Statistics of fermions in a $d$-dimensional box near a hard wall

Abstract

We study $N$ noninteracting fermions in a domain bounded by a hard wall potential in $d\ge 1$ dimensions. We show that for large $N$, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth potentials. We compute this $d$ dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closest to the wall. Our results are then extended in several directions, including non-smooth boundaries such as a wedge, and also to finite temperature.

5 pages + 14 pages (Supp. Mat.), 6 figures

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• Wigner function for noninteracting fermions in hard wall potentials
• Quench dynamics of noninteracting fermions with a delta impurity
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Truncated quantum observables and their semiclassical limit