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arXiv:cond-mat/0306597 · Journal of Physics B Atomic Molecular and Optical Physics

Friedel oscillations in a gas of interacting one-dimensional fermionic atoms confined in a harmonic trap

S N Artemenko, Gao Xianlong, W Wonneberger

Published Mar 24, 2004• 11 citations
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Abstract

Using an asymptotic phase representation of the particle density operator in the one-dimensional harmonic trap, the part which describes the Friedel oscillations is extracted. The expectation value with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga–Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent ν = ( K +1)/2 away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.

DOI
10.1088/0953-4075/37/7/052

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Using an asymptotic phase representation of the particle density operator in the one-dimensional harmonic trap, the part which describes the Friedel oscillations is extracted. The expectation value with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga–Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent ν = ( K +1)/2 away from the classical boundaries. This value differs from that known for open boundary conditions or strong pinning impurities. The soft boundary in the present case thus modifies the decay of Friedel oscillations. The case of two components is also discussed.
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.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution n0(z,N).{n}_{0}(z,N).n0​(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near 2kF,{2k}_{F},2kF​, with kF{k}_{F}kF​ being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
A phase operator formulation for a recent model of interacting one-dimensional fermions in a harmonic trap is developed. The resulting theory is similar to the corresponding approach for the Luttinger model with open boundary conditions (OBC). However, in place of the spatial coordinate zzz, a dimensionless variable uuu defined on the unit circle appears as argument of the phase fields and uuu is non-linearly related to zzz. Furthermore, form factors appear which reflect the harmonic trap geometry. The theory is applied to calculate one-particle correlation functions. In a properly defined thermodynamic limit, bulk and boundary critical exponents are calculated for the static two-point correlation function and the dynamic local correlation function. The local spectral density is also considered. The critical exponents found are in agreement with those known for OBC with the exception of the boundary scaling exponent Δ⊥Δ_\perpΔ⊥​.
Using a bosonization and variational Gaussian wave-functional method we analytically calculate the Friedel oscillations induced by a single impurity in one-dimensional electron systems with long-range Coulomb interaction. It is shown that the Friedel oscillations are drastically different from those in electron systems with short-range interaction. First, in the long-range case the oscillatory part of the Friedel oscillations is not periodic as that in the short-range case. Second, in the long-range case the envelope of the Friedel oscillations decays as $\mathrm{exp}[\ensuremath{-}\mathrm{const}(\mathrm{ln}{x)}^{1/2}]$ far away from the impurity, which is much slower than the power law in the short-range case.
The Wigner function W ( z , k ) for a model of interacting one-dimensional fermions confined to a harmonic trap is evaluated at zero temperature. The model considers two hyperfine states of the same fermionic species and treats the dominant interactions between the two using the bosonization method. Interactions substantially modify the shape of the Wigner function. Irrespective of the sign of the coupling constant, the Friedel oscillations inherent in the Wigner function are enhanced in the k -direction and suppressed in the z -direction.
We study Friedel oscillations in one-dimensional electron liquids for arbitrary electron-electron interaction and arbitrary impurity strength. Explicit results for spinless as well as spin- 12\frac{1}{2}21​ electrons are given. In the case of Luttinger-liquid leads, the Friedel oscillations decay as ${x}^{\ensuremath{-}g}$ far away from the impurity where ggg is the interaction constant. For a weak scatterer, a slower decay is found at small-to-intermediate distances from the impurity, with a crossover to the asymptotic ${x}^{\ensuremath{-}g}$ decay.