BEC-BCS Crossover

BCS superfluidity

Superfluidity of boson was first discovered in $^{4}He$ system at a critical temperature of $T_{C}\sim 2.2K$. This was connected to the formation of $^{4}He$ condensates. Superfluidity of fermions, the electrons, was first discovered in Mecury at a transition temperature $T_{C}\sim 4.2K$, which is known as the `superconductivity' of metals. \\ In the early age, there are two major confusions about the fermionic superfluidity

• what is the mechanism for superfluidity of fermions (electrons)?
  • It is intuitive to suggest that two electrons could form tightly bounded pairs (Schafroth pairs) and then form condensates. However, there was no known interaction which is strong enough to overcome the Coulomb repulsion.
• why does it happen at such low temperature compared with $T_{F}$ (typically $\sim 10^{4}K$ in metal)?
  • For bosonic case in $^{4}He$, we can estimate the transition temperature $T_{BEC}$ (assuming phase space density 1 and typical Helium density) to be $T_{BEC}\sim 3K$ which is consistent with the experimental findings. However, the fermi temperature in a fermionic system in Mercury is much higher (10^4) than the observed superfluidity transition temperature.

The two puzzles remain unresolved until 1956 when Bardeen, Cooper and Schrieffer proposed the BCS theory. In short:

• It is correct to think of fermion (electron) pairs. However, instead of the tightly bound pairs, the pair here is the loosely bound BCS pair of electrons formed due to the effective attractive interaction mediated by the hosting lattice.
• The temperature scale is the Debye temperature because of the involvement of the hosting lattice in the pairing mechanism. This temperature is further modified by the pairing energy and the density of states on the Fermi sea.

Characteristic Temperature Scale

It is not clear until the discovery of

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File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

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