We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasiparticle residue $Z$ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with a varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasiparticle 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, however, 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.

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• Fermi gases
• Polarons
• 1-dimensional systems
• Bethe ansatz
• Diagrammatic methods
• Variational approach