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.

arXiv comment5 pages + 14 pages (Supp. Mat.), 6 figures

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• Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap
• Non-interacting fermions in hard-edge potentials
• Third-order phase transition: random matrices and screened Coulomb gas with hard walls
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• Wigner function for noninteracting fermions in hard wall potentials
• Free fermions and $\alpha$-determinantal processes
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
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• Kernels for noninteracting fermions via a Green's function approach with applications to step potentials
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Truncated quantum observables and their semiclassical limit