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Exact formulas for the form factors of local operators in the Lieb-Liniger model

We present exact formulas for the form factors of local operators in the repulsive Lieb-Liniger model at finite size. These are essential ingredients for both numerical and analytical calculations. From the theory of Algebraic Bethe Ansatz, it is known that the form factors of local operators satisfy a particular type of recursive relations. We show that in some cases these relations can be used directly to derive compact expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size. Our main results are determinant formulas for the form factors of the operators $(Ψ^{\dagger}(0))^2Ψ^2(0)$ and $Ψ^{R}(0)$, for arbitrary integer $R$, where $Ψ$, $Ψ^{\dagger}$ are the usual field operators. From these expressions, we also derive the infinite size limit of the form factors of these local operators in the attractive regime.