Hopfions in the Lee-Huang-Yang superfluids

TL;DR

This work addresses the existence and stability of Hopfions—three-dimensional topological toroidal modes—in a Lee-Huang-Yang (LHY) superfluid, realized when mean-field interactions cancel in a binary Bose-Einstein condensate. By deriving a dimensionless, single-component Gross-Pitaevskii equation with a quartic $|\,\psi\,|^3$ nonlinearity and a harmonic-oscillator trap, the authors analyze stationary Hopfion states characterized by a hidden twist $s$ and a vertical vorticity $m$, focusing on the case $s=1$ with $m=0$–$4$. Linear stability analysis via Bogoliubov–de Gennes equations reveals instability intervals that shrink with norm; nevertheless, Hopfions with $m$ up to $4$ remain stable in wide ranges of the chemical potential provided the norm exceeds a threshold, with higher $m$ enlarging the instability domain but not prohibiting stability for sufficiently large $N$. The study includes detailed numerical profiles, phase structures, and dynamic evolutions, and provides experimentally realistic parameter estimates for $^{39}$K mixtures, indicating that these 3D topological solitons are within reach of current BEC experiments. Overall, the work demonstrates that LHY-dominated quantum fluids can sustain robust Hopfions, offering new avenues for topological excitations in quantum fluids and guiding future experiments.

Abstract

It is known that, under appropriate conditions, mean-field interactions can be canceled in binary BEC, leading to the formation of the Lee-Huang-Yang (LHY) superfluid, in which the nonlinearity is solely represented by the quartic LHY term. In this work we systematically investigate the existence, stability and evolution of hopfion states in this species of quantum matter. They are characterized by two independent topological winding numbers: inner twist $s$ of the vortex-ring core and overall vorticity $m$. The interplay between the LHY self-repulsion and a trapping harmonic-oscillator potential results in stability of the hopfions with $s = 1$ and $m$ ranging from $0$ to $4$. The hopfions exhibit distinct topological phase distributions along the vertical axis and the radial direction in the horizontal plane. Their effective radius and peak density increase with the chemical potential, along with expansion of the vortex-ring core. Although the instability domain of the hopfion modes broadens with the increase of $m$, stable hopfions persist in a wide range of the chemical potential, up to $m=4$, at least, provided that the norm exceeds a certain threshold value. The predictions are experimentally accessible in currently used BEC setups.

Figures (5)

• Figure 1: (a) Top, central, and bottom rows: The hopfions with chemical potential $\mu =1.2$, intrinsic winding number $s=1$, and vorticities $m=0,1,2$, maintained by the HO potential with $\omega =0.1$. The first column shows the isosurface of $|\phi (x,y,z)|=0.45|\phi _{\text{max}}|$. The second and third columns display $|\phi |$ and $\arg (\phi )$ in the cross section $y=0$. The fourth and fifth columns show $|\phi |$ and $\arg (\phi )$ in the cross section $z=0$. The spatial domain in all the 2D plots is [$-20,+20$]. Here and in other 2D figures, blue and red regions in plots of $|\phi |$ correspond to lower and higher values, respectively.
• Figure 2: Families of the twisted toroidal states (hopfions) with intrinsic winding number $s=1$ and different values of vorticity $m$: (a) the norm vs. the chemical potential; (b) the effective width vs. the norm; (c) radius $R$ of the coiled vortex ring vs. the norm. (d) The dependence of the hopfion's amplitude, $A=\text{max}(|\phi |)$, on $\mu$. The inset in (a) is the zoom-in, taken close to the border points separating stable (solid) and unstable (dashed) subfamilies. For the clarity's sake, the curves with $m=0$ and $2$ (black and blues ones, respectively) in (b, c) are shifted by $\Delta W(\Delta R)=-1$ and $+1$, respectively.
• Figure 3: The real part of the instability growth rate, for different values of azimuthal perturbation index $k$, versus the chemical potential. The instability domains are $\mu \in [0.25, 0.85]$ for $m=0$ (a), $[0.35, 1.02]$ for $m=1$ (b) and $[0.45, 1.30]$ for $m=2$ (c).
• Figure 4: Isosurfaces $|\phi \left( x,y,z\right) |=0.55\phi _{\text{max}}$ illustrating profiles of the hopfions with $m=4,s=1$ at $\mu =1.1$ (a) and $2.6$ (b). The 2D profiles on the walls and bottoms of the 3D boxes in (a) and (b) show the wave function in cross sections $x=0$, $y=0$, and $z=0$, respectively. (c, d) Phase distributions of $\phi$ in cross sections $y=0$ and $z=0$. The spatial domains are $(x,z)\in \lbrack -28,+28]$ in (c), and $(x,y)\in \lbrack -28,+28]$ in (d). (e) Norm $N$ vs. chemical potential $\mu$ for the hopfions with $m=4,s=1$. (f) The instability growth rates vs. $\mu$ for different values of azimuthal perturbation index $k$.
• Figure 5: Isosurfaces $\left\vert \phi \left( x,y,z\right) \right\vert =0.35|\phi |_{\text{max}}$ and $0.45|\phi |_{\text{max}}$ in the first three rows, and in the bottom one, show, respectively, the unstable evolution of the hopfions with $m=0,\mu=0.6$ and $m=4,\mu =1.5$ (the first and third rows, respectively), and the stable evolution for $m=2,\mu =1.5$ and $m=4,\mu =2.16$ (the second and fourth rows). Broadband random perturbations are applied to all hopfions at $t=0$. In all cases, the twist topological charge is $s=1$.