We consider a mobile impurity coupled to an ideal Fermi gas in one spatial
dimension through an attractive contact interaction. We calculate the
quasi-particle residue $Z$ exactly, based on Bethe Ansatz and diagrammatic
Monte Carlo methods, and with varational Ansatz up to one particle-hole
excitation of the Fermi sea. We find that the exact quasi-particle residue
vanishes in the thermodynamic limit as a power law in the number of particles,
consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid
theory. The variational Ansatz, cependant, predicts a finite value of $Z$, even
in the thermodynamic limit. We also study how the presence of the impurity
affects the density of the spin-up sea by calculating the pair correlation
function. Subtracting the homogeneous background and integrating over all
distances gives the charge $Q$. This charge turns out to grow continuously from
0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz
predicts $Q=0$ at all couplings. So, although the variational Ansatz has been
shown to be remarkably accurate for the energy and the effective mass, it fails
even qualitatively when predicting $Z$ and the pair correlation function in the
thermodynamic limit.

Cet article explore les excursions dans le temps et leurs implications.

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