Stable hopfion-like state in trapped quantum droplets

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, viz., vorticity S and winding number M of the intrinsic twist, whose product determines the Hopf number, Q_H=MS, which is the basic characteristic of the hopfions. We construct hopfion-like states (HLSs) 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). Although our system exhibits two independent topological charges, it does not fully conform to the Hopf map structure, and therefore does not constitute hopfions in the complete sense. 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, HLSs with S=1,M=1-7, which correspond, accordingly, to Q_H=1-7, 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 HLSs 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 HLSs by means of an elementary geometric picture. For Q_H=0, circles which represent the preimage of the full state do not intersect. On the contrary, for Q_H\ge 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.

Graphical abstract

Keywords

hopfions / quantum droplets / toroidal potential / Bose−Einstein condensates / Hopf number

1 Introduction

In 1867, Kelvin [1] proposed the hypothesis of vortex atoms, sparking mathematical studies of knots. However, it was not until 1997 that Faddeev and Niemi [2] suggested that knots might exist as stable solitons in the three-dimensional classical field theory, thereby initiating research of physical realizations of knot-like structures. Numerical studies of the Faddeev−Skyrme version of the O(3) sigma model have demonstrated the existence of stable toroidal solutions in it [3–9]. Unlike ordinary vortex-torus solutions, which are characterized by the single winding number (topological charge) M, those ones possess two independent winding numbers, M and S, with S determining the inner twist of the vortex torus [see Eqs. (14) and (15) below]. The corresponding Hopf number is defined as

(1)Q_H=MS,

which is the fundamental topological characteristic of the intrinsically twisted vortex-soliton states, known as hopfions.

Hopfions, which constitute a class of knot-like solitons, may be elegantly presented by dint of the Hopf fibration, which is a mapping from the unit sphere in the four-dimensional space onto unit sphere in 3D, i.e., S^3\to S^2 [10]. Various species of hopfions have been extensively studied across many fields [11-30].

In optics, scalar hopfions with toroidal vortex structures have been proposed as approximate solutions to the Maxwell’s equations [31]. Additionally, the use of structured light fields, featuring pronounced spatial variations in their polarization, phase, and amplitude, have made it possible to demonstrate an experimental realization of optical hopfions [32]. Recent studies employing high-order harmonics have also produced optical hopfions [33]. In liquid crystals, material structures in the form of hopfions and various bound states built of them have been widely investigated too, both theoretically and experimentally [34–39]. In cosmology, topological defects are utilized to model the large-scale structure of the universe, with studies of knot-like hopfions providing valuable insights into the topology of cosmic formations [40–44]. These findings highlight the significance of hopfions in unraveling complex topological phenomena across diverse disciplines.

Ultracold atomic Bose−Einstein condensates (BECs), with their tunable intrinsic interactions, offer an expedient platform for the realization of knot-shaped solitons. Various types of vortex knot solitons have been extensively studied in BECs, including trefoil vortex knots, Solomon vortex links, and various other vortex rings and lines [45–57]. Topological transitions from vortex rings to knots and links were explored too [58]. In particular, encircling a vortex line by a vortex ring gives rise to composite hopfions [59]. Previous studies have demonstrated that hopfions can stably exist in a rotating BEC confined by an oblate harmonic-oscillator trap [59]. Moreover, in the 3D free space, hopfions may be stable in a single-component BEC with self-repulsion whose strength is made spatially modulated, growing, as a function of radial coordinate r, faster than r^3 [60].

Quantum droplets (QDs), i.e., self-trapped states filled by an ultradilute superfluid, are maintained by the balance between mean-field effects and corrections to them induced by quantum fluctuations [61, 62]. In terms of the corresponding Gross−Pitaevskii equations (GPEs), the corrections are represented by the Lee−Huang−Yang (LHY) terms [63, 64]. Actually, QDs represent a new species of quantum matter, as well as a novel form of self-localized states [65, 66]. Recent detailed studies predict that stable QDs may exist, in the free space, not only in the form of a self-trapped ground-state (GS) mode, but also as robust excited states with embedded vorticity [67–77]. In addition to that, stable vortex QDs with high values of the topological charge (vorticity), as well as multipole QDs, can be maintained by a Gaussian-shaped toroidal potential (TP) [78]. However, three-dimensional (3D) vortex states with complex topological structures, such as hopfions, have yet to be investigated in the realm of QDs.

In this paper, we produce stable solutions with two independent topological charges constructed using a TP. Since authentic hopfions require strict adherence to the Hopf map topology, we characterize our solutions as hopfion-like states (HLSs), recognizing their topological resemblance while distinguishing them from true hopfions that fully satisfy the Hopf map structure. Unlike the above-mentioned one trapping potential, which was employed in Ref. [78], the TP considered here is one of the harmonic-oscillator type, rather than Gaussian-shaped potential structure. Numerical analysis indicates that the HLSs with Q_H=0 (they may exist with M=0 and S\ne 0, see Eq. (1) and Ref. [70], as toroidal patterns whose preimages (“skeletons”) [35] are composed of non-intersecting concentric circles) are stable, in a typical experimentally relevant setup, at values of the TP’s inner radius R_0\ge 1.1 μm, an instability region appearing below this threshold. Furthermore, HLSs with high values of the Hopf number, up to Q_H=7, are also found to be stable. In particular, in contrast to the results of Ref. [59], the inclusion of the LHY term allows for the existence of stable HLSs with S=1 and M>1 (hence they have Q_H>1, as per Eq. (1)). Similar to the Q_H=0 solutions, these HLSs exhibit a double-ring pattern, which may serve as an experimental signature for identifying HLSs. These HLSs correspond to preimages which include intersections. In the horizontal plane, the self-intersecting preimages form petal-like structures, see Fig.1 below. As Q_H increases, the number of petals increases accordingly. These findings provide direct insights into the geometric and topological structures of HLSs in QDs and suggest a potential pathway for their experimental realization.

The subsequent presentation is organized as follows. Section 2 introduces the 3D model including the TP. Numerical results for stationary solutions, including the HLSs with Q_H=0 and Q_H\ne 0, are reported in Section 3. Additionally, an elementary geometric representation of the Hopf number is introduced and analyzed in that section. The paper is concluded by Section 4.

2 The model

The binary BEC in the 3D space with coordinates \left(X,Y,Z\right) is modeled by the system of nonlinearly-coupled GPEs which include the cubic mean-field terms and quartic LHY ones [61]:

(2)\begin{array}{rl}\mathrm{i}\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }T}{\mathrm{\Psi }}_1=& -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}_{XYZ}^2{\mathrm{\Psi }}_1+\left(G_{11}|{\mathrm{\Psi }}_1{|}^2+G_{12}|{\mathrm{\Psi }}_2{|}^2\right){\mathrm{\Psi }}_1\\ & +\mathrm{\Gamma }{\left(|{\mathrm{\Psi }}_1{|}^2+|{\mathrm{\Psi }}_2{|}^2\right)}^{\frac{3}{2}}{\mathrm{\Psi }}_1+V\left(R,Z\right){\mathrm{\Psi }}_1,\end{array}

(3)\begin{array}{rl}\mathrm{i}\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }T}{\mathrm{\Psi }}_2=& -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}_{XYZ}^2{\mathrm{\Psi }}_2+\left(G_{22}|{\mathrm{\Psi }}_2{|}^2+G_{21}|{\mathrm{\Psi }}_1{|}^2\right){\mathrm{\Psi }}_2\\ & +\mathrm{\Gamma }{\left(|{\mathrm{\Psi }}_1{|}^2+|{\mathrm{\Psi }}_2{|}^2\right)}^{\frac{3}{2}}{\mathrm{\Psi }}_2+V\left(R,Z\right){\mathrm{\Psi }}_2,\end{array}

where G_{11}=G_{22}=4\pi {\hslash }^{2}a/m and G_{12}=G_{21}=4\pi {\hslash }^2{a}^{\mathrm{\prime }}/m are the self- and cross-interaction strengths, with atomic mass m, a and a^{\mathrm{\prime }} being the intra- and inter-species scattering lengths, respectively. The TP potential is defined as

V\left(R,Z\right)={\displaystyle \frac{1}{2}}m{\mathrm{\Omega }}^2\left[{\left(R-R_0\right)}^2+Z^2\right],

where R=\sqrt{X^2+Y^2} is the radial coordinate in the 2D plane, R_0 is the inner radius of the toroid, and \mathrm{\Omega } is the TP trapping frequency. The coefficient of the LHY correction is [61]

(4)\mathrm{\Gamma }={\displaystyle \frac{4m^{\frac{3}{2}}{G}^{\frac{5}{2}}}{3{\pi }^2{\hslash }^3}}={\displaystyle \frac{128\sqrt{\pi }}{3m}}{\hslash }^2{a}^{\frac{5}{2}}.

For symmetric states with

(5){\mathrm{\Psi }}_1={\mathrm{\Psi }}_2\equiv \mathrm{\Psi }/\sqrt{2},

coupled GPEs (2) and (3) admit the reduction to a single equation,

(6)\begin{array}{rl}\mathrm{i}\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }T}\mathrm{\Psi }=& -\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}_{\mathrm{X}\mathrm{Y}\mathrm{Z}}^2\mathrm{\Psi }+\frac{\delta G}{2}{\left|\mathrm{\Psi }\right|}^2\mathrm{\Psi }\\ & +\mathrm{\Gamma }{\left|\mathrm{\Psi }\right|}^3\mathrm{\Psi }+\frac{1}{2}m{\mathrm{\Omega }}^2\left[{\left(R-R_0\right)}^2+Z^2\right]\mathrm{\Psi },\end{array}

where \delta G=\left(4\pi {\hslash }^2/m\right)\left(a^{\mathrm{\prime }}+a\right)\equiv \left(4\pi {\hslash }^2/m\right)\delta a. The total number of atoms in the system is

(7)N={\displaystyle \int \left({\left|{\mathrm{\Psi }}_1\right|}^2+{\left|{\mathrm{\Psi }}_2\right|}^2\right){\mathrm{d}}^3\mathbf{R}={\displaystyle \int {\left|\mathrm{\Psi }\right|}^2{\mathrm{d}}^3\mathbf{R}.}}

The set of the system’s control parameters includes a, a^{\mathrm{\prime }} and N.

By means of rescaling,

(8)\begin{array}{rl}& T=t\cdot t_0,\left(X,Y,Z\right)=\left(x,y,z\right)\cdot l_0,\\ & \mathrm{\Psi }=l_0^{-\frac{3}{2}}\cdot \psi ,\frac{\hslash t_0}{m{l}_0^2}=1,\end{array}

where t_0 and l_0 are time and length scales, Eq. (6) is cast in the dimensionless form:

(9)\begin{array}{rl}\mathrm{i}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\psi =& -\frac{1}{2}{\mathrm{\nabla }}^2\psi +g{\left|\psi \right|}^2\psi +\gamma {\left|\psi \right|}^3\psi \\ & +\frac{1}{2}\omega \left[{\left(r-r_0\right)}^2+z^2\right]\psi ,\end{array}

where we define the scaled strengths of the contact interaction, LHY correction, and TP as, respectively,

(10)\begin{array}{rl}& g=2\pi \frac{\delta a}{l_0}=2\pi \delta \tilde{a},\\ & \gamma =\frac{128\sqrt{\pi }}{3}{\left(\frac{a}{l_0}\right)}^{\frac{5}{2}}=\frac{128\sqrt{\pi }}{3}{\tilde{a}}^{\frac{5}{2}},\\ & \omega ={\left(t_0\mathrm{\Omega }\right)}^2.\end{array}

In the further analysis, we refer to parameters of the {}^{39}\mathrm{K} atomic gas, selecting l_0=0.1 μm. The Hamiltonian (energy) corresponding to Eq. (9) is

(11)\begin{array}{rl}E=& \int (\frac{1}{2}{\left|\mathrm{\nabla }\psi \right|}^2+\frac{1}{2}g{\left|\psi \right|}^4+\frac{2}{5}\gamma {\left|\psi \right|}^5\\ & +\frac{1}{2}\omega \left[{\left(r-r_0\right)}^2+z^2\right]{\left|\psi \right|}^2){\mathrm{d}}^3\mathit{r}.\end{array}

Stationary solutions of Eq. (9) with chemical potential \mu are looked for, below, as

(12)\psi =\varphi (r)\exp (-\mathrm{i}\mu t),

with function \varphi obeying the stationary GPE:

(13)\begin{array}{rl}\mu \varphi =& -\frac{1}{2}{\mathrm{\nabla }}^2\varphi +g{\left|\varphi \right|}^2\varphi +\gamma {\left|\varphi \right|}^3\varphi \\ & +\frac{1}{2}\omega \left[{\left(r-r_0\right)}^2+z^2\right]\varphi .\end{array}

3 Stationary solutions

Hopfions are defined as knot-like solitons that can be mapped into the Hopf fibration [79], being characterized by two independent winding numbers [80, 81]. The numerical solution of Eq. (9) for HLSs was performed by means of the Newton-conjugate-gradient method, starting with the ansatz (initial guess)

(14)\varphi \left(r,z\right)={\left(r^{\mathrm{\prime }}\right)}^S\exp \left(-{\displaystyle \frac{{\left(r^{\mathrm{\prime }}\right)}^2}{A}}+\mathrm{i}M\theta +\mathrm{i}S\phi \right),

where A>0 is a real constant and the special coordinates are defined as

(15)\left(r^{\mathrm{\prime }},\theta ,\phi \right)=\left[\sqrt{{\left(r-r_0\right)}^2+z^2},\arctan {\displaystyle \frac{y}{x}},\arctan {\displaystyle \frac{r-r_0}{z}}\right]

(i.e., \theta is the usual angular coordinate in the horizontal plane \left(x,y\right)). In contrast with the usual vortex tori in the 3D space, which feature the single winding number M (vorticity) in the (x,y) plane, here the second winding number S, which is defined in the (r,z) plane, determines the inner twist of the vortex torus. Ansatz (14) implies a structure like a twisted toroidal vortex tube nested in the 3D solution, coiling up around the vertical (z) axis. This shape is typical for solitons of the Faddeev−Skyrme model, with the triplet of real scalar fields realizing the Hopf map, \mathbf{\Phi }:R^3\to S^2 [2, 3, 82–86], therefore such states are named hopfions, which are characterized by the Hopf number (topological invariant) [Eq. (1)].

3.1 Hopfion-like states (HLSs) with Q_H=0

A typical stable HLS with zero vorticity, i.e., S=1,M=0 (hence it has Q_H=0, according to Eq. (1)), which is supported by the TP, is displayed in Fig.2. Direct simulations of its perturbed evolution demonstrate that this HLS is stable, at least, up to T=10 ms. Its toroidal shape produces a double-ring pattern in the horizontal plane Z=0, as shown in Fig.2(b), where density |\varphi {|}^2 of the inner ring is slightly higher than in the outer ring. The position of the pivots (phase singularities) of the internal vortex rings at Y=0, corresponding to the location of the inner ring, is indicated by the orange dashed line in Fig.2. This double-ring pattern plays the role similar to that of the HLS’s preimage [35], and may serve as a signature for the its experimental observation.

Basic properties of the hopfions with S=1,M=0 are summarized in Fig.3. It displays the total atom number N(\mu ) and energy E(\mu ) for the HLSs with different values of a^{\mathrm{\prime }} and R_0, with solid lines and dashed segments referring to stable and unstable solutions, respectively. It is evident that, for a fixed scattering length a, increase in the TP radius R_0 leads to higher atom number N and energy E. This fact suggests that a larger radius requires a higher number of atoms for the formation of HLSs. Additionally, an instability region arises when R_0 is below 1.1 μm. This observation implies that, for a fixed interaction strength g, the HLSs suffers destabilization when the inner TP radius R_0 falls below a critical value, R_c=1.1 μm. A more detailed investigation reveals that HLSs with S=1,M=0 cease to exist when R_0 falls below 0.4 μm. This indicates that in the case of a spherically symmetric harmonic trap, corresponding to R_0=0, such HLS structures cannot exist in a stable form.

Note that all the N(\mu ) dependences plotted in Fig.3(a) satisfy the Vakhitov−Kolokolov criterion, , which is the well-known necessary stability criterion [88, 89]. On the other hand, it is seen too that the criterion is not sufficient for the full stability, as the curves include unstable segments. The same conclusions are valid for the N(\mu ) dependences for the HLS families with Q_H\ne 0, which are plotted below in Fig.4.

The simulated evolution of an unstable HLS with S=1, M=0 is shown in Fig.5. In the course of the evolution, it loses the phase structure and eventually decay into the GS. Although Fig.5 illustrates the evolution up to T=12\mathrm{m}\mathrm{s}, the ring-like structure of weakly unstable HLSs can be preserved for a relatively long time and does not split until T=40\mathrm{m}\mathrm{s}. Moreover, for the same values of a and TP radius R_0, increase in |a^{\mathrm{\prime }}|, which corresponds to enhancement of the interaction strength g, results in an increase in both the number of atoms N and energy E. As we adopt , it follows from here that we have , indicating that the increase of |a^{\mathrm{\prime }}| strengthens the attractive interaction. Thus, stronger attraction leads to the formation of HLSs with higher atom numbers and energy. Actually, the HLSs with S=1,M=0 exhibit the lowest atom-number threshold, below which the HLSs do not exist. This threshold increases with the TP radius R_0 and interaction strength g.

As mentioned above, HLSs are a class of solitons of the Faddeev−Skyrme model and realize the Hopf map, \mathbf{\Phi }:R^3\to S^2. For finite-energy solutions, one requires \mathbf{\Phi }\to \mathit{n} as |\mathit{r}|\to \mathrm{\infty }, where \mathit{n} is a constant unit vector. Thus R^3 can be compactified to S^3 and the map reduces to

(16)\mathbf{\Phi }:S^3\to S^2.

Since {\pi }_3\left(S^2\right)=\mathbb{Z}, where different integers from \mathbb{Z} are the Hopf numbers Q_H, corresponding to different realizations of the maps [3].

The Hopf number Q_H also has an elementary geometric interpretation. The preimage of every point of the target space S^2 is isomorphic to a circle. These circles are all linked, meaning that any given circle intersects the disk spanned by any other circle. Thus, the Hopf number represents the linkage number, which quantifies the degree of the linkage between any two arbitrary circles [3, 48, 87].

For the HLSs with Q_H=0, which we address in this section, different circles corresponding to different points in the parameter space S^2 do not link, resulting in concentric circles. To illustrate this property, Fig.6 shows contour plots of different HLSs with R_0=0.9 μm and 1.1 μm. Solid circles in the figure represent contours where both the real and imaginary parts of wave function \varphi are constant (i.e., \mathrm{R}\mathrm{e}(\varphi )=C_1, \mathrm{I}\mathrm{m}(\varphi )=C_2), different colors corresponding to different values of C_1 and C_2. It is clearly observed that these circles are concentric and do not intersect, which is consistent with Q_H=0. The real and imaginary parts of the wave function represent two degrees of freedom in the parameter space S^2, with different values corresponding to different points on the S^2 sphere. Solid circles in the figure represent the projection of a point from the S^2 sphere onto the real space. The projection is realized through the Hopf map and stereographic projection.

3.2 HLSs with Q_H\ne 0

HLSs with nonzero Hopf numbers are topologically nontrivial states. Examples of HLSs with S=1,M=1-7 are shown in Fig.7. Accordingly, they carry Q_H=1-7. The first row of Fig.7 illustrates density isosurfaces of the HLSs, revealing that they all are toroidal modes. The second and third rows display the phase distributions of the wave functions in the horizontal (Z=0) and vertical (Y=0) planes, respectively. The latter rows demonstrate that the HLSs indeed have the twist number S=1, while their vorticities in the horizontal plane range from M=1 to M=7. The stability of these HLSs was verified by simulation of their perturbed evolution, in the framework of Eq. (9).

The N(\mu ) and E(\mu ) dependences for HLSs with Q_H=1-7 are shown in Fig.4, where solid and dashed segments again represent stable and unstable solutions, respectively. It is observed that, as the twist winding number M increases, the atom number N and energy E of the HLSs decrease. Similar to the above results for Q_H=0, the HLSs with Q_H=1-7 also exhibit a minimum-N threshold, which increases with the value of Q_H. HLSs with higher Hopf numbers Q_H display intermittent instability regions. In general, the instability region becomes broader as the Hopf number increases.

The evolution of unstable HLSs is illustrated in Fig.8(a1)−(a3). It is observed that the unstable HLSs lose their topological structure in the course of the evolution. The vortex ring in the center is gradually destroyed, and the HLS eventually degenerates into the GS. Similar to the unstable evolution reported in Ref. [59], the vortex ring in the center splits into fragments of vortex lines. Additionally, we examined the evolution of a stable HLS with parameters S=1, M=1, \mathrm{\Omega }=51000\mathrm{H}\mathrm{z}, and N=111130. As shown in Fig.8(b1)−(b3), the HLS becomes unstable when the TP strength falls below the critical value, {\mathrm{\Omega }}_c=15405\mathrm{H}\mathrm{z}. Furthermore, as displayed in Fig.8(c1)−(c3), if the TP is removed at the start of the evolution, the previously stable HLS rapidly becomes unstable on a short timescale \simeq 0.02ms. This observation suggests that the HLSs, characterized by their complex topological structure, are unlikely to remain stable in the free space, without the support of the TP.

Unlike the previously discussed case of Q_H=0, where the curves corresponding to constant values of (\mathrm{R}\mathrm{e}(\varphi ),\mathrm{I}\mathrm{m}(\varphi )) in the 3D space do not intersect, the HLSs with Q_H\ne 0 exhibit intersection of these curves, resulting in a knot with a linking number equal to Q_H. Fig.1 offers the elementary geometric representations of the HLSs with Q_H=1-7. In the figure, red and blue curves correspond to distinct constant values of (\mathrm{R}\mathrm{e}(\varphi ),\mathrm{I}\mathrm{m}(\varphi )). To better highlight the structure of the HLSs, the figure focuses on the shape of the knots in the horizontal plane, Z=0, where the linking number can be identified by observing which curve passes over which one. Note that these are 3D curves, but not flat loops. When viewed in the Z=0 cross-section, these knots exhibit a petal-like structure, with the number of petals increasing with the Hopf number, Q_H.

It is natural to expect that HLSs with Q_H\ne 0 cannot be stable in the absence of the LHY correction, i.e., setting \gamma =0 in Eq. (9). Indeed, our numerical analysis readily demonstrates that, while the combination of the mean-field self-attractive cubic nonlinearity and TP trap creates HLSs as solutions of Eq. (9), they are completely unstable. In particular, in the case of \gamma =0 the dependence N(\mu ) for the HLS family with S=1,M=1 (Q_H=1), displayed in Fig.9(a) fully contradicts the VK criterion, which confirms its complete instability.

The unstable evolution of the HLSs in this case is illustrated in Fig.10. It is seen that the unstable HLS with a relatively small norm suffers complete destruction (in panel (a)), while a HLS with a large norm splits into a set of fragments, in panel (b).

Additionally, we examined the special case in which the nonlinearity is represented solely by the LHY term, i.e., g=0 in Eq. (9) (the so-called LHY superfluid [63, 90]). In this case, we find that HLS families with Q_H=1 obey the VK criterion and are chiefly stable, including an unstable subfamily, as demonstrated by the blue curves in Fig.9. The figure also shows that, for the same value of strength \gamma of the LHY term and g=0, the HLSs exhibit lower norm N and energy E, in comparison to their counterparts found in the presence of the mean-field self-attraction (). However, the instability region is slightly larger in the case of g=0 than in the presence of .

4 Conclusion

The objective of the work is to construct stable QDs (quantum droplets) carrying the highly nontrivial topological structure which makes it possible to identify them as hopfions. Nonetheless, these states are not hopfions in the strict mathematical sense, as the Hopf map is not explicitly involved, but rather hopfion-like states (HLSs). The stability of (a part of) these states, which is a crucially important issue, is provided by the action of TP (toroidal potential), which is included in the corresponding system of the 3D GPEs (Gross−Pitaevskii equations). The interaction in the GPE system is represented by the self-attractive mean-field cubic terms and repulsive LHY (Lee−Huang−Yang) quartic ones, which represent the effect of quantum fluctuations around the mean-field configurations. The numerical analysis demonstrates that not only the modes with zero Hopf number, Q_H=0 (they have zero vorticity, M=0, and the twist topological charge S=1) can stably exist in this system, but also the HLSs in the range of Q_H\equiv MS=1-7 exhibit stability regions. The intensity profiles of these solutions in the horizontal (Z=0) plane reveal a characteristic double-ring preimage of the HLS, which may serve as a distinct signature for identifying HLSs in the experiment. Naturally, the stability regions gradually shrink with the increase of Q_H. In the course of their evolution, unstable HLSs lose their topological structures, eventually decaying into the GS (ground state). The HLSs cannot be stable if the confining TP is absent, i.e., they cannot represent stable states in the free space. To further elucidate the geometry and topology of the HLSs (in particular, of the stable ones), we have displayed their elementary geometric representation by means of the corresponding Hopf map (of the parameter manifold into the real space) and stereographic projection. For Q_H=0, the representation reveals a series of non-intersecting concentric rings in the real space. In contrast, the solutions with Q_H=1-7 exhibit intricate knots, with the linking number equal to Q_H. In the horizontal plane, these knots exhibit a petal-like structure, with the number of petals also equal to Q_H. Furthermore, the role of the LHY term is crucial in determining the stability of HLSs. When the LHY term is absent ( in Eq. (9)), HLSs with Q_H\ne 0 cannot stably exist. In contrast, when only the LHY nonlinear term is present (g=0,\gamma \ne 0 in Eq. (9)), HLSs with Q_H\ne 0 can stably exist, although the instability region is slightly larger compared to the case with .

References

[1] L. Kelvin , On vortex atoms, Proc. R. Soc. Edin 6, 94 (1867)

[2] L. Faddeev and A. J. Niemi , Stable knot-like structures in classical field theory, Nature 387(6628), 58 (1997)

[3] J. Gladikowski and M. Hellmund , Static solitons with nonzero Hopf number, Phys. Rev. D 56(8), 5194 (1997)

[4] R. A. Battye and P. M. Sutcliffe , Knots as stable soliton solutions in a three-dimensional classical field theory, Phys. Rev. Lett. 81(22), 4798 (1998)

[5] H. Aratyn , L. A. Ferreira , and A. H. Zimerman , Toroidal solitons in 3+1 dimensional integrable theories, Phys. Lett. B 456(2−4), 162 (1999)

[6] J. Hietarinta and P. Salo , Faddeev−Hopf knots: Dynamics of linked un-knots, Phys. Lett. B 451(1−2), 60 (1999)

[7] H. Aratyn , L. A. Ferreira , and A. H. Zimerman , Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers, Phys. Rev. Lett. 83(9), 1723 (1999)

[8] R. S. Ward , Skyrmions and Faddeev−Hopf solitons, Phys. Rev. D 70(6), 061701 (2004)

[9] P. Sutcliffe , Knots in the Skyrme−Faddeev model, Proc. Royal Soc. Math. Phys. Eng. Sci. 463(2087), 3001 (2007)

[10] E. Radu and M. S. Volkov , Stationary ring solitons in field theory – Knots and vortons, Phys. Rep. 468(4), 101 (2008)

[11] B. Göbel , I. Mertig , and O. A. Tretiakov , Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles, Phys. Rep. 895(12), 1 (2021)

[12] M. Sallermann , H. Jónsson , and S. Blügel , Stability of hopfions in bulk magnets with competing exchange interactions, Phys. Rev. B 107(10), 104404 (2023)

[13] D. Raftrey and P. Fischer , Field-driven dynamics of magnetic Hopfions, Phys. Rev. Lett. 127(25), 257201 (2021)

[14] Y. Liu , W. Hou , X. Han , and J. Zang , Three-dimensional dynamics of a magnetic hopfion driven by spin transfer torque, Phys. Rev. Lett. 124(12), 127204 (2020)

[15] S. B. Gudnason and M. Nitta , Baryonic torii: Toroidal baryons in a generalized Skyrme model, Phys. Rev. D 91(4), 045027 (2015)

[16] Y. Shen , Q. Zhang , P. Shi , . Optical skyrmions and other topological quasiparticles of light, Nat. Photon. 18(1), 15 (2024)

[17] D. Sugic , R. Droop , E. Otte , D. Ehrmanntraut , F. Nori , J. Ruostekoski , C. Denz , and M. R. Dennis , Particle-like topologies in light, Nat. Commun. 12(1), 6785 (2021)

[18] N. A. Veretenov , S. V. Fedorov , and N. N. Rosanov , Topological vortex and knotted dissipative optical 3D solitons generated by 2D vortex solitons, Phys. Rev. Lett. 119(26), 263901 (2017)

[19] I. Luk’yanchuk , Y. Tikhonov , A. Razumnaya , and V. M. Vinokur , Hopfions emerge in ferroelectrics, Nat. Commun. 11(1), 2433 (2020)

[20] P. J. Ackerman and I. I. Smalyukh , Diversity of Knot solitons in liquid crystals manifested by linking of preimages in torons and hopfions, Phys. Rev. X 7(1), 011006 (2017)

[21] D. Kleckner and W. T. M. Irvine , Creation and dynamics of knotted vortices, Nat. Phys. 9(4), 253 (2013)

[22] S. Bolognesi and M. Shifman , Hopf Skyrmion in QCD with adjoint quarks, Phys. Rev. D 75(6), 065020 (2007)

[23] M. Ezawa , Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon’s knot, trefoil knot, and other linked nodal varieties, Phys. Rev. B 96(4), 041202 (2017)

[24] P. Sutcliffe , Hopfions in chiral magnets, J. Phys. A 51(37), 375401 (2018)

[25] M. Kobayashi and M. Nitta , Torus knots as Hopfions, Phys. Lett. B 728(20), 314 (2014)

[26] X. S. Wang , A. Qaiumzadeh , and A. Brataas , Current-driven dynamics of magnetic Hopfions, Phys. Rev. Lett. 123(14), 147203 (2019)

[27] X. Yu , Y. Liu , K. V. Iakoubovskii , K. Nakajima , N. Kanazawa , N. Nagaosa , and Y. Tokura , Realization and current-driven dynamics of fractional Hopfions and their ensembles in a helimagnet FeGe, Adv. Mater. 35(20), 2210646 (2023)

[28] D. Ehrmanntraut , R. Droop , D. Sugic , E. Otte , M. R. Dennis , and C. Denz , Optical second-order skyrmionic hopfion, Optica 10(6), 725 (2023)

[29] H. Wang and S. Fan , Photonic spin hopfions and monopole loops, Phys. Rev. Lett. 131(26), 263801 (2023)

[30] F. N. Rybakov , N. S. Kiselev , A. B. Borisov , L. Döring , C. Melcher , and S. Blügel , Magnetic hopfions in solids, APL Mater. 10(11), 111113 (2022)

[31] C. Wan , Y. Shen , A. Chong , . Scalar optical hopfions, eLight 2(1), 22 (2022)

[32] Y. Shen , B. Yu , H. Wu , C. Li , Z. Zhu , and A. V. Zayats , Topological transformation and free-space transport of photonic hopfions, Adv. Photonics 5(1), 015001 (2023)

[33] Z. Lyu , Y. Fang , and Y. Liu , Formation and controlling of optical hopfions in high harmonic generation, Phys. Rev. Lett. 133(13), 133801 (2024)

[34] P. J. Ackerman , J. van de Lagemaat , and I. I. Smalyukh , Self-assembly and electrostriction of arrays and chains of hopfion articles in chiral liquid crystals, Nat. Commun. 6(1), 6012 (2015)

[35] I. I. Smalyukh , Review: knots and other new topological effects in liquid crystals and colloids, Rep. Prog. Phys. 83(10), 106601 (2020)

[36] J. S. B. Tai and I. I. Smalyukh , Three-dimensional crystals of adaptive knots, Science 365(6460), 1449 (2019)

[37] J. S. B. Tai , P. J. Ackerman , and I. I. Smalyukh , Topological transformations of Hopf solitons in chiral ferromagnets and liquid crystals, Proc. Natl. Acad. Sci. USA 115(5), 921 (2018)

[38] H. R. O. Sohn , P. J. Ackerman , T. J. Boyle , G. H. Sheetah , B. Fornberg , and I. I. Smalyukh , Dynamics of topological solitons, knotted streamlines, and transport of cargo in liquid crystals, Phys. Rev. E 97(5), 052701 (2018)

[39] H. Zhao , B. A. Malomed , and I. I. Smalyukh , Topological solitonic macromolecules, Nat. Commun. 14(1), 4581 (2023)

[40] A. Vilenkin,E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, Cambridge, England, 1994

[41] S. A. Hojman and F. A. Asenjo , Cosmological electro-magnetic hopfions, Phys. Scr. 99(5), 055514 (2024)

[42] D. Foster , Massive hopfions, Phys. Rev. D 83(8), 085026 (2011)

[43] A. Thompson , A. Wickes , J. Swearngin , and D. Bouwmeester , Classification of electromagnetic and gravitational hopfions by algebraic type, J. Phys. A 48(20), 205202 (2015)

[44] T. Smołka and J. Jezierski , Simple description of generalized electromagnetic and gravitational hopfions, Class. Quantum Gravity 35(24), 245010 (2018)

[45] D. Proment , M. Onorato , and C. F. Barenghi , Vortex knots in a Bose−Einstein condensate, Phys. Rev. E 85(3), 036306 (2012)

[46] H. B. Luo , L. Li , and W. M. Liu , Three-dimensional skyrmions with arbitrary topological number in a ferromagnetic spin-1 Bose−Einstein condensate, Sci. Rep. 9(1), 18804 (2019)

[47] S. Zou , W. K. Bai , T. Yang , and W. M. Liu , Formation of vortex rings and hopfions in trapped Bose-Einstein condensates, Phys. Fluids 33(2), 027105 (2021)

[48] Y. Kawaguchi , M. Nitta , and M. Ueda , Knots in a spinor Bose−Einstein condensate, Phys. Rev. Lett. 100(18), 180403 (2008)

[49] D. S. Hall , M. W. Ray , K. Tiurev , E. Ruokokoski , A. H. Gheorghe , and M. Möttönen , Tying quantum knots, Nat. Phys. 12(5), 478 (2016)

[50] R. N. Bisset , W. Wang , C. Ticknor , R. Carretero-González , D. J. Frantzeskakis , L. A. Collins , and P. G. Kevrekidis , Robust vortex lines, vortex rings, and hopfions in three-dimensional Bose−Einstein condensates, Phys. Rev. A 92(6), 063611 (2015)

[51] T. Ollikainen , S. Masuda , M. Möttönen , and M. Nakahara , Quantum knots in Bose-Einstein condensates created by counter-diabatic control, Phys. Rev. A 96(6), 063609 (2017)

[52] Y. K. Liu , S. J. Yang , G. H. Yang , and J. J. Zhang , Interlocked knot in spinor Bose−Einstein condensates, Chaos Solitons Fractals 140, 110209 (2020)

[53] B. Jackson , J. F. McCann , and C. S. Adams , Vortex line and ring dynamics in trapped Bose−Einstein condensates, Phys. Rev. A 61(1), 013604 (1999)

[54] W. Wang , R. N. Bisset , C. Ticknor , R. Carretero-González , D. J. Frantzeskakis , L. A. Collins , and P. G. Kevrekidis , Single and multiple vortex rings in three-dimensional Bose−Einstein condensates: Existence, stability, and dynamics, Phys. Rev. A 95(4), 043638 (2017)

[55] C. Ticknor , W. Wang , and P. G. Kevrekidis , Spectral and dynamical analysis of a single vortex ring in anisotropic harmonically trapped three-dimensional Bose−Einstein condensates, Phys. Rev. A 98(3), 033609 (2018)

[56] V. P. Ruban , W. Wang , C. Ticknor , and P. G. Kevrekidis , Instabilities of a vortex-ring-bright soliton in trapped binary three-dimensional Bose−Einstein condensates, Phys. Rev. A 105(1), 013319 (2022)

[57] M. Abad , M. Guilleumas , R. Mayol , and M. Pi , Vortex rings in toroidal Bose−Einstein condensates, Laser Phys. 18(5), 648 (2008)

[58] W. K. Bai , T. Yang , and W. M. Liu , Topological transition from superfluid vortex rings to isolated knots and links, Phys. Rev. A 102(6), 063318 (2020)

[59] Y. M. Bidasyuk , A. V. Chumachenko , O. O. Prikhodko , S. I. Vilchinskii , M. Weyrauch , and A. I. Yakimenko , Stable Hopf solitons in rotating Bose−Einstein condensates, Phys. Rev. A 92(5), 053603 (2015)

[60] Y. V. Kartashov , B. A. Malomed , Y. Shnir , and L. Torner , Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity, Phys. Rev. Lett. 113(26), 264101 (2014)

[61] D. S. Petrov , Quantum mechanical stabilization of a collapsing Bose−Bose mixture, Phys. Rev. Lett. 115(15), 155302 (2015)

[62] D. S. Petrov and G. E. Astrakharchik , Ultradilute low-dimensional liquids, Phys. Rev. Lett. 117(10), 100401 (2016)

[63] N. B. Jørgensen , G. M. Bruun , and J. J. Arlt , Dilute fluid governed by quantum fluctuations, Phys. Rev. Lett. 121(17), 173403 (2018)

[64] T. D. Lee , K. Huang , and C. N. Yang , Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev. 106(6), 1135 (1957)

[65] Z. Luo , W. Pang , B. Liu , Y. Y. Li , and B. A. Malomed , A new form of liquid matter: Quantum droplets, Front. Phys. (Beijing) 16(3), 32201 (2021)

[66] M. Guo and T. Pfau , A new state of matter of quantum droplets, Front. Phys. (Beijing) 16(3), 32202 (2021)

[67] Y. Li , Z. Chen , Z. Luo , C. Huang , H. Tan , W. Pang , and B. A. Malomed , Two-dimensional vortex quantum droplets, Phys. Rev. A 98(6), 063602 (2018)

[68] G. Li , Z. Zhao , X. Jiang , Z. Chen , B. Liu , B. A. Malomed , and Y. Li , Strongly anisotropic vortices in dipolar quantum droplets, Phys. Rev. Lett. 133(5), 053804 (2024)

[69] X. Zhang , X. Xu , Y. Zheng , Z. Chen , B. Liu , C. Huang , B. A. Malomed , and Y. Li , Semidiscrete quantum droplets and vortices, Phys. Rev. Lett. 123(13), 133901 (2019)

[70] Y. V. Kartashov , B. A. Malomed , L. Tarruell , and L. Torner , Three-dimensional droplets of swirling superfluids, Phys. Rev. A 98(1), 013612 (2018)

[71] L. Dong and Y. V. Kartashov , Rotating multidimensional quantum droplets, Phys. Rev. Lett. 126(24), 244101 (2021)

[72] Z. Lin , X. Xu , Z. Chen , Z. Yan , Z. Mai , and B. Liu , Two-dimensional vortex quantum droplets get thick, Commun. Nonlinear Sci. Numer. Simul. 93, 105536 (2021)

[73] T. A. Yoğurt , U. Tanyeri , A. Keleş , and M. Ö. Oktel , Vortex lattices in strongly confined quantum droplets, Phys. Rev. A 108(3), 033315 (2023)

[74] G. E. Astrakharchik and B. A. Malomed , Dynamics of one-dimensional quantum droplets, Phys. Rev. A 98(1), 013631 (2018)

[75] C. D’Errico , A. Burchianti , M. Prevedelli , L. Salasnich , F. Ancilotto , M. Modugno , F. Minardi , and C. Fort , Observation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Res. 1(3), 033155 (2019)

[76] I. Ferrier-Barbut , H. Kadau , M. Schmitt , M. Wenzel , and T. Pfau , Observation of quantum droplets in a strongly dipolar Bose gas, Phys. Rev. Lett. 116(21), 215301 (2016)

[77] R. N. Bisset , L. A. P. Ardila , and L. Santos , Quantum droplets of dipolar mixtures, Phys. Rev. Lett. 126(2), 025301 (2021)

[78] L. Dong , M. Fan , and B. A. Malomed , Three-dimensional vortex and multipole quantum droplets in a toroidal potential, Chaos Solitons Fractals 188, 115499 (2024)

[79] H. Hopf, Selecta Heinz Hopf: Herausgegeben zu seinem 70. Geburtstag von der Eidgenössischen Technischen Hochschule Zürich, Springer, Berlin, Heidelberg, 1964

[80] D. W. Lyons , An elementary introduction to the Hopf fibration, Math. Mag. 76(2), 87 (2003)

[81] J. H. C. Whitehead , An expression of Hopf’s invariant as an integral, Proc. Natl. Acad. Sci. USA 33(5), 117 (1947)

[82] L. Faddeev and A. J. Niemi , Partially dual variables in SU(2) Yang−Mills theory, Phys. Rev. Lett. 82(8), 1624 (1999)

[83] J. Jäykkä , J. Hietarinta , and P. Salo , Topologically nontrivial configurations associated with Hopf charges investigated in the two-component Ginzburg−Landau model, Phys. Rev. B 77(9), 094509 (2008)

[84] E. Babaev , L. D. Faddeev , and A. J. Niemi , Hidden symmetry and knot solitons in a charged two-condensate Bose system, Phys. Rev. B 65(10), 100512 (2002)

[85] J. M. Speight , Supercurrent coupling in the Faddeev−Skyrme model, J. Geom. Phys. 60(4), 599 (2010)

[86] J. Jäykkä and J. M. Speight , Supercurrent coupling destabilizes knot solitons, Phys. Rev. D 84(12), 125035 (2011)

[87] Z. Lyu , Y. Fang , and Y. Liu , Formation and controlling of optical Hopfions in high harmonic generation, Phys. Rev. Lett. 133(13), 133801 (2024)

[88] N. G. Vakhitov and A. A. Kolokolov , Stationary solutions of the wave equation in a medium with nonlinearity saturation, Radiophys. Quantum Electron. 16(7), 783 (1973)

[89] L. Bergé , Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep. 303(5−6), 259 (1998)

[90] T. G. Skov , M. G. Skou , N. B. Jørgensen , and J. J. Arlt , Lee−Huang−Yang Fluid, Phys. Rev. Lett. 126(23), 230404 (2021)

RIGHTS & PERMISSIONS

Higher Education Press