Long-distance behavior of temperature correlation functions in the one-dimensional Bose gas

We describe a Bethe ansatz based method to derive, starting from a multiple integral representation, the long-distance asymptotic behavior at finite temperature of the density-density correlation function in the interacting one-dimensional Bose gas. We compute the correlation lengths in terms of solutions of non-linear integral equations of the thermodynamic Bethe ansatz type. Finally, we establish a connection between the results obtained in our approach with the correlation lengths stemming from the quantum transfer matrix method.

💡 Research Summary

The paper presents a comprehensive Bethe‑ansatz based framework for deriving the long‑distance asymptotics of the finite‑temperature density‑density correlation function in the one‑dimensional interacting Bose gas (the Lieb‑Liniger model). Starting from the known multiple‑integral representation of the correlation function, the authors recast the integrals in terms of Bethe‑ansatz rapidities and subsequently formulate a set of nonlinear integral equations (NLIE) of the thermodynamic Bethe‑ansatz (TBA) type that describe excited‑state contributions. While the conventional TBA focuses on the ground‑state free energy, the present NLIE incorporates two complex parameters ({\lambda_j^{\pm}}) whose solutions determine the subleading eigenvalue (\Lambda) of the quantum transfer matrix (QTM). The correlation length (\xi) and the oscillatory phase (\phi) are extracted from this eigenvalue via \