Deriving Ginzburg-Landau Theory from BCS: A Simplified Approach

[Demystifier]

Is there a simple derivation of Ginzburg-Landau theory of superconductivity, with emphasis on simple, from the BCS theory?

 

[Fred Wright]

I think that you are asking for the impossible unless you consider quantum field theory simple. The following is an historical overview of the process of developing the BCS microscopic theory from the Landau-Ginzberg free energy model by its original author, Lev Gor'kov: Gor'kov paper (LG to BCS)

 

[ZapperZ]

I think that you are asking for the impossible unless you consider quantum field theory simple. The following is an historical overview of the process of developing the BCS microscopic theory from the Landau-Ginzberg free energy model by its original author, Lev Gor'kov: Gor'kov paper (LG to BCS)

I think he wants it the other way, i.e. from BCS to derive GL theory.

Still, that's no consolation since I do not think there is also a "simple" way to do the derivation.

Zz.

 

[Demystifier]

I think that you are asking for the impossible unless you consider quantum field theory simple.

Quantum field theory, or second quantization, is simple in my view. But the way how it is used, for instance, in the book by Abrikosov, Gorkov and Dzyaloshinskii is not simple. On the other hand, the book by Mattuck is much simpler. An even simpler book on QFT is the one by Lancaster and Blundell (QFT for the Gifted Amateur), which in fact does give a rough simple idea of how to get GL from BCS and something more complete on that level would be very desirable. I hope it helps to get a picture of what do I mean by "simple".

 

[dRic2]

Try "Quantum theory of many-particle system" by by Fetter and Walecka. I'm no physicist, but I enrolled in a course on many-body physics for fun and this is the book used as a reference. I went through the first 4 chapters only, but I looked at the index of contents and I found this:

(about BCS theory)

then some pages after, this one:

This is above my current understanding, so I don't really know if that's what you are looking for. Anyway the book is very good and it cost only 20€ (Europe).

 

[Demystifier]

This is above my current understanding, so I don't really know if that's what you are looking for.

That's the original Gorkov's derivation, which is anything but simple.

 

[MathematicalPhysicist]

Hi @Demystifier I assume you find Gorkov's difficult just like I find BBGKY difficult, tried to understand it from Kardar's textbook; for the life of me this is difficult.

And the equations just keep on coming...


Edit: There's a saying that in maths and physics you don't understand the theories you just get accustomed to them. I believe this is attributed to John Von-Neuman but I say this from memory.

 

[Demystifier]

I find BBGKY difficult, tried to understand it from Kardar's textbook; for the life of me this is difficult.

The simplest explanation of BBGKY I am aware is presented in Tong's lectures:
http://www.damtp.cam.ac.uk/user/tong/kinetic.html

 

[MathematicalPhysicist]

BTW, is there a problem between Ginzburg and Landau?

Since I see some texts call it GL and others LG. Obviously there's a competition between the two who is the first to be attributed this theory...

 

[vanhees71]

My favorite for this kind of topics is

A. Schmitt, Introduction to Superfluidity, Springer 2015
https://doi.org/10.1007/978-3-319-07947-9

 

[dRic2]

A. Schmitt, Introduction to Superfluidity, Springer 2015

It appears to be available on arxiv.

I know it is a bit off topic, but it is something I wanted to ask for some time now. I'll put it in between spoilers tag.

Spoiler

So, as I said, I am no physicist and I am not familiar with QFT at all. All I "know" (well, you know what I mean... ) is classical QM. I first encounter this so-called "field" formalism reading the book I was talking about by Fetter and Walecka. The approach explained in the book to attack a many-body system can be over-simplified as follow:
1) you want to calculate the green function for a (ground) state of interacting particles
2) you switch to the interaction picture
3) you use Gell-Mann and Low's theorem along with Wick's theorem to get a perturbation expansions in which each term is an integral of greens function for the non-interacting system
4) the end.
My question is the following: every time I hear talking about field-theory the are some Lagrangians involved, but in the book I mentioned there are no Lagrangians. Also in the book by Schmitt it seems that this Lagrangian approach is present. Is there a book that explain how these two methods are related to each other ?

 

[vanhees71]

I was not aware that the book by Andreas Schmitt is available on arXiv too, which is of coarse great:

https://arxiv.org/abs/1404.1284

I think Fetter-Walecka is among the best books about non-relativistic QFT, because it emphasizes the equivalence between the "1st and 2nd quantization formulation" of many-body systems in the special case that you deal with situations of conserved particle numbers. It becomes clear rather quickly that also in this case where both formulations are equivalent the field-theory formulation is simpler, because it takes automatically care of the symmetrization (boson) and antisymmetrization (fermion) of the multi-particle wave functions, which is a complicated issue in the 1st-quantization formalism.

The 2nd quantization formalism goes beyond this when it comes to the quasiparticle approach, where the quasiparticle numbers are not conserved, and the quasiparticle approach is obviously very important in many-body physics.

Now it turns out that you get directly to the field-theoretical formulation by starting with a classical field theory, described in terms of the least-action principle, i.e., with a Lagrangian and using the canonical quantization approach to the fields. That's the same approach as Dirac's formulation of the 1st-quantization formalism using the least-action principle for classical point-particle systems.

From a more formal systematic point of view the great advantage of the least-action principle approach to physics is that it let's you formulate symmetry principles (aka Noether's theorems) in a simple way, and that is the key for a systematic understanding, where the entire formalism of QT comes from.

Another very important aspect is that the action principles opens the door to the formulation of QT and particularly also QFT in terms of the Feynman path-integral approach and other related functional approaches. Particularly for gauge theories this is a great simplification compared to the pure operator formalism.

 

[dRic2]

Thanks a lot. One final question: I have the book by Sakurai and Napolitano and there is a chapter about Feynman path integral. I skimmed through it and I noticed lots of similarities like (propagators and Green's function). If one day I will study that chapter with more care do you think is enough to get the connection ? Or should I study directly from a book about QFT starting with Dirac's method ?

 

[vanhees71]

The chapter in Sakurai and Napolitano is a very good introduction to the path-integral formalism in the "1st-quantization formalism". I would study it in any case before starting with the path-integral method for fields.

I'm not so familiar with non-relativistic many-body textbooks. One I know and like is

A. Altland, B. Simons, Condensed Matter Field Theory, Cambridge University press 2010

It contains both 2nd quantization in the operator and the path-integral formalism.