Explicit proof of Anderson's orthogonality catastrophe for the one-dimensional Fermi polaron with attractive interaction

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 $n=N\mathrm{/}L$, 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 quasi-particle residue $Z$ decays algebraically, $Z=W{N}^{-\theta }$. We confirm that the Anderson exponent $\theta$ is equal to $2{\delta }_F^2\mathrm{/}{\pi }^2$, where ${\delta }_F$ is the Bethe-ansatz phase shift at the Fermi edge. The prefactor $W$ is obtained numerically as a function of the interaction parameter.