscopic crystal about every one second a single molecule (yes, just one undergoes a conformational change)!

Note that the activation energy distribution D(E) is peaked around 230 meV. This is the same energy that is deduced from studies of the activation energy for the glassy behaviour seen in NMR, specific heat, and thermal expansion. Moreover it is also the energy barrier calculated from quantum chemistry for the transition between the two conformations of the ethylene end groups (staggered vs. eclipsed) that I discussed in a recent post.

The reference given above also gives an explanation using ab initio calculations as to why the presence of the glass transition depends on the chemical identity of the anion X in kappa-(BEDT-TTF)2X. It relates to the relative strength of the bonding between X and the ethylene end groups of the BEDT-TTF molecules.

One thing that is not clear is what determines the width of the distribution D(E).

There are subtleties that I have glossed over here and other interesting things but the aim of this post is to focus on the big picture and some of my basic puzzles.

I thank Jens Muller for a very helpful discussion about his work.

Tutorial on bad metals

After yesterday's colloquium a large group of IISER students (both undergraduate and graduate) expressed an interest in having a tutorial on more of the subject of emergent quantum matter.
It is today at 6pm after they are done with the days lectures. This tells you something about the quality of the students and institution!

I am going to give a tutorial about bad metals. I will probably cover half of these slides. Hopefully there will be lots of questions and side discussions on the blackboard.

Emergent quantum matter talk at IISER Pune

On monday I am giving a talk at the Indian Institute for Science Education and Research (IISER ) Pune.
Here is the current version of the slides.

A minor detail that matters in organic charge transfer salts

One helpful way to think about condensed matter is in terms of relative energy scales. This can help one decide what is important and what is not.
However, this does not always work, particularly in complex systems where new low energy scales can emerge.

For a long time there has been a "minor detail" about organic charge transfer salts based on the BEDT-TTF molecule that I have found rather annoying and puzzling.
It concerns the role of ethylene end groups on the molecule and their possible different conformations (eclipsed vs. staggered).

Why should the conformations matter?

I would think not. The overlap of the relevant electronic molecular orbitals which are largely centred on sulphur atoms are negligible as seen below in the HOMO (Highest Occupied Molecular Orbital) for a BEDT-TTF dimer.

The figures are taken from this paper by Edan Scriven and Ben Powell.

However, things are more subtle than I would have thought.

Here are some of the significant effects that result from these two different conformations. They have different energies and by thermal annealing in a crystal you can convert between them.
As a result disorder in a crystal can be controlled by varying the cooling rate.
In some materials there is even a glass transition around 80 Kelvin.

Examples of the dramatic effects of the disorder can be seen.

Resistance vs. temperature curve (see for example the figure below taken from here).

Suppression of the superconducting transition temperature.
This can be seen in the curves above.

Electrical noise experiments

Another dramatic effect of the ethylene groups that is much larger than most people expect is
Isotopic substitution of the hydrogen with deuterium in the ethylene groups can drive the Mott metal-insulator transition. 
This somehow arises from a geometrical isotope effect associated with hydrogen bonds between the ethylene groups and the anion.

It turns out that changing the conformation of the end group can have a significant effect on the parameters in the Hubbard model, that is the simplest possible effective Hamiltonian for these materials.
This is shown in this recent paper which estimates these parameters using DFT-based electronic structure calculations and Wannier orbitals to map onto a tight-binding model.

Influence of molecular conformations on the electronic structure of organic charge transfer salts Daniel Guterding, Roser Valentí, and Harald O. Jeschke .

In particular in going from Eclipsed (E) to Staggered (S) or visa versa is enough to cross the Mott insulator-metal phase boundary.
This provides a framework to understand the experimental puzzles discussed above.

One minor quibble. 
The authors estimate the Hubbard paper U (Coulomb interaction) for two holes on a BEDT-TTF dimer with a formula which is only valid in a particular limit.
The general formula for the energy of  two electrons on a two site Hubbard model is

where Um is the Hubbard interaction on a single dimer, V is the inter site Coulomb repulsion and t is the intersite hopping. The authors are assuming that Um - Vm is much larger than 4t which Scriven and Powell argue is not the case.
This will lead to quantitative changes but not change the main point that the conformational changes can produce a significant change in the Hubbard model parameters; particularly a large enough change to cross the Mott insulator-metal phase boundary.

Later I will write about the noise measurements (which I puzzled about before) which turn out to be a very sensitive probe of these two molecular conformations and their interconversion.

I thank Jens Muller for very helpful discussions about this work.

SciPost is a great initiative towards restoring science to journals

"SciPost is a a complete scientific publication portal managed by active professional scientists."

It is worth checking out.
SciPost addresses many concerns I have about the current sorry state of science and publishing. These include that journals becoming redundant and counter productive and so we need alternative publication models, particularly not involving for-profit companies.

One thing I particularly like is the transparency. All the referee reports are public and referees have the option of being anonymous or not. Furthermore, anyone can write a report. Authors responses to the reports are also public.
I think this public accountability may raise standards significantly.

I hope you (and I) will consider supporting it by
-  submitting articles
- writing referee reports (either on request or volunteering)
- writing commentaries
- being willing to serve on the Editorial College

Jean-Sebastien Caux is to be commended for all the work he has put into this.
I thank Matt Davis for bringing this significant initiative to my attention.

A basic quantum concept: energy level repulsion (avoided crossings)

When I learnt and later taught basic quantum mechanics I don't think the notion of energy level repulsion (or equivalently avoided crossings) was emphasised (or even discussed?).

Much later I encountered the idea in advanced topics in theoretical physics such as random matrix theory and in theoretical chemistry  (non-adiabatic transitions and conical intersections).

Yet level repulsion is a very simple phenomena that can be illustrated with just a two by two matrix describing two coupled quantum states, as nicely discussed on the Wikipedia page.

Last semester when I was teaching Solid State Physics I realised just how central and basic the phenomena is and that the students did not appreciate this.

Level repulsion is the origin of several key phenomena in chemistry and physics.

In solid state physics, it is the origin of the appearance of band gaps at the zone boundary and thus the all important distinction between metals and insulators.

Previously, I posted how Chemistry is quantum science because chemical bonding (the lowering of energy due to interacting atoms) arises due to the superposition principle. This could also be viewed as level repulsion.

Another key idea in chemistry is that of transition states and activation energies for chemical reactions. When one uses a diabatic state picture, particularly as emphasised by Shaik and Warshel, the transition state emerges naturally in terms of level repulsion.

The figure is taken from here.

Can you think of any other nice examples?

Relating the Hall coefficient to thermodynamic quantities

Previously, I have posted about how in certain contexts one can relate non-equilibrium transport quantities to equilibrium thermodynamic quantities. This is particularly nice because for theorists it is usually a lot easier to calculate the latter than the former.
But, it should be stressed that all of these results are an approximation or only hold in certain limits.

Here are some examples.

The thermoelectric power can be related to the temperature derivative of the chemical potential through the Kelvin formula (illuminated by Peterson and Shastry).

A paper argues that the Weidemann-Franz ratio in a non-Fermi liquid can be related to the ratio of two different susceptibilities.

Work of Shastry showing that the high frequency limit of the Hall coefficient, Lorenz ratio and thermopower can be related to equilibrium correlation functions.

It has been suggested that the transverse thermoelectric conductivity (Nernst signal) due to superconducting fluctuations is closely related to the magnetisation.

Haerter, Pederson, and Shastry conjectured that for a doped Mott insulator on a triangular lattice that the Hall coefficient can be related to the temperature dependence of diamagnetic susceptibility.

Here I want to discuss some interesting results for the Hall coefficient R_H.
First,  remember that in a simple Fermi liquid (or a classical Drude model) with only one species of charge carrier of charge q and density n that

R_H=1/q n

Clearly this is a case where a rather complex transport quantity (which is actually a correlation function involving three currents) reduces to a simple thermodynamic quantity.

But, what about in strongly correlated systems?

There is a rarely cited paper from 1993

Sign of equilibrium Hall conductivity in strongly correlated systems 
 A. G. Rojo, Gabriel Kotliar, and G. S. Canright

It gives an argument of just a few lines that relates the Hall coefficient to the orbital magnetic susceptibility (Landau diamagnetism)

BTW. I think there is a typo in the very last equation. It should also contain a factor of the charge compressibility.

I am a bit puzzled by the derivation, because it appears to be completely rigorous and general. The derivation of the Kelvin formula for thermopower, also has this deceptive general validity. It turns out that "devil in the details" turns out to be that the two limits of sending frequency and wave vector to zero do not commute.

A related paper shows that with a certain limiting procedure the Hall response (at zero temperature) is related to the derivative of the Drude weight with respect to the density.

Reactive Hall Response
X. Zotos, F. Naef, M. Long, and P. Prelovšek

This is valuable because it gives a simple explanation of why in a doped Mott insulator the Hall coefficient can change sign as the doping changes.

Mental illness IS irrational

That is the point!

Mentally healthy people are rational and reasonable (within the bounds of human fallibility!).
They are not driven or paralysed by amplified anxieties, phobias, mood swings, suicidal thoughts, depression, "black clouds", ...

Yet, often people struggle to understand and/or be empathetic with someone suffering from mental illness because they are not thinking and/or acting in a "rational" manner.

This dismissal or diminishing of what is going on can even be done by a sufferer themself, "I know I am having all these crazy thoughts and feelings, but I know they are crazy so it does not matter.... I don't need to get help."
or...
The person is so far gone that they actually think that their irrational thoughts are rational. It is everyone else who is crazy...

Here two good recent articles about mental health by physicists.

The plight of the postdocs: Academia and mental health

There’s an awful cost to getting a PhD that no one talks about

Talk advice, especially for the inexperienced

There is nothing new in this post. But the issue  keeps coming up.
I have written many posts about this before.
My last one was Advice to undergrads giving research talks.
Perhaps the following basic point gets lost in all the suggestions.

Keep it simple.
In almost every situation, most of your audience knows very little about your specific research topic. In some cases, they know virtually nothing about your actual research field.

Thus, you need to cut out almost all the technical details and give plenty of background and motivation.

But again, you need to realistic. Don't kid yourself that in 5 minutes you are going to teach them Density Functional Theory or Two-dimensional NMR spectroscopy.

Have modest goals.
Teach the audience some interesting science.
Convince them that your topic/field is interesting and important.
Show them you have achieved something concrete and interesting.
Don't bore people.

But perhaps, these are ambitious goals because most talks I hear don't achieve them!

Science is broken III

I find it surprising that is almost three years since I my last post on this topic. I guess I try not to be too negative too often...

Vox has a very long article, The 7 biggest problems facing science, according to 270 scientists. It is worth reading and I largely agree with it. Possible concrete solutions are discussed. Many of the issues I have discussed on this blog and so it is both encouraging and discouraging to see others voice the same concerns.

I think a significant positive change would occur if the luxury journals (especially the pernicious Nature Publishing Group) were banned/boycotted and put out of business.

Funding agencies could simply say you can't use our money to do research that you publish in them. And in future applications you can't list any such publications from before 2017 on your CV.

I know it is not going to happen. But we do need to at least acknowledge that if it did that a lot of problems would be diminished.

Emergence and fire on the prairie

My son brought to my attention an Econtalk podcast  that has an interesting discussion of emergence. Here is some of the transcript.

I've become so fascinated by the prairie as a metaphor for emergent order. And some things have to emerge and grow in an organic way. And I've referenced this book--it's by Shona Brown and Kathleen Eisenhardt; it's called Competing on the Edge. It's actually a management book. It's not about emergent order per se, and it's not about wildlife or prairies literally. 

[Here is a long but beautiful quote from the book].

Imagine yourself at O'Hare at a far different time, not in 1998 but in 1898, or even in 1798, before the patchwork quilt of roads, fences, and farms had changed the Midwestern landscape forever. Around you would be an abundance of plants, long grasses of various colors, a pallette of flowers, some trees. You'd see the original "amber waves of grain." If waited a little while, you'd also see a variety of animals going about their daily ritual. You would be enjoying a living, breathing prairie that stretched a thousand miles to the Rocky Mountains. It's an ecological system that today is virtually extinct. 

Suppose you were given the task of recreating that prairie as it was 200 years ago. Assume you have no budget constraints. Also assume you cannot buy the prairie, but rather that you have to create your own. As you think about how you'd approach the problem, jot down the key steps to your solution. 

If you are like most people, at least our friends, you probably came up with a list that is something like this: 

Step 1. Buy a plot of land where prairie likely thrived in the past. For example, on the outskirts of Chicago--O'Hare. 

Step 2. Check the libraries. Look for old photos of the prairie. Obtain the most complete list available of all the plant and animal members of a prairie ecosystem. 

Step 3. Collect samples of all the relevant species, e.g., seeds of plants, male and female pairs of animals. 

Step 4. Clear the plot of land and plant your seeds, along with a few trees. 

Step 5. Release the animals into the plot of land. 

Step 6. Watch and wait. 

Perhaps you added a few steps with more intervention, like fertilization or watering. But overall, you likely suggested some kind of approach that we will loosely call "Assemble." That is, the steps you listed were to clear out your workspace, get the component plants and animals, lay out the blueprint, follow the directions, and start assembling. You'd then piece together the various components of the prairie and hope that somehow a prairie emerges. The approach is quite reasonable. It seems intuitively correct. If you were assembling a car, house, or a toaster, it would probably work. All you'd have to do is to assemble the components of the desired system on a reasonably attractive plot of land, and eventually a prairie would emerge. It makes sense, right? 

Wrong. Assembly doesn't work. At least, not for a prairie. A prairie is something that grows. It has to start small. It has pieces that interact and build on each other. Once it's up and running, the prairie works as a complex system. It is dependent on the intricate interaction of all the components of the system. A prairie cannot be brought to life with one abracadabra, one wave of the magic wand. 

Ecologists have in fact experimented with trying to grow prairies. Early experimenters took the assemble approach. But they ran into complications. Urban weeds are one such complication. Relative to most prairie species, these noxious weeds are aggressive and fast-growing. Given a chance, these tough weeds will muscle out the more timid prairie species and prohibit them from thriving. Knowing this, early ecologists began their work by clearing their field of weeds and then planting prairie grass seeds. Then the prairie flourished, right? Wrong. The prairie never emerged from these cleared plots. What happened? The problem with this logic is that the first plants to sprout and grow in a freshly cleared field, the most aggressive, fast-growing weeds. So even though the prairie seeds are planted first, the urban weeds that go over the cleared so and the prairie never took hold. 

... In later experiments, ecologists extended the grow-not-assemble approach. In particular, one successful experiment in Illinois, a college grew a prairie savannah--a prairie with trees--that began by planting a sample of choice prairie savannah weeds, seeds, and wooded weed-filled fields on the outskirts of Chicago. 

Butterflies came back, with the savannah.... As the experiments continued, ecologists learned more lessons about recreating prairies. They learned that order matters. Reversing the introduction of one species and another, e.g., reversing the order of entry of two predators, alters the ecosystem that emerges. Adding or subtracting of species also alters the system, affecting its final states and its resilience to change. Perhaps the most subtle lesson to learn was that not all of the essential ingredients to growing a prairie savannah are visible at the end. Ecologists learned this lesson when they were stymied in their efforts. They were close to creating a prairie, but something was not quite right. Half-breed prairies were being created. A mixture of prairie and non-prairie species. The experiments didn't seem to be capable of evolving into that final step of a pure prairie. Ecologists searched for key components of a prairie that might be missing. But this was the whole problem: they were searching for something that they thought should be there but wasn't. Instead, ecologists should have been looking for something that had been there, but did not stay. In other words, a fleeting member of the prairie system, a missing link. 

Was there a missing link that was not present in the mature prairie but was essential to growing it? Yes--that missing link was fire. Initially, ecologists failed to introduce fire into their experimental prairie systems because its presence is not explicit in the final product. It was not an immediately obvious candidate to be deliberately added. Moreover, although ecologists were trying to mimic nature and minimally manage the fields of emerging prairie, the incidence of wildfires was far lower than it would have been in a true, natural setting. Without fire, ecologists could not create the elusive, pure-bred prairie. Fire triggers certain prairie seeds to sprout and eliminates many fire-intolerant urban plants. Without fire, there is no prairie.

What do we learn about emergence?
The sum is greater than the parts.
The prairie is a lot more than the individual components that you see today.

Are there some lessons here for self assembly of material systems and biomimetics?

The mysterious origin of resistivity in Fermi liquids

It is hard to believe that we really don't understand the basic issues that I am going to discuss.

Resitivity occurs in a metal because scattering causes decay of charge currents. This means that the total momentum of the electrons in the presence of an electric field decays.
However, in a Fermi liquid metal with strong electron-electron interactions the main scattering of electrons is due to electron-electron scattering. But, in such collisions the total momentum of the two electrons is the same before and after the collision.
One can calculate the life time of the quasi-particles and it is inversely proportional to the temperature squared. The quasi-particle scattering rate ~ T^2.
Suppose one makes the relaxation time approximation in the Boltzmann equation or equivalently, neglects vertex corrections in the corresponding current-current correlation function associated with the Kubo formula for the conductivity. Then the resistivity is proportional to the quasi-particle scattering rate and one has resistivity ~ T^2. We say the transport lifetime is the same as the quasi-particle lifetime.
However, these are approximations, and strictly speaking there is no decay of the total electron momentum (or current) by electron-electron scattering and so the resistivity should be zero!
One way to save the situation is when there is Umklapp scattering. However, this requires a special relation between the shape of the Fermi surface and the Brillouin zone, as illustrated below.

These issues and puzzles are highlighted in a beautiful paper

Scalable T^2 resistivity in a small single-component Fermi surface 
Xiao Lin, Benoît Fauqué, Kamran Behnia

By chemical doping they tune the charge density and Fermi energy by several orders of magnitude, with the size of the Fermi surface increasing from some very small fraction of the Brilloiun zone.
In all cases the resistivity equals A T^2, characteristic of electron-electron scattering.
The figure below shows how the proportionality factor A scales with the density.
They also find A scales with the inverse of the effective mass squared as one expects from the Kadowaki-Woods ratio.

Yet for small densities (and Fermi surfaces) it is just not clear how one can have electron-electron scattering since Umklapp scattering is not relevant.

This major puzzle awaits an explanation.

I thank David Cavanagh, Jure Kokalj, Jernez Mravlje, and Peter Prevlosek for stimulating discussions about this topic.

Note added. The theoretical issues are nicely reviewed in
Resistivity of non-Galilean-invariant Fermi- and non-Fermi liquids 
 H. K. Pal, V. I. Yudson, D. L. Maslov

Viscosity talk in Belgrade

Today I am giving a seminar at the Institute of Physics in Belgrade.
My host is Darko Tanaskovic who has done nice work using DMFT to study bad metals and quantum criticality.

Here is the current version of the slides.
The main results are in this paper.