In a free Fermi gas at temperature
T much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability
P(R,T), i.e. the probability that a region of size
R is devoid of particles, in dimension
d, and on the case of a spherical region of large radius
R. We show that at low temperature it takes the scaling form
P(R,T)∼exp[−(kFR)d+1Φd(u=2RT/kF)], where
kF is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function
Φd(u) in any dimension. Remarkably, it exhibits a transition of order
23(d+1) at the universal critical value
uc=2/π, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.
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 consider the real time dynamics of
N noninteracting fermions in
d=1. They evolve in a trapping potential
V(x), starting from the equilibrium state in a potential
V0(x). We study the time evolution of the Wigner function
W(x,p,t) in the phase space
(x,p), and the associated kernel which encodes all correlation functions. At
t=0 the Wigner function for large
N is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale
eN being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several
V0(x). In the large
N limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time dependent rescaling. For non-harmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a non-trivial time and space dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the "classical" regime
ℏ∼1/N. Using the diagonal ensemble we compute the Wigner function in the quantum case (large
N, fixed
ℏ) and show that it agrees with the GGE. We also obtain the higher order (non-local) correlations in the diagonal ensemble.
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 consider the real-time dynamics of N noninteracting fermions in d = 1. They evolve in a trapping potential V ( x ), starting from the equilibrium state in a potential V 0 ( x ). We study the time evolution of the Wigner function W ( x , p , t ) in the phase space ( x , p ), and the associated kernel which encodes all correlation functions. At t = 0 the Wigner function for large N is uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf over a scale e N being described by a universal scaling function related to the Airy function. We obtain exact solutions for the Wigner function, the density, and the correlations in the case of harmonic and inverse square potentials, for several V 0 ( x ). In the large- N limit, near the edges where the density vanishes, we obtain limiting kernels (of the Airy or Bessel types) that retain the form found in equilibrium, up to a time-dependent rescaling. For nonharmonic traps the evolution of the Fermi volume is more complex. Nevertheless we show that, for intermediate times, the Fermi surf is still described by the same equilibrium scaling function, with a nontrivial time- and space-dependent width which we compute analytically. We discuss the multi-time correlations and obtain their explicit scaling forms valid near the edge for the harmonic oscillator. Finally, we address the large-time limit where relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the “classical” regime . Using the diagonal ensemble we compute the Wigner function in the quantum case (large N , fixed ℏ ) and show that it agrees with the GGE. We also obtain the higher order (nonlocal) correlations in the diagonal ensemble.