${\alpha }^{\mathrm{\prime }}$-expansion of Anti-Symmetric Wilson Loops in $\mathcal{N}=4$ SYM from Fermi Gas

We study the large 't Hooft coupling expansion of 1/2 BPS Wilson loops in the anti-symmetric representation in $\mathcal{N}=4$ super Yang-Mills (SYM) theory at the leading order in the 1/N expansion. Via AdS/CFT correspondence, this expansion corresponds to the ${\alpha }^{\mathrm{\prime }}$ expansion in bulk type IIB string theory. We show that this expansion can be systematically computed by using the low temperature expansion of Fermi distribution function, known as the Sommerfeld expansion in statistical mechanics. We check numerically that our expansion agrees with the exact result of anti-symmetric Wilson loops recently found by Fiol and Torrents.