Wigner function of noninteracting trapped fermions

We study analytically the Wigner function W N (x,p) of N noninteracting fermions trapped in a smooth confining potential V (x) in d dimensions. At zero temperature, W N (x,p) is constant over a finite support in the phase space (x,p) and vanishes outside. Near the edge of this support, we find a universal scaling behavior of W N (x,p) for large N. The associated scaling function is independent of the precise shape of the potential as well as the spatial dimension d. We further generalize our results to finite temperature T > 0. We show that there exists a low-temperature regime T ∼ e N /b, where e N is an energy scale that depends on N and the confining potential V (x), where the Wigner function at the edge again takes a universal scaling form with a b-dependent scaling function. This temperature-dependent scaling function is also independent of the potential as well as the dimension d. Our results generalize to any d 1 and T 0 the d = 1 and T = 0 results obtained by Bettelheim and Wiegman [Phys. Rev. B 84, 085102 (2011)] (see also the earlier paper by Balazs and Zipfel [Ann. Phys. (NY) 77, 139 (1973)]).Matrices aléatoires et fermions piégé