Stable hopfions in trapped quantum droplets

TL;DR

This paper demonstrates the existence and stability of three-dimensional hopfions in trapped quantum droplets modeled by coupled Gross-Pitaevskii equations, highlighting the role of quantum fluctuations and trapping potential in stabilizing these topological solitons.

Contribution

The study constructs and analyzes stable and partly stable hopfions with various topological charges in binary Bose-Einstein condensates, emphasizing the importance of the Lee-Huang-Yang term for stability.

Findings

Stable hopfions with zero Hopf number exist under certain trap conditions.

True hopfions with nonzero Hopf number are partly stable when quantum fluctuations are included.

Hopfions are unstable without the Lee-Huang-Yang quantum fluctuation term.

Abstract

Hopfions are a class of three-dimensional (3D) solitons which are built as vortex tori carrying intrinsic twist of the toroidal core. They are characterized by two independent topological charges, \textit{viz}., vorticity $S$ and winding number $M$ of the intrinsic twist, whose product determines the \textit{Hopf number}, $Q_{H}=MS$, which is the basic characteristic of the hopfions. We construct hopfions as solutions of the 3D Gross-Pitaevskii equations (GPEs) for Bose-Einstein condensates in binary atomic gases. The GPE system includes the cubic mean-field self-attraction, competing with the quartic self-repulsive Lee-Huang-Yang (LHY) term, which represents effects of quantum fluctuations around the mean-field state, and a trapping toroidal potential (TP). A systematic numerical analysis demonstrates that families of the states with $S=1,M=0$, i.e., $Q_{H}=0$, are stable, provided that the inner TP\ radius $R_{0}$ exceeds a critical value. Furthermore, true hopfions with $S=1,M=1\sim 7$, which correspond, accordingly, to $Q_{H}=1\sim 7$, also form partly stable families, including the case of the LHY\ superfluid, in which the nonlinearity is represented solely by the LHY term. On the other hand, the hopfion family is completely unstable in the absence of the LHY term, when only the mean-field nonlinearity is present. We illustrate the knot-like structure of the hopfions by means of an elementary geometric picture. For $Q_{H}=0$, circles which represent the \textit{preimage} of the full state do not intersect. On the contrary, for $Q_{H}\geq 1$ they intersect at points whose number is identical to $Q_{H}$. The intersecting curves form multi-petal structures with the number of petals also equal to $Q_{H}$.