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

TL;DR

This paper derives exact determinant formulas for the form factors of local operators in the finite-size repulsive Lieb-Liniger model, facilitating precise analytical and numerical computations.

Contribution

It provides the first explicit determinant formulas for specific local operator form factors in the Lieb-Liniger model, including their infinite size limits in the attractive regime.

Findings

Determinant formulas for form factors of $( ext{field operators})^2$ and $ ext{field operators}^R$

Compact expressions scale linearly with system size

Infinite size limit expressions in the attractive regime

Abstract

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 $(\Psi^{\dagger}(0))^2\Psi^2(0)$ and $\Psi^{R}(0)$, for arbitrary integer $R$, where $\Psi$, $\Psi^{\dagger}$ 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.