Gas-to-soliton transition of attractive bosons on a spherical surface

We investigate the ground state properties of $N$ bosons with attractive zero-range interactions characterized by the scattering length $a>0$ and confined to the surface of a sphere of radius $R$. We present the analytic solution of the problem for $N=2$, mean-field analysis for $N\to \mathrm{\infty }$, and exact diffusion Monte-Carlo results for intermediate $N$. For finite $N$, we observe a smooth crossover from the uniform state in the limit $a\mathrm{/}R\gg 1$ (weak attraction) to a localized state at small $a\mathrm{/}R$ (strong attraction). With increasing $N$ this crossover narrows down to a discontinuous transition from the uniform state to a soliton of size $\sim R\mathrm{/}\sqrt{N}$. The two states are separated by an energy barrier, tunneling under which is exponentially suppressed at large $N$. The system behavior is marked by a peculiar competition between space-curvature effects and beyond-mean-field terms, both breaking the scaling invariance of a two-dimensional mean-field theory.