Breakdown of the scale invariance in a near-Tonks-Girardeau gas: some exact results and beyond

In this Letter, we consider the elementary monopole excitations of the harmonically trapped Bose gas in the vicinity of Tonks-Girardeau limit. Using Girardeau's Fermi-Bose duality and subsequently, an effective fermion-fermion odd-wave interaction, we obtain the dominant correction to the scaleinvariance- protected value of the excitation frequency. We produce a series of diffusion Monte Carlo results that confirm our analytic perturbative value for three particles. And less expectedly, our result stands in an excellent agreement with the result of a hydrodynamic simulation of the collective excitations in the limit of a large number of atoms (with the Lieb-Liniger equation of state as an input). The sub-leading term in the near-Tonks-Girardeau expansion of the sum rule upper bound to the monopole frequency, by Menotti and Stringari [Phys. Rev. A 66, 043610 (2002)], also gives the same number. Surprisingly it was found that the usually successful hydrodynamic perturbation theory predicts a shift that is 9/4 higher than its ab initio numerical counterpart. We conjecture that the sharp boundary of the cloud in local density approximation-characterized by an infinite density gradient-renders the perturbation theory for the collective excitation frequencies inapplicable.