Hole probability for non-interacting fermions in a d-dimensional trap

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability P (R) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P (R) is described by a universal scaling function of kF R, for which we obtain an exact formula (kF being the local Fermi wave-vector). It exhibits a super exponential tail P (R) / e-κd(kF R)d+1 where κdis a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

Impurities in systems of noninteracting trapped fermions

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.

The influence of the electroneutrality of the metal layer on the plasmon spectrum in "dielectric-metal-dielectric" structures

This paper proposes a model that takes into account the discretization of the Fermi wave vector and energy levels, as well as the condition of electroneutrality when investigating the influence of metal thickness on the spectrum of SPPs waves in heterogeneous dielectric-metal-dielectric structures.

Doped Kondo chain, a heavy Luttinger liquid

The doped 1D Kondo Lattice describes complex competition between itinerant and magnetic ordering. The numerically computed wave vector-dependent charge and spin susceptibilities give insights into its low-energy properties. Similar to the prediction of the large N approximation, gapless spin and charge modes appear at the large Fermi wave vector. The highly suppressed spin velocity is a manifestation of “heavy” Luttinger liquid quasiparticles. A low-energy hybridization gap is detected at the small (conduction band) Fermi wave vector. In contrast to the exponential suppression of the Fermi velocity in the large-N approximation, we fit the spin velocity by a density-dependent power law of the Kondo coupling. The differences between the large-N theory and our numerical results are associated with the emergent magnetic Ruderman–Kittel–Kasuya–Yosida interactions.

SPIN-CHARGE STRUCTURE OF QUANTUM WIRES COUPLED TO ELECTRON RESERVOIRS

We report the correlated charge and spin density distributions in a quantum wire coupled to electron reservoirs. It is found that charging the wire because of the electron density redistribution between the wire and reservoirs results in the increase of the critical electron density, below which the spontaneous spin polarization appears. The distributions of the electron densities with spin up and spin down along the wire have components oscillating in opposite phases with the wave vector 2kF, kF being the Fermi wave vector. As a result the antiferromagnetic spin order appears, with one of the spin components spontaneously predominating. The charge density distribution is close to the Wigner order with the small amplitude of the 4kF charge-density waves.

CONSTRAINT AND CONFINEMENT IN STRONGLY CORRELATED FERMION SYSTEMS

We discuss in this paper the low energy properties of a liquid of fermions coupling to a U(1) gauge field at wavevectors q<Λ≪k F at dimensions larger than one, where Λ≪k F is a high momentum cutoff and k F is the Fermi wave vector. In particular, we shall consider the e2→∞ limit where charge and current fluctuations at wave vectors q<Λ are forbidden, and the problem reduces to the problem of imposing constraint that no charge and current fluctuations are allowed in the liquid of fermions. Within a bosonization approximation, we show that the low energy properties of the system can be described as a Fermi liquid of chargeless quasiparticles which has vanishing wavefunction overlap with the bare fermion's in the system. The case of a two component system (t–J model) will also be discussed.