Breakdown of scale invariance in the vicinity of the Tonks-Girardeau limit

In this article, we consider the monopole excitations of a harmonically trapped Bose gas in the vicinity of the 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 scale-invariance-protected value of the excitation frequency, for microscopically small excitation amplitudes. We produce a series of diffusion Monte Carlo results that confirm our analytic prediction for three particles. And less expectedly, our result stands in excellent agreement with the result of a hydrodynamic simulation (with the Lieb-Liniger equation of state as an input) of the microscopically large but macroscopically small excitations. We also show that the frequency we obtain coincides with the upper bound derived by Menotti and Stringari using sum rules. Surprisingly, however, we 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\char22{}characterized by an infinite density gradient\char22{}renders the perturbation inapplicable. All our results also directly apply to the three-dimensional $p$-wave-interacting waveguide-confined fermions.