Wigner function of noninteracting trapped fermions

TL;DR

This paper analytically investigates the Wigner function of large numbers of noninteracting trapped fermions, revealing universal scaling behaviors at the edges of phase space both at zero and finite temperatures across various dimensions.

Contribution

It provides a universal scaling description of the Wigner function near the phase space edge for large N fermions at zero and finite temperatures, independent of potential shape and dimension.

Findings

Universal edge scaling function at zero temperature

Finite temperature edge scaling behavior with temperature-dependent universality

Results applicable to any dimension d ≥ 1 and temperature T ≥ 0

Abstract

We study analytically the Wigner function $W_N({\bf x},{\bf p})$ of $N$ noninteracting fermions trapped in a smooth confining potential $V({\bf x})$ in $d$ dimensions. At zero temperature, $W_N({\bf x},{\bf p})$ is constant over a finite support in the phase space $({\bf x},{\bf p})$ and vanishes outside. Near the edge of this support, we find a universal scaling behavior of $W_N({\bf x},{\bf p})$ 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 $V({\bf x})$, 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 $d\geq 1$ and $T \geq 0$ the $d=1$ and $T=0$ results obtained by Bettelheim and Wiegman [Phys. Rev. B ${\bf 84}$, 085102 (2011)].