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 investigate the entanglement entropy of a class of
N-particle excited state of fermions confined in a two-dimensional harmonic trap rotating at an angular frequency
Ω. The excited state is constructed by filling a particular set of
N single-particle energy levels. We analytically compute the Rényi entropies of order
q in a disc of radius
r around the centre of the trap, and the cumulants corresponding to number fluctuations of fermions within the disc. We found that the area law scaling of entanglement entropy holds even for a class of excited states. We also verified the well-known series expansion of entanglement entropy in terms of the particle number cumulants for non-interacting fermions. We further derive the centered cumulant generating function, demonstrating that the associated probability distribution function in the disc has identical scaling properties, up to a variable shift, to the known ground-state result. Finally, we extend our result to an annular region, showing that both the Rényi entropy and particle number cumulants of the annulus decompose into sums of the corresponding quantities of the two bounding discs. These additive relations hold as long as the width of the annulus is sufficiently large.
We study the distribution of particle number in extended subsystems of a one-dimensional noninteracting Fermi gas confined in a potential well at zero temperature. Universal features are identified in the scaled bulk and edge regions of the trapped gas where the full counting statistics are given by the corresponding limits of the eigenvalue statistics in Gaussian unitary random matrix ensembles. The universal limiting behavior is confirmed by the bulk and edge scaling of the particle number fluctuations and the entanglement entropy.
Abstract We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions , the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the ‘Airy kernel’) and hard edge (described by the ‘Bessel kernel’) universality classes. We further discuss extensions of these results to higher dimensions and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.
We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.
We study the average density of
N spinless noninteracting fermions in a two-dimensional harmonic trap rotating with a constant frequency $\mathrm{\ensuremath{\Omega}}$ and in the presence of an additional repulsive central potential $\ensuremath{\gamma}/{r}^{2}$. The average density at zero temperature was recently studied in Phys. Rev. A 103, 033321 (2021), and an interesting multilayered ``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. We show that, while the bulk density profile gets smeared significantly at a temperature $T\ensuremath{\sim}O(1)$, the edge profile already acquires significant smearing at a much lower temperature $T\ensuremath{\sim}O(1/\sqrt{N})$. 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 we demonstrate the universality of the associated scaling functions both in the bulk and at the edges of the ``wedding cake.''