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

TL;DR

This paper explores how attractive bosons confined to a spherical surface transition from a uniform state to a localized soliton, revealing a smooth crossover at small N and a sharp transition at large N, influenced by curvature and quantum effects.

Contribution

It provides an analytic solution for two bosons, mean-field analysis for large N, and Monte Carlo results for intermediate N, elucidating the gas-to-soliton transition on a spherical surface.

Findings

Smooth crossover from uniform to localized state at finite N

Discontinuous transition at large N with a soliton size ~ R/√N

Energy barrier suppresses tunneling between states at large N

Abstract

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 $N\rightarrow \infty$, and exact diffusion Monte-Carlo results for intermediate $N$. For finite $N$, we observe a smooth crossover from the uniform state in the limit $a/R\gg 1$ (weak attraction) to a localized state at small $a/R$ (strong attraction). With increasing $N$ this crossover narrows down to a discontinuous transition from the uniform state to a soliton of size $\sim R/\sqrt{N}$. 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.