The name is absent

32

to be revised because it assumes that the gap function remains constant during each time
step to preserve the orthonormal nature of the u’s and v’s. This assumption breaks the self-
consistency of the equation. Some technique has been proposed to circumvent this problem
and conserve the mean number of the particles [36].

3.4.4 Beyond mean-field

Although the BdG calculations we are performing have exhibited some novel properties
which LDA fails to describe, it is still based on mean-field approximation, which does
not properly account for some many-body effects such as the Gorkov-Melik-Barkhudarov
corrections [38] that may lead to significant decreases in the pairing gap. And as stated
above, we neglected the Hartree terms (the unsuppressed normal correlation energy) in
mean-field approximation at unitary, which could be crucial in searching FFLO state and
CC limit. For the latter one, we haven’t observed such CC limit in our calculation even for
very large polarization in spherical trap while it has been verified in MIT’s experiment at P
around 0.7.

A DFT based scheme, the so-called asymmetric superfluid local density approxima-
tion,has been proposed by A. Bulgac et.al [37] to go beyond these limitation of mean-field
approximation. The ASLDA leads to a HFB∕BdG equation which can be viewed as the gen-
eralized Kohn-Sham equation [39] in the standard DFT. The basic equations of ASLDA is
as follows

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