Beyond-mean-field analysis of the Townes soliton and its breathing mode

By using the Bogoliubov perturbation theory we describe the self-bound ground state and excited breathing states of $N$ two-dimensional bosons with zero-range attractive interactions. Our results for the ground state energy $B_N$ and size $R_N$ improve previously known large-$N$ asymptotes and we better understand the crossover to the few-body regime. The breathing oscillations, absent at the mean-field level, result from the quantum-mechanical breaking of the mean-field scale symmetry. The breathing-mode frequency scales as $\mathrm{\Omega }\propto \mathrm{\mid }{B}_N\mathrm{\mid }\mathrm{/}\sqrt{N}$ at large $N$.