Superfluidity & BEC: Connected Phenomena

[tirrel]

Hi...

reading Landau (statystical physics part2, second chapter) I see that in a first moment he explains superfluidity simply observing the spectrum of quasi-particles... no word about BEC...


then all of a sudden in a paragraph ("wave function of the condensate")... he talks aboout condensation in a state with p not equal to zero and from its velocity he findes the velocity of the superfluid...


are the two things connected? and where is it demonstrated in the book that this "strange" condensation (strange because it's in a state with p not equal to zero) happens?

 

[tirrel]

maybe I've got an idea... could it be that Landau's theory explains the phenomenon viewing what happens through an effective "one-quasi-particle theory" (he draws even a band diagram!), whereas the BEC is son of an interacting theory developed (not in detail) on the Fock space where "actually-particles" live? (but actually, the celebrated BEC function describing the superfluid doesn't belong to the fock space, even if it comes up from that)...

?

 

[Minich]

Landau rejected BEC as the origin of helium superfluidity.
In his main paper on superfluid helium he critisized Tisza for his proposal of BEC as an explanation for superfluidity.
Landau all his life long never accepted BEC in superfluid helium (i have read all Landau papers on superfluidity in russian and in english).

Last Landau-Lifgarbages textbook (statystical physics part2, second chapter) was rewritten by L. P. PITAEVSKII after Landau had died.

STATISTICAL PHYSICS
Part 2
Theory of the Condensed State
by
E. M. LIFSHITZ and L. P. PITAEVSKII
Institute of Physical Problems, U.S.S.R. Academy of Sciences
Volume 9 of Course of Theoretical Physics
Translated from the Russian by
J. B. SYKES and M. J. KEARSLEY

Everybody now knows

Gross-Pitaevski equation and resonances in Bose-Einstein condensates

)

 

[tirrel]

Ok... thanks...

but what I can do with Landau's theory using bosonic-tipe spectrum of quasi-particles to describe excitations?
Is it still valid at least as a theory of low energy exctitations on the "ground state" described by Pitaevskii? (and so results like Landau limit velocity can be taken as "correct")

 

[Minich]

Forum rules don't give me the opportunity answer your question sincerely.

Excitation spectrum in helium Landau received from Migdal (see acknolegements from Landau to Migdal in the first Landau paper on superfluid helium).

But in the second paper Landau corrected spectrum for rotons, when he had got the results of Bogoliubov on unideal bose gas.

Pitaevski is the pupil of Landau and Pitaevski couldn't change Landau's view on BEC (that there is no BEC in superfluid helium) in LANDAU textbook. Landau is a god to his pupils )

Is Landau wrong or not: ask better moderators of this forum, can i answer you what i think about it without getting infraction. I've got a lot already.
As you know Landau had got Nobel prize for superfluid helium theory.

 

[xsemoi]

Hi,

nowadays superfluidity and BEC are most often seen as been "almost equal". I strongly recommend you to read the APPENDIX of "Reviews of Modern Physics volume 80, page 885, year 2008". Alternative, Leggett wrote a nice book about this. It is entitled "Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed Matter Systems". You might find it in a library.


Jonas

 

[DrDu]

Bose Einstein condensation is a phenomenon which a priori primarily defined for non-interacting particles. A generalization, which is meaningful also in the case of interacting particles, is Off Diagonal Long Range Order (ODLRO). See, e.g.
@article{yang1962concept,
title={Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors},
author={Yang, C.N.},
journal={Reviews of Modern Physics (US)},
volume={34},
year={1962},
publisher={Inst. for Advanced Study, Princeton, NJ}
}

 

[EmpaDoc]

We know now that both Landau and Fritz London (who proposed that BEC was the reason for 4He superfluidity) found a bit of the truth, although their views seemed contradictory at the time.
Landau was correct in that the *direct* reason for superfluidity is the spectrum of the elementary excitations: If there exists a critical velocity, then we have superfluidity.
We also know that BEC can cause such a spectrum, and it does so in 4He. But it has been found, since then, that other systems that do not have BEC, can still be superfluid. Because of the form of their elementary excitation spectrum.
So in 4He:
BEC -> spectrum with critical velocity -> superfluidity.

 

[DrDu]

We also know that BEC can cause such a spectrum, and it does so in 4He. But it has been found, since then, that other systems that do not have BEC, can still be superfluid. Because of the form of their elementary excitation spectrum.

On the other hand, not every BEC is superfluid, e.g. a BEC of non-interacting bosons.

 

[EmpaDoc]

On the other hand, not every BEC is superfluid, e.g. a BEC of non-interacting bosons.

Precisely. The logic goes a bit like: "In this case, the butler was the murderer. But not all murderers are butlers, and not all butlers are murderers."

 

[Minich]

Hi,
nowadays superfluidity and BEC are most often seen as been "almost equal".

I can't see those subjects equal. So coudn't Landau )

Lambda points for He4 and for BEC are of DIFFERENT nature!

Can you calculate lambda point curve for He4?
Even Feynman couldn't do it!

 

[DrDu]

I can't see those subjects equal. So coudn't Landau )

Lambda points for He4 and for BEC are of DIFFERENT nature!

Can you calculate lambda point curve for He4?
Even Feynman couldn't do it!

I think the lambda point is quite independent from superfluidity. It is the sign of a phase transition which is characterized by the breaking of U1 symmetry of the extended Galilean group. The characteristic sign of this symmetry breaking is the appearance of a Goldstone mode, namely second sound. On the other hand, the breaking of a gauge symmetry is equivalent under very general assumptions to BEC, see, e.g., http://arXiv.org/abs/0711.0848v1
To show theoretically that He-II is superfluid does not follow from it being a BEC alone, but needs further information on it's excitation spectrum.

 

[Minich]

I think the lambda point is quite independent from superfluidity. It is the sign of a phase transition which is characterized by the breaking of U1 symmetry of the extended Galilean group. The characteristic sign of this symmetry breaking is the appearance of a Goldstone mode, namely second sound. On the other hand, the breaking of a gauge symmetry is equivalent under very general assumptions to BEC, see, e.g., http://arXiv.org/abs/0711.0848v1
To show theoretically that He-II is superfluid does not follow from it being a BEC alone, but needs further information on it's excitation spectrum.

4 days ago i settled all problems with journal's editor and peer reviewers so i sign final paper version of my unified theory of superfluidity and superconductivity (for "conventional" and "unconventional" SC) by the New Year eve. The publishing date is settled 29 february 2012. So, DrDu, in three months we can discuss this thread more thoroughly )

Especially why Landau and Bogoluibov theory are mainly incorrect as SUPERFLUID theories.
The main problem in superfluid theories is not quasiparticles and its spectrum, but what is it superfluid motion?
Can i ask You to read Lieb famous work on 1 dimensional bose gas with delta potentials before this date and try answer the question, why couldn't Lieb find SUPERFLUID solution of his model of 1D bose gas and couldn't find TRUE superfluid ground states?

You can find Lieb paper in
http://arxiv.org/PS_cache/cond-mat/pdf/0610/0610117v1.pdf
Appendix B
An Exactly Soluble Model page 179

I found superfluid solution for this model (exactly solvable) and ground state with less energy than Lieb's.This solution also be published by 29feb2012.

 

[DrDu]

4
Can i ask You to read Lieb famous work on 1 dimensional bose gas with delta potentials before this date and try answer the question, why couldn't Lieb find SUPERFLUID solution of his model of 1D bose gas and couldn't find TRUE superfluid ground states?

I found superfluid solution for this model (exactly solvable) and ground state with less energy than Lieb's.This solution also be published by 29feb2012.

But Lieb explicitly sais that this model has a superfluid ground state on p 46?
Anyhow I am looking forward to your paper.

 

[Minich]

There is a conflict in page numbering
there is page numbering in Acrobat reader
there is page numbering in document

For example Appendix B start page in Acromat reader is 179
in document it is 173

Your number on 46 is ?
Where you can find the solution at 46 page if the model begins at 173?

 

[DrDu]

a) I'm not using acrobat.
b) People tend to discuss also at least briefly what they are doing in the appendix.
As you also mention that one of the main questions is how to define superfluidity the discussion on p.46 could have hardly slipped your attention.

 

[Minich]

a) I'm not using acrobat.
b) People tend to discuss also at least briefly what they are doing in the appendix.
As you also mention that one of the main questions is how to define superfluidity the discussion on p.46 could have hardly slipped your attention.

If You mean the text in 46 page:
2The ground state with v = 0 remains an eigenstate of the Hamiltonian with arbitrary v (but
not necessarily a ground state) since its total momentum is zero. Its energy is 1
2mNv
2 above
the ground state energy for v = 0. Since in a finite box the spectrum of the Hamiltonian for
arbitrary v is discrete and the energy gap above the ground state is bounded away from zero
for v small, the ground state for v = 0 is at the same time the ground state of the Hamiltonian
with v if 1
2mNv
2 is smaller than the gap.


You can easely determine that this is not true for every case )))
See how many words are used and by the way every word is impotant!
What is it finite box?
Is ion lattice finite box? (for example...)

What do you think about words:

ground state with v = 0 remains an eigenstate of the Hamiltonian with arbitrary v (but
not necessarily a ground state)

))

 

[DrDu]

No, I was referring to:

"A simple example illustrating the fact that BEC is not necessary
for superfluidity is the 1D hard-core Bose gas. This system is well known to have a
spectrum like that of an ideal Fermi gas [Gi2] (see also Chapter 8 and Appendix B),
and it is easy to see that it is superfluid in its ground state in the sense of (5.18).
On the other hand, it has no BEC [Le, PiSt]. The definition of the superfluid
velocity as the gradient of the phase of the condensate wave function [HoM, Bm]
is clearly not applicable in such cases."

Btw. he somewhere specifies that he considers boxes with periodic boundary conditions.