Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas

Abstract

The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential $\ensuremath{-}{r}^{\ensuremath{-}2}$) is a well-known issue in quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem that differs from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation, resulting in a redefinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schr\"odinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schr\"odinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counterintuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential ${r}^{\ensuremath{-}2}$. The 2D vortical modes avoid the phase singularity at the pivot ($r=0$) by having the amplitude diverging at $r\ensuremath{\rightarrow}0$ instead of the usual situation with the amplitude of the vortical mode vanishing at $r\ensuremath{\rightarrow}0$ (the norm of the mode converges despite of the singularity of the amplitude at $r\ensuremath{\rightarrow}0$). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential ${r}^{\ensuremath{-}2}$ gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form and in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.