Wigner function of noninteracting trapped fermions

Abstract

We study analytically the Wigner function $W_N(\mathbf{x},\mathbf{p})$ of $N$ noninteracting fermions trapped in a smooth confining potential $V(\mathbf{x})$ 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 $V(\mathbf{x})$, 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 $d\ge 1$ and $T\ge 0$ the $d=1$ and $T=0$ results obtained by Bettelheim and Wiegman [Phys. Rev. B $84$, 085102 (2011)].

16 pages, 4 figures

References in corpus (5)

• Many-Body Physics with Ultracold Gases
• Theory of ultracold Fermi gases
• The entanglement entropy of one-dimensional gases
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Phase transitions and edge scaling of number variance in Gaussian random matrices

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