Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap

TL;DR

This paper provides an exact analysis of the spatial density profile of noninteracting fermions in a rotating 2D harmonic trap with a central potential, revealing complex layered structures and edge behaviors in the large particle limit.

Contribution

It introduces a detailed, exact characterization of the density profile, including multi-layered structures and edge kinks, in a rotating fermionic system with a central potential, expanding understanding of quantum many-body systems.

Findings

Bulk density exhibits a hole and layered 'wedding cake' structure.

Edge density profiles have kinks at Hermite polynomial zeros.

Large $k$ limit of edge profile resembles a propagating front.

Abstract

We compute exactly the average spatial density for $N$ spinless noninteracting fermions in a $2d$ harmonic trap rotating with a constant frequency $\Omega$ in the presence of an additional repulsive central potential $\gamma/r^2$. We find that, in the large $N$ limit, the bulk density has a rich and nontrivial profile -- with a hole at the center of the trap and surrounded by a multi-layered "wedding cake" structure. The number of layers depends on $N$ and on the two parameters $\Omega$ and $\gamma$ leading to a rich phase diagram. Zooming in on the edge of the $k^{\rm th}$ layer, we find that the edge density profile exhibits $k$ kinks located at the zeroes of the $k^{\rm th}$ Hermite polynomial. Interestingly, in the large $k$ limit, we show that the edge density profile approaches a limiting form, which resembles the shape of a propagating front, found in the unitary evolution of certain quantum spin chains. We also study how a newly formed droplet grows in size on top of the last layer as one changes the parameters.