We study analytically the Wigner function $W_N(\mathbf{x},\mathbf{p})$ of $N$ noninteracting fermions trapped in a smooth confining potential $V\left(\mathbf{x}\right)$ in $d$ dimensions. At zero temperature, $W_N(\mathbf{x},\mathbf{p})$ is constant over a finite support in the phase space $(\mathbf{x},\mathbf{p})$ and vanishes outside. Near the edge of this support, we find a universal scaling behavior of $W_N(\mathbf{x},\mathbf{p})$ 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 $T\sim e_N/b$, where $e_N$ is an energy scale that depends on $N$ and the confining potential $V\left(\mathbf{x}\right)$, 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 $d\ge 1$ and $T\ge 0$ 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)]).

• Received 18 January 2018

DOI:https://doi.org/10.1103/PhysRevA.97.063614