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 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.
The Wigner function
WN(x,p) is a useful quantity to characterize the quantum fluctuations of an
N-body system in its phase space. Here we study
WN(x,p) for
N noninteracting spinless fermions in a
d-dimensional spherical hard box of radius
R at temperature
T=0. In the large-
N limit, the local-density approximation predicts that ${W}_{N}(\mathbf{x},\mathbf{p})\ensuremath{\approx}1/{(2\ensuremath{\pi}\ensuremath{\hbar})}^{d}$ inside a finite region of the
(x,p) plane, namely, for
∣x∣<R and
∣p∣<kF, where
kF is the Fermi momentum, while
WN(x,p) vanishes outside this region, or droplet, on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)$ and find a striking algebraic tail for very large momenta ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)\ensuremath{\propto}1/{p}^{4}$, well beyond
kF, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher
d and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension
d.
Abstract Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box. We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles. Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps). In non-interacting systems, the characteristic length scale of the oscillations is set by ( 2 k F ) − 1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations of period ( 4 k F ) − 1 . Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix. While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number. We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface. In addition, we determine the spatially varying on-site density–density correlation, which in turn yields Tan’s contact density and, upon integration, Tan’s contact. As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.
Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box.We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles.Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps).In non-interacting systems, the characteristic length scale of the oscillations is set by (2k F ) -1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wignercrystal oscillations of period (4k F ) -1 .Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix.While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number.We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface.In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan's contact density and, upon integration, Tan's contact.As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.
We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.