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

In this article, we consider the monopole excitations of the harmonically trapped Bose gas in the vicinity of the Tonks-Girardeau limit. Using Girardeaus 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 of the microscopically large but macroscopically small excitations.