# Tag Archives: Bose-Einstein Condensation

## Reflecting on General Ideas

In condensed matter physics, it is easy to get lost in the details of one’s day-to-day work. It is important to sometimes take the time to reflect upon what you’ve done and learned and think about what it all means. In this spirit, below is a list of some of the most important ideas related to condensed matter physics that I picked up during my time as an undergraduate and graduate student. This is of course personal, and I hope that in time I will add to the list.

1. Relationship between measurements and correlation functions
2. Relationship between equilibrium fluctuations and non-equilibrium dissipative channels (i.e. the fluctuation-dissipation theorem)
3. Principle of entropy maximization/free-energy minimization for matter in equilibrium
4. Concept of the quasi-particle and screening
5. Concept of Berry phase and the corresponding topological and geometrical consequences
6. Broken symmetry, the Landau paradigm of phase classification and the idea of an order parameter
7. Sum rules and the corresponding constraints placed on both microscopic theories and experimental spectra
8. Bose-Einstein and Cooper Pair condensation and their spectacular properties
9. Logical independence of physical theories on the theory of everything
10. Effects of long-range vs. short-range interactions on macroscopic properties of solids
11. Role of dimensionality in observing qualitatively different physical properties and phases of matter

The first two items on the list are well-explained in Forster’s Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions without the use of Green’s functions and other advanced theoretical techniques. Although not yet a condensed matter phenomenon, Bell’s theorem and non-locality rank among the most startling consequences of quantum mechanics that I learned in graduate school. I suspect that its influence will be observed in a condensed matter setting in due time.

## Balibar and his Beef with Science Magazine’s Depiction of a BEC

I’m coming to the end of reading Sebastian Balibar’s physics book (intended for a general audience) entitled The Atom and the Apple. Thematically, the book works by asking a basic question at the beginning of each chapter and seeing the wondrous science that has to be understood to answer such a basic question. The author pulls on a dangling thread and watches entire garments unravel. The book is interspersed with personal anecdotes, which gives the physics some semblance of humanity.

Just to whet your appetite a little, let me recount one of the amusing stories Balibar relays. In 1995, following the discovery of Bose-Einstein Condensation by the Colorado and MIT groups, Science had the following image on the cover (sorry, I couldn’t find a large picture in color anywhere, it’s supposed to be primarily blue if that helps). It shows marching soldiers, a supposed metaphor for BEC.

Balibar takes issue with this representation and goes onto say about the cover:

No, what bothered me was actually that march—those orderly
troops. True, I had bad memories of my own experiences with military marches. Before May 1968, the hierarchy at the École Polytechnique had little patience for the antics of its rebel students, and my deviance had cost me fifteen days in prison and gotten me barred from marching with my fellow students on the Champs Élysées. But regardless of these youthful memories, I saw in that march of atoms a basic error of interpretation as to what the recently discovered “Bose-Einstein” condensation was. The order of the actual condensate seemed radically different to me from the regimentation depicted on the magazine cover.
The artist at Science hadn’t just dreamt up this march all by him or herself, though: the military analogy had been hanging around in the public scientific discourse for a long time. Nevertheless, in becoming famous, this image threatened to distort the understanding of the discovery. I intended to denounce it firmly.

What Balibar had in mind about BEC is much more accurately depicted in this great little video.

The book also has some anecdotes about the history and controversy surrounding the Nobel Prizes awarded for superfluidity in liquid helium-4 among many other interesting historical detours. The discussion on radioactivity is also noteworthy. This charming little book is written with an approach that I feel more popular physics books should take, or Balibar could write a couple more himself.

## 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.

## Macroscopic Wavefunctions, Off-Diagonal Long Range Order and U(1) Symmetry Breaking

Steven Weinberg wrote a piece a while ago entitled Superconductivity for Particular Theorists (pdf!). Although I have to admit that I have not followed the entire mathematical treatment in this document, I much appreciate the conceptual approach he takes in asking the following question:

How can one possibly use such approximations (BCS theory and Ginzburg-Landau theory) to derive predictions about superconducting phenomena that are essentially of unlimited accuracy?

He answers the question by stating that the general features of superconductivity can be explained using the fact that there is a spontaneous breakdown of electromagnetic gauge invariance. The general features he demonstrates are due to broken gauge invariance are the following:

1. The Meissner Effect
2. Flux Quantization
3. Infinite Conductivity
4. The AC Josephson Effect
5. Vortex Lines

Although not related to this post per se, he also makes the following (somewhat controversial) comment that I have to admit I am quoting a little out of context:

“…superconductivity is not macroscopic quantum mechanics; it is the classical field theory of  a Nambu-Goldstone field”

Now, while it may be true that one can derive the phenomena in the list above using the formalism outlined by Weinberg, I do think that there are other ways to obtain similar results that may be just as general. One way to do this is to assume the existence of a macroscopic wave function. This method is outlined in this (illuminatingly simple) set of lecture notes by M. Beasley (pdf!).

Another general formalism is outlined by C.N. Yang in this RMP, where he defines the concept of off-diagonal long range order for a tw0-particle density matrix. ODLRO can be defined for a single-particle density matrix in the following way:

$\lim_{|r-r'| \to \infty} \rho(r,r') \neq 0$

This can be easily extended to the case of a two-particle density matrix appropriate for Cooper pairing (see Yang).

Lastly, there is a formalism similar to that of Yang’s as outlined by Leggett in his book Quantum Liquids, which was first developed by Penrose and Onsager. They conclude that many properties of Bose-Einstein Condensation can be obtained from again examining the diagonalized density matrix:

$\rho(\textbf{r},\textbf{r}';t) = \sum_i n_i(t)\chi_i^*(\textbf{r},t)\chi_i(\textbf{r}',t)$

Leggett then goes onto say

“If there is exactly one eigenvalue of order N, with the rest all of order unity, then we say the system exhibits simple BEC.”

Again, this can easily be extended to the case of a two-particle density matrix when considering Cooper pairing.

The 5-point list of properties of superconductors itemized above can then be subsequently derived using any of these general frameworks:

1. Broken Electromagnetic Gauge Invariance
2. Macroscopic Wavefunction
3. Off-Diagonal Long Range Order in the Two-Particle Density Matrix
4. Macroscopically Large Eigenvalue of Two-Particle Density Matrix

These are all model-independent formulations able to describe general properties associated with superconductivity. Items 3 and 4, and to some extent 2, overlap in their concepts. However, 1 seems quite different to me. It seems to me that 2, 3 & 4 more easily relate the concepts of Bose-Einstein condensation to BCS -type condensation, and I appreciate this element of generality. However, I am not sure at this point which is a more general formulation and which is the most useful. I do have a preference, however, for items 2 and 4 because they are the easiest for me to grasp intuitively.

Please feel free to comment, as this post was intended to raise a question rather than to answer it (which I cannot do at present!). I will continue to think about this question and will hopefully make a more thoughtful post with a concrete answer in the future.