We study the average density of
N spinless noninteracting fermions in a
2d harmonic trap rotating with a constant frequency
Ω and in the presence of an additional repulsive central potential
γ/r2. The average density at zero temperature was recently studied in Phys. Rev. A
103, 033321 (2021) and an interesting multi-layered "wedding cake" structure with a "hole" at the center was found for the density in the large
N limit. In this paper, we study the average density at finite temperature. We demonstrate how this "wedding-cake" structure is modified at finite temperature. These large
N results warrant going much beyond the standard Local Density Approximation. We also generalize our results to a wide variety of trapping potentials and demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the "wedding-cake".
We compute exactly the average spatial density for
N spinless noninteracting fermions in a
2d harmonic trap rotating with a constant frequency $\mathrm{\ensuremath{\Omega}}$ in the presence of an additional repulsive central potential $\ensuremath{\gamma}/{r}^{2}$. 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 multilayered ``wedding cake'' structure. The number of layers depends on
N and on the two parameters $\mathrm{\ensuremath{\Omega}}$ and $\ensuremath{\gamma}$ leading to a rich phase diagram. Zooming in on the edge of the
kth layer, we find that the edge density profile exhibits
k kinks located at the zeros of the
kth 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.
We study a system of
N noninteracting spinless fermions trapped in a confining potential, in arbitrary dimensions
d and arbitrary temperature
T. The presence of the confining trap breaks the translational invariance and introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so-called ``bulk regime''), where the fermions do not feel the curvature of the trap, physical properties of the fermions have traditionally been understood using the local density (or Thomas-Fermi) approximation. However, these approximations drastically fail near the edge where the density vanishes and thermal and quantum fluctuations are thus enhanced. The main goal of this paper is to show that, even near the edge, novel universal properties emerge, independently of the details of the shape of the confining potential. We present a unified framework to investigate both the bulk and the edge properties of the fermions. We show that for large
N, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any
n-point correlation function, including the average density profile, can be expressed as an $n\ifmmode\times\else\texttimes\fi{}n$ determinant whose entry is called the kernel, a central object for such processes. Near the edge, we derive the large-
N scaling form of the kernels, parametrized by
d and
T. In
d=1 and
T=0, this reduces to the so-called Airy kernel, that appears in the Gaussian unitary ensemble (GUE) of random matrix theory. In
d=1 and
T>0 we show a remarkable connection between our kernel and the one appearing in the (
1+1)-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently, our result provides a finite-
T generalization of the Tracy-Widom distribution, that describes the fluctuations of the position of the rightmost fermion at
T=0, or those of the largest single-fermion momentum. In
d>1 and $T\ensuremath{\ge}0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the
1d Airy kernel at
T=0 obtained in random matrix theory. Some of our finite-temperature results should be testable in present-day cold-atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.
We study the ground state of $N\ensuremath{\gg}1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $\mathrm{\ensuremath{\Omega}}>0$. The support of the density of the Fermi gas is a disk of radius
Re. We calculate the variance of the number of fermions,
NR, inside a disk of radius
R centered at the origin for
R in the bulk of the Fermi gas. We find rich and interesting behaviors in two different scaling regimes, (i) $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}<1$ and (ii) $1\ensuremath{-}\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1/N)$, where $\ensuremath{\omega}$ is the angular frequency of the oscillator. In the first regime we find that $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}\ensuremath{\simeq}(AlnN+B)\sqrt{N}$ and we calculate
A and
B as functions of
R/Re, $\mathrm{\ensuremath{\Omega}}$, and $\ensuremath{\omega}$. We also predict the higher cumulants of
NR and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime, the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling
k successive Landau levels, as found in previous studies. Here, we show that
VarNR is a discontinuous piecewise linear function of $\ensuremath{\sim}(R/{R}_{e})\sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any
k. We argue that a similar piecewise linear behavior extends to all the cumulants of
NR and to the entanglement entropy. We show that these results match smoothly at large
k with the above results for $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of
VarNR near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the
z direction.
We establish an exact mapping between the positions of
N noninteracting fermions in a two-dimensional rotating harmonic trap in its ground state and the eigenvalues of the $N\ifmmode\times\else\texttimes\fi{}N$ complex Ginibre ensemble of random matrix theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite
N, the R\'enyi entanglement entropy and the number variance of a disk of radius
r in the ground state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large
N, we provide exact expressions for the scaling functions associated with these two observables.
We extend to finite temperature a Green's-function method that was previously proposed to evaluate ground-state properties of mesoscopic clouds of noninteracting fermions moving under harmonic confinement in one dimension. By calculations of the particle and kinetic-energy density profiles, we illustrate the role of thermal excitations in smoothing out the quantum shell structure of the cloud and in spreading the particle spill out from quantum tunnel at the edges. We also discuss the approach of the exact density profiles to the predictions of a semiclassical model often used in the theory of confined atomic gases at finite temperature.