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.

Characteristic Temperature Scale

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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