Luttinger-model approach to interacting one-dimensional fermions in a harmonic trap

Abstract

A model of interacting one-dimensional fermions confined to a harmonic trap is proposed. The model is treated analytically to all orders of the coupling constant by a method analogous to that used for the Luttinger model. As a first application, the particle density is evaluated and the behavior of Friedel oscillations under the influence of interactions is studied. It is found that attractive interactions tend to suppress the Friedel oscillations while strong repulsive interactions enhance the Friedel oscillations significantly. The momentum distribution function and the relation of the model interaction to realistic pair interactions are also discussed.

Charge and Spin Dynamics of Interacting Fermions in a One-Dimensional Harmonic Trap

“…A parabolic trap potential, however, causes the particle density to vary along the trap. To treat such an inhomogeneous gas cloud, new Boson representations of the Fermi operators have been introduced [13]. Here, we follow the other route put forward recently by Recati et al [9] and consider an inhomogeneous TLL [14] with x-dependent parameters, assuming a trap potential which is slowly varying on the scale of the Fermi wavelength.…”

“…This is adequate for slow variations of V compared to the Fermi wavelength and for large particle numbers, N 1, where we can ignore Friedel oscillations occurring on the scale of the Fermi wavelength =2k F [13]. The chemical potential ensures that N R dxnx and determines the length 2R of the gas cloud through VR , independent…”

Quantum shock waves and population inversion in collisions of ultracold atomic clouds

“…A feature that hydrodynamics is unable to capture are the small oscillations in the density profile visible in the TDMRG data for any t ≥ 0 in the strongly interacting limit. These are called shell effects [78][79][80][81][82][83] and are a feature of the ground state that persists during the evolution. We stress that the agreement between the results obtained with two completely different methods such as TDMRG and hydrodynamics is a strong check of the accuracy of our simulations.…”

Relaxation and Dephasing in a Many-Fermion Generalization of the Caldeira-Leggett Model

“…In contrast to an analogous system of free fermions [7], the bath leads to transitions between levels, with strong effects of the Pauli principle. This model might also prove relevant to the discussion of cold fermionic atoms in a 1d harmonic trap [8,9] under the influence of fluctuations of the trapping potential.We rewrite and solve the Hamiltonian using the method of bosonization, for the case of large particle numbers. This enables us to evaluate Green's functions and to describe relaxation and dephasing of an extra particle added above the Fermi sea.…”

“…excitations are confined to the region near the Fermi level. Then we may employ the method of bosonization, rewriting the energy of fermions as a sum over boson modes [9]. This is possible since the energies of the oscillator levels increase linearly with quantum number, just as the kinetic energy in the Luttinger model of interacting electrons in one dimension (for recent reviews see [11]).…”