We provide a fully analytical derivation of Anderson's orthogonality catastrophe for the one dimensional Fermi polaron integrable model, describing a system of $N$ spin-up fermions, with fixed density $n=N/L$, interacting with a single spin-down fermion via an attractive contact potential. The proof combines the determinant representations of the norm of the many-body wave function and of its scalar product with the noninteracting ground state, obtained from the Bethe-ansatz solution, with the special properties of Cauchy matrices. We derive the leading asymptotics of the two determinants in the thermodynamic limit and show that the quasiparticle residue $Z$ decays algebraically, $Z=W{N}^{-\theta }$. We confirm that the Anderson exponent $\theta$ is equal to $2{\delta }_F^2/{\pi }^2$, where ${\delta }_F$ is the Bethe-ansatz phase shift at the Fermi edge. We also derive the asymptotic behavior of the prefactor $W$ at strong coupling and compare it against exact numerics.

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• Fermi gases
• Polarons
• 1-dimensional systems
• Atomic gases
• Quantum many-body systems
• Strongly correlated systems
• Bethe ansatz