# Tag Archives: Toy Model

## Why Was BCS So Important?

BCS theory, which provides a microscopic framework to understand superconductivity, made us realize that a phenomenon similar to Bose-Einstein condensation was possible for fermions. This is far from a trivial statement, though we sometimes think of it as so in present times.

A cartoon-y scheme to understand it is in the following way. We know that if you put a few fermions together, you can get a boson, such as 4-helium. It was also known well before BCS theory, that one gets a phenomenon reminiscent of Bose-Einstein condensation, known as superfluidity, in 4-helium below 2.17K. The view of 4-helium as a Bose-Einstein condensate (BEC) was advocated strongly by Fritz London, who was perhaps the first to think of it in such a way.

Now let us think of another type of boson, a diatomic molecule, as seen in gas form below:

Even if the individual atoms were fermions, one would then predict that if this bosonic diatomic gas molecule could remain in the gas phase all the way down to low temperature, that at some point, this diatomic gas would condense into a BEC. This idea is correct and this is indeed what is observed.

However, the idea of a BEC becomes a little more cloudy when one considers a less dilute diatomic gas where the atoms are not so strongly bound together. In that case, the cartoon starts to look something like this:
Here the “diatomic molecules” are overlapping, and it is not easy to see which atoms are paired together to form the diatomic molecule, if one can even ascribe this trait to them. In this case, it is no longer simple to see whether or not BEC will occur and indeed if there is a limit in distance between the molecules that will necessarily give rise to BEC.

This is the question that BCS theory so profoundly addresses. It says that the “diatomic molecules” or Cooper pairs can span a great distance. In superconducting aluminum, this distance is ~16,000 Angstroms, which means the Cooper pairs are wildly overlapping. In fact, in this limit, the Cooper pair is no longer strictly even a boson, in the sense that Cooper pair creation and annihilation operators do not obey Bose-Einstein commutation relations.

However, the Cooper pair can still qualitatively thought of as a pseudo-boson that undergoes pseud0-BEC, and this picture is indeed  very useful. It enabled the prediction of pseud0-BEC in neutron stars, liquid 3-helium and ultra-cold fermionic gases — predictions which now have firm experimental backing.

An interesting note is that one can study this BCS-to-BEC crossover in ultracold Fermi gases and go from the overlapping to non-overlapping limit by tuning the interaction between atoms and I’ll try to write a post about this in the near future.

So while BCS theory has many attributes that make it important, to my mind, the most profound thing is that it presents a mechanism by which weakly interacting fermionic pairs can condense into a pseudo-BEC. This is not at all obvious, but indeed what happens in nature.

Update: In light of the description above, it seems surprising that the temperature at which Cooper pairs form is the same temperature at which they seem to condense into a pseudo-BEC. Why this is the case is not obvious and I think is an open question, especially with regards to the cuprates and in particular the pseudogap.

## Excitons in Cuprous Oxide

There is an excellent pair of papers here and here demonstrating the existence of a Rydberg series of excitons in Cu$_2$O, a material that has long been the poster child for observing excitons. The data in both papers are pretty stunning — in the first one, they see a Rydberg series up to n=25! In the second, they see higher angular momentum ($f$-like and $h$-like) excitons apart from the usual $p$-like excitons.

Excitons can occur in semiconductors and insulators and are the result of the Coulomb interaction between electrons and holes. For those familiar with superconductivity, an exciton, in many ways, bears a qualitative resemblance to Cooper pairs, except here the quasi-particles are of opposite charge. Hence, the binding between the electron and hole arises from the usual attractive Coulomb interaction. However, it helps to think of the exciton (similar to Cooper pairs) as possessing a center of mass wavefunction and a relative wavefunction.

The relative wavefunction is then qualitatively and quantitatively similar to the quantum mechanical solution of the hydrogen atom. The main differences are that (1) the reduced mass of the electron-hole pair is usually much less than the reduced mass of the electron-proton system and also that (2) there exists a background dielectric constant, $\epsilon$, that further reduces the energy of the Rydberg series in an insulator. In the simple hydrogen-like picture of the exciton, the energy levels (with center of mass momentum taken to be zero) then have the following energies:

$E_n =- \frac{\mu}{m\epsilon^2}\frac{R}{n^2}$

where $\mu$ is the reduced mass of the electron-hole system, $m$ is the proton mass, and $R$ is the hydrogenic Rydberg.

It turns out that this perfectly spherically symmetric model does not encapsulate completely the exciton in a solid, however, because the potential in a crystal is not perfectly rotation-symmetric. Because of the lattice, the “angular momentum” of the exciton cannot be labeled using an $s, p, d, f$, etc. kind of scheme and we must resort to some group theory.

For the case of Cu$_2$O discussed in the papers above, the symmetry of the crystal is cubic — therefore the symmetry is still relatively high. However, because continuous rotational symmetry is broken, the light used to detect the excitons in this experiment are no longer forbidden from exciting $f$-like and $h$-like excitons and it is remarkable that they are observable!

Even though the observations in these papers can be accounted for in a quite simple theoretical framework, the experimental results are nonetheless quite elegant and remarkable.

As an interesting aside, the reason that this experiment is possible is because of the extreme purity of the Cu$_2$O crystals used for these studies. It turns out the cleanest of these samples are naturally-occurring, rather than man-made.

## A Critical Ingredient for Cooper Pairing in Superconductors within the BCS Picture

I’m sure that readers that are experts in superconductivity are aware of this fact already, but there is a point that I feel is not stressed enough by textbooks on superconductivity. This is the issue of reduced dimensionality in BCS theory. In a previous post, I’ve shown the usefulness of the thinking about the Cooper problem instead of the full-blown BCS solution, so I’ll take this approach here as well. In the Cooper problem, one assumes a 3-dimensional spherical Fermi surface like so:

3D Fermi Surface

What subtly happens when one solves the Cooper problem, however, is the reduction from three dimensions to two dimensions. Because only the electrons near the Fermi surface condense, one is really working in a shell around the Fermi surface like so, where the black portion does not participate in the formation of Cooper pairs:

Effective 2D Topology Associated with the Cooper Problem

Therefore, when solving the Cooper problem, one goes from working in a 3D solid sphere (the entire Fermi sea), to working on the surface of the sphere, effectively a 2D manifold. Because one is now confined to just the surface, it enables one of the most crucial steps in the Cooper problem: assuming that the density of states ($N(E)$) at the Fermi energy is a constant so that one can pull it out of the integral (see, for example, equation 9 in this set of lecture notes by P. Hirschfeld).

The more important role of dimensionality, though, is in the bound state solution. If one solves the Schrodinger equation for the delta-function potential (i.e. $V(x,y)= -\alpha\delta(x)\delta(y)$) in 2D one sees a quite stunning (but expected) resemblance to the Cooper problem. It tuns out that the solution to obtain a bound state takes the following form:

$E \sim \exp{(-\textrm{const.}/\alpha)}$.

Note that this is exactly the same function that appears in the solution to the Cooper problem, and this is of course not a coincidence. This function is not expandable in terms of a Taylor series, as is so often stressed when solving the Cooper problem and is therefore not amenable to perturbation methods. Note, also, that there is a bound state solution to this problem whenever $\alpha$ is finite, again similar to the case of the Cooper problem. That there exists a bound state solution for any $\alpha >0$ no matter how small, is only true in dimensions two or less. This is why reduced dimensionality is so critical to the Cooper problem.

Furthermore, it is well-known to solid-state physicists that for a Fermi gas/liquid, in 3D $N(E) \sim \sqrt{E}$, in 2D $N(E) \sim$const., while in 1D $N(E) \sim 1/\sqrt{E}$. Hence, if one is confined to two-dimensions in the Cooper problem, one is able to treat the density of states as a constant, and pull this term out of the integral (see equation 9 here again) even if the states are not confined to the Fermi energy.

This of course raises the question of what happens in an actual 2D or quasi-2D solid. Naively, it seems like in 2D, a solid should be more susceptible to the formation of Cooper pairs including all the electrons in the Fermi sea, as opposed to the ones constrained to be close to the Fermi surface.

If any readers have any more insight to share with respect to the role of dimensionality in superconductivity, please feel free to comment below.

## Another Lesson from the Coupled Oscillator: Fano Lineshape

Every once in a while, I read a paper that exhibits a rather odd spectral signature. This odd spectral signature is asymmetric and looks something like this (taken from Wikipedia):

Fano Resonance

This is called a Fano lineshape and occurs when an excitation interferes with a background process. “Background process” is pretty vague, but I hope that the notion becomes clearer below. Here are a couple papers that have observed Fano lineshapes in different contexts, including a review paper:

The reason I chose the first two papers was because a couple peaks in their respective spectra exhibit a Fano lineshape as a function temperature. In the Raman paper, after the charge density wave gap in the single-particle spectrum opens up, the lineshapes of the phonons are no longer Fano-like, but become Lorentzian. This indicates that the Fano lineshape likely resulted from strong electron-phonon coupling.

Anyway, there is a pedagogical paper on the Fano resonance that is worth a read here (pdf!), and is where the example below comes from. This paper in fact illustrates the simplest and most enlightening case of the Fano resonance I’ve come across. The basic idea is that one has a pair of coupled oscillators, and someone is driving only one of the oscillators (ball 1 below). This is pictured below for clarity:

Only the ball 1 is driven

Now, if one looks at the amplitude of the first oscillator, one gets the following plot:

The first oscillator was chosen to have a natural resonance frequency of 1, while the second oscillator was chosen to have a resonance frequency of 1.5. You can see that this is not exactly where the peaks show up, however. This is because of the level repulsion due to the coupling between the two oscillators.

Now the \$64k question: why does the asymmetric lineshape appear? Well, there are a few ingredients. The first ingredient is that the first oscillator is being driven by an extrenal source and the second oscillator is not. Secondly, there is a strong coupling between the first oscillator and second one. As one decreases the coupling between the two, the amplitude of the second (asymmetric) peak starts to shrink. Thirdly, the natural frequencies of the oscillators need to be relatively close to one another otherwise again the second peak decreases in amplitude.

What is remarkable is that this simple coupled oscillator model is able to exhibit the Fano lineshape while serving as a very instructive toy model. Because of this simplicity, the Fano lineshape is extremely general and is used to explain data in a wide variety of contexts across all areas of physics. A spectroscopist should always keep a look out for an asymmetric lineshape, as it is usually the result of a non-trivial coupling or interaction with another degree of freedom.

## Thoughts on Consistency and Physical Approximations

Ever since the series of posts about the Theory of Everything (ToE) and Upward Heritability (see here, here, here and here), I have felt like perhaps my original perspective lacked some clarity of thought. I recently re-read the chapter entitled Physics on a Human Scale in A.J. Leggett’s The Problems of Physics. In it, he takes a bird’s eye view on the framework under which most of us in condensed matter physics operate, and in doing so, untied several of the persisting mental knots I had after that series of blog posts.

The basic message is this: in condensed matter physics, we create models that are not logically dependent on the underlying ToE. This means that one does not deduce the models in the mathematical sense, but the models must be consistent with the ToE. For example, in the study of magnetism, the models must be consistent with the microscopically-derived Bohr-van Leeuwen theorem.

When one goes from the ToE to an actual “physical” model, one is selecting relevant features, rather than making rigorous approximations (so-called physical approximations). This requires a certain amount of guesswork based on experience/inspiration. For example, in writing down the BCS Hamiltonian, one neglects all interaction terms apart from the “pairing interaction”.

Leggett then makes an intuitive analogy, which provides further clarity. If one is building a transportation map of say, Bangkok, Thailand, one could do this in two ways: (i) One could take a series of images from a plane/helicopter and then resize the images to fit on a map or (ii) one could draw schematically the relevant transportation features on a piece of paper that would have to be consistent with the city’s topography. Generally, scheme (ii) will give us a much better representation of the transportation routes in Bangkok than the complicated images in scheme (i). This is the process of selecting relevant features consistent with the underlying system.  This is usually the process undertaken in condensed matter physics. Note this process is not one of approximation, but one of selection while retaining consistency.

With respect to the previous posts on this topic, I stand by the following: (1) I do still think that Laughlin and Pines’ claim that certain effects (such as the Josephson quantum) cannot be derived from the ToE to be quite speculative. It is difficult to prove either (mine or L&P’s) viewpoint, but I  take the more conservative (and what I would think is the simpler) option and suggest that in principle one could obtain such effects from the ToE. (2) Upward heritability, while also speculative, is a hunch that claims that concepts in particle physics and condensed matter physics (such as broken symmetry) may result from a yet undiscovered connection between the two realms of physics. I still consider this a plausible idea, though it could be just a coincidence.

Previously, I was under the assumption that the views expressed in the L&P article and the Wilzcek article were somehow mutually exclusive. However, upon further reflection, I no longer think that this is so and have realized that in fact they are quite compatible with each other. This is probably where my main misunderstanding laid in the previous posts, and I apologize for any confusion this may have caused.

## Excitonic Insulator

The state of matter dubbed the excitonic insulator was first qualitatively discussed by Mott, Keldysh and Kopaev, and others and then expanded upon more systematically by Jerome, Rice and Kohn.

The excitonic insulating state can be considered from two normal states (pictured below). Either the system must be a small-gap semiconductor or a small indirect overlap semimetal. In fact, Mott had first considered the semimetallic limit, while Kohn and others had considered the semiconducting limit.

Intuitively, one can consider the following two heuristic arguments from the different limits, as presented in the article by Rossnagel, which was cited in the previous post:

1. Semiconducting limit: If one can somehow reduce the band gap energy, $E_G$, then at some point, the binding energy to form an exciton, $E_B$, will exceed $E_G$, and the system will unstable to the spontaneous formation excitons.
2. Semimetallic limit: In this case, one considers screening effects. If one decreases the band overlap, a characteristic energy, $E_1$, will be reached such that particle-hole pairs will be insufficiently screened, leading to a localization of the charge carriers.

Therefore, in the regime of $E_1$$E_G$ <$E_B$, the excitonic insulator state is expected. Properties of the excitonic insulator state are presented pedagogically in a Les Houches lecture by Kohn in this book, which is very difficult to find!

In a solid state context, it has been difficult to establish whether the excitonic insulator state has been realized because a lattice distortion is expected to accompany the transition to the excitonic insulator ground state. Therefore, it is difficult to isolate the driving mechanism behind the transition (this difficulty will be familiar to those who study high T-c superconductivity!).

There are a few materials suspected to possess excitonic insulator ground states in a solid state setting: 1T-TiSe$_2$, Ta$_2$NiSe$_5$ and TmSe$_{0.45}$Te$_{0.55}$. In my personal opinion, the case for 1T-TiSe$_2$ is probably the strongest purely because there have been far more experiments on this material than the other candidate materials.

Though this state of matter was considered almost 50 years ago, it still remains relevant today. As Landau once said,

Unfortunately, everything that is new is not interesting, and everything which is interesting, is not new.

## Plasmons, the Coulomb Interaction and a Gap

In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance,  the dispersion relation for surface gravity waves on water (pdf!) is:

$\omega^2(k)=gk \tanh kh$.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is $\textbf{E} = \frac{\sigma\hat{x}}{\epsilon_0}$. This expression is surprisingly independent of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like $\textbf{E}\sim\frac{1}{\textbf{r}}$. Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at $\textbf{q}=0$. Similarly, for the 1D case, the electric field from the points bounding the solid falls of as $\frac{1}{\textbf{r}^2}$, and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at $\textbf{q}=0$ for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.