Tan's contact scaling behaviour for trapped Lieb-Liniger bosons: from two to many

TL;DR

This paper demonstrates that the contact parameter for trapped 1D bosons can be accurately approximated by rescaling the two-body solution, simplifying the analysis of many-body effects and highlighting the contact's fundamental importance.

Contribution

The authors introduce a simple rescaling method of the two-boson solution to accurately predict the contact parameter for many-body trapped bosons, bridging two- and many-body physics.

Findings

Rescaled two-body solution closely matches many-body results.

Analytical approach agrees with DMRG calculations across interaction strengths.

Highlights the fundamental role of the contact parameter in quantum systems.

Abstract

We show that the contact parameter of N harmonically-trapped interacting 1D bosons at zero temperature can be analytically and accurately obtained by a simple rescaling of the exact two-boson solution, and that N-body effects can be almost factorized. The small deviations observed between our analytical results and DMRG calculations are more pronounced when the interaction energy is maximal (i.e. at intermediate interaction strengths) but they remain bounded by the large-N local-density approximation obtained from the Lieb-Liniger equation of state stemming from the Bethe Ansatz. The rescaled two-body solution is so close to the exact ones, that is possible, within a simple expression interpolating the rescaled two-boson result to the local-density, to obtain N-boson contact and ground state energy functions in very good agreement with DMRG calculations. Our results suggest a change of paradigm in the study of interacting quantum systems, giving to the contact parameter a more fundamental role than energy.