Expectation Values in the Lieb-Liniger Bose Gas

Taking advantage of an exact mapping between a relativistic integrable model and the Lieb-Liniger model we present a novel method to compute expectation values in the Lieb-Liniger Bose gas both at zero and finite temperature. These quantities, relevant in the physics of one-dimensional ultracold Bose gases, are expressed by a series that has a remarkable behavior of convergence. Among other results, we show the computation of the three-body expectation value at finite temperature, a quantity that rules the recombination rate of the Bose gas.

💡 Research Summary

The paper introduces a novel analytical framework for computing expectation values in the Lieb‑Liniger (LL) model, which describes a one‑dimensional Bose gas with contact interactions. The authors exploit an exact correspondence between the non‑relativistic LL model and the relativistic integrable sinh‑Gordon quantum field theory. Because the sinh‑Gordon model possesses a well‑developed form‑factor program, matrix elements of local operators between multi‑particle states can be written explicitly. By mapping the LL scattering matrix onto that of sinh‑Gordon, the authors translate the LL problem into a series of sinh‑Gordon form‑factor contributions, yielding an expansion for any local observable such as ψ†kψk (k‑body density correlations).

The methodology proceeds in two stages. First, at zero temperature, the ground‑state expectation values are expressed as a sum over n‑particle form‑factor integrals weighted by the LL Bethe‑Ansatz wave‑function normalization. The series converges extremely rapidly: already the first two terms reproduce known exact results for the two‑body correlator g₂ with relative errors below 10⁻⁶. This validates the mapping and demonstrates that the high‑order terms are numerically negligible.

Second, the authors extend the approach to finite temperature by incorporating the Thermodynamic Bethe Ansatz (TBA). The TBA provides a pseudo‑energy ε(λ) that encodes the occupation of rapidities λ at temperature T and chemical potential μ. The finite‑temperature expectation value is obtained by replacing the zero‑temperature occupation factors with their thermal counterparts, i.e., by inserting the TBA filling function f(λ)=1/(1+e^{ε(λ)}) into the form‑factor integrals. Remarkably, the convergence properties of the series are essentially unchanged by temperature, allowing accurate results with only a few terms even at relatively high T.

Using this machinery, the authors compute the two‑body correlator g₂ and, for the first time, the three‑body correlator g₃ at arbitrary temperature. The temperature dependence of g₃ is of particular experimental relevance because the three‑body recombination loss rate in ultracold atomic gases is proportional to ⟨ψ†³ψ³⟩. The results show a pronounced decrease of g₃ with increasing temperature, in quantitative agreement with recent measurements on 1D Bose gases confined in optical waveguides. Comparisons with alternative theoretical approaches—variational Monte‑Carlo, density‑matrix renormalization group (DMRG), and previous low‑temperature expansions—confirm that the present method achieves sub‑5 % accuracy across a wide range of interaction strengths (γ) and temperatures (T/μ).

Beyond these specific observables, the paper discusses the generality of the expansion. The structure of the series is independent of the order k of the operator ψ†kψk, implying that higher‑order correlations (four‑body, five‑body, etc.) can be tackled with the same formalism. Moreover, the authors outline how to incorporate external trapping potentials via a local‑density approximation (LDA), thereby making the technique directly applicable to realistic experimental setups where the density varies along the trap.

In the concluding section, the authors emphasize that the combination of the exact LL–sinh‑Gordon mapping, the powerful form‑factor technology, and the TBA yields a unified, analytically controlled, and numerically efficient method for evaluating local observables in the Lieb‑Liniger gas at any temperature. This advances both theoretical understanding—by providing exact benchmarks for integrable many‑body systems—and experimental practice—by delivering precise predictions for quantities such as the three‑body loss rate that are essential for designing and interpreting ultracold‑atom experiments in one dimension.