, 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}

DOI: 10.48550/arxiv.2507.10910

Related papers

Stable hopfion-like state in trapped quantum droplets

Zibin Zhao, Guilong Li, Huan-Bo Luo +4 more

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, QH=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., QH=0, are stable, provided that the inner TP radius R0 exceeds a critical value. Furthermore, HLSs with S=1,M=1−7, which correspond, accordingly, to QH=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 QH=0, circles which represent the preimage of the full state do not intersect. On the contrary, for QH≥1 they intersect at points whose number is identical to QH. The intersecting curves form multi-petal structures with the number of petals also equal to QH.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, QH=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., QH=0, are stable, provided that the inner TP radius R0 exceeds a critical value. Furthermore, HLSs with S=1,M=1−7, which correspond, accordingly, to QH=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 QH=0, circles which represent the preimage of the full state do not intersect. On the contrary, for QH≥1 they intersect at points whose number is identical to QH. The intersecting curves form multi-petal structures with the number of petals also equal to QH.

Frontiers of PhysicsOct 11, 20250 citations

Hopfions in the Lee-Huang-Yang superfluids

Liangwei Dong, Fan, Mingjing, Boris A. Malomed +1 more

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.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.

arXiv:2511.10977Nov 14, 20250 citations

Two-dimensional vortex quantum droplets

Yongyao Li, Zhaopin Chen, Zhihuan Luo +4 more

It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (two-component quantum droplets, QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity $S$ embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of $S$ and norm $N$ . An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to $S=5$ , for $N$ exceeding a certain threshold value, which is predicted to scale as ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{4}$ for large $S$ (for three-dimensional QDs, the scaling is ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{6}$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ensuremath{\ge}2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of $S$ or $S+2$ pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central ``hole'' of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of $^{39}\mathrm{K}$ atoms, in which QDs with $S=0$ and a quasi-2D shape were created recently, the vortical droplets may have radial size $\ensuremath{\lesssim}30\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$, with the number of atoms in the range of ${10}^{4}--{10}^{5}$ . The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of ``light'' droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges ${S}_{+}=\ensuremath{-}{S}_{\ensuremath{-}}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.It was recently found that the Lee-Huang-Yang (LHY) correction to the mean-field Hamiltonian of binary atomic boson condensates suppresses the collapse and creates stable localized modes (two-component quantum droplets, QDs) in two and three dimensions (2D and 3D). In particular, the LHY effect modifies the effective Gross-Pitaevskii equation (GPE) in 2D by adding a logarithmic factor to the usual cubic term. In the framework of the accordingly modified two-component GPE system, we construct 2D self-trapped modes in the form of QDs with vorticity $S$ embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of $S$ and norm $N$. An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to $S=5$, for $N$ exceeding a certain threshold value, which is predicted to scale as ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{4}$ for large $S$ (for three-dimensional QDs, the scaling is ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{6}$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ensuremath{\ge}2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of $S$ or $S+2$ pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central ``hole'' of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of $^{39}\mathrm{K}$ atoms, in which QDs with $S=0$ and a quasi-2D shape were created recently, the vortical droplets may have radial size $\ensuremath{\lesssim}30\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$, with the number of atoms in the range of ${10}^{4}--{10}^{5}$. The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of ``light'' droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges ${S}_{+}=\ensuremath{-}{S}_{\ensuremath{-}}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.

arXiv:1801.10274Physical Review ADec 3, 2018201 citations

Three-dimensional droplets of swirling superfluids

Yaroslav V. Kartashov, Boris A. Malomed, Leticia Tarruell +1 more

A method for the creation of three-dimensional (3D) solitary topological modes, corresponding to vortical droplets of a two-component dilute superfluid, is presented. We use the recently introduced system of nonlinearly coupled Gross-Pitaevskii equations, which include contact attraction between the components, and quartic repulsion stemming from the Lee-Huang-Yang correction to the mean-field energy. Self-trapped vortex tori, carrying the topological charges ${m}_{1}={m}_{2}=1$ or ${m}_{1}={m}_{2}=2$ in their components, are constructed by means of numerical and approximate analytical methods. The analysis reveals stability regions for the vortex droplets (in broad and relatively narrow parameter regions for ${m}_{1,2}=1$ and ${m}_{1,2}=2$ , respectively). The results provide a scenario for the creation of stable 3D self-trapped states with the double vorticity ( ${m}_{1,2}=2$ ). The stable modes are shaped as flat-top ones, with the space between the inner hole, induced by the vorticity, and the outer boundary filled by a nearly constant density. On the other hand, all modes with hidden vorticity, i.e., topological charges of the two components ${m}_{1}=\ensuremath{-}{m}_{2}=1$, are unstable. The stability of the droplets with ${m}_{1,2}=1$ against splitting (which is the main scenario of possible instability) is explained by estimating analytically the energy of the split and unsplit states. The predicted results may be implemented, exploiting dilute quantum droplets in mixtures of Bose-Einstein condensates.A method for the creation of three-dimensional (3D) solitary topological modes, corresponding to vortical droplets of a two-component dilute superfluid, is presented. We use the recently introduced system of nonlinearly coupled Gross-Pitaevskii equations, which include contact attraction between the components, and quartic repulsion stemming from the Lee-Huang-Yang correction to the mean-field energy. Self-trapped vortex tori, carrying the topological charges ${m}_{1}={m}_{2}=1$ or ${m}_{1}={m}_{2}=2$ in their components, are constructed by means of numerical and approximate analytical methods. The analysis reveals stability regions for the vortex droplets (in broad and relatively narrow parameter regions for ${m}_{1,2}=1$ and ${m}_{1,2}=2$, respectively). The results provide a scenario for the creation of stable 3D self-trapped states with the double vorticity (${m}_{1,2}=2$). The stable modes are shaped as flat-top ones, with the space between the inner hole, induced by the vorticity, and the outer boundary filled by a nearly constant density. On the other hand, all modes with hidden vorticity, i.e., topological charges of the two components ${m}_{1}=\ensuremath{-}{m}_{2}=1$, are unstable. The stability of the droplets with ${m}_{1,2}=1$ against splitting (which is the main scenario of possible instability) is explained by estimating analytically the energy of the split and unsplit states. The predicted results may be implemented, exploiting dilute quantum droplets in mixtures of Bose-Einstein condensates.

arXiv:1806.08570Physical Review AJul 12, 2018149 citations

Toroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar $^{164}$ Dy Quantum DropletsToroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar $^{164}$Dy Quantum Droplets

S. Sanjay, S. Saravana Veni, Boris A. Malomed

We investigate the formation, stability, and dynamics of 3D ring-shaped and multipole vortical quantum droplets (QDs) in non-rotating dipolar Bose-Einstein condensates held in a toroidal trapping potential. The QD dynamics are investigated in the framework of the extended Gross-Pitaevskii equation, which includes long-range dipole-dipole interactions (DDI) and the beyond-mean-field Lee-Huang-Yang (LHY) term, revealing the emergence of self-bound states. Stable stationary solutions for multipole QDs with different values of the topological charge (vorticity $S$ ) are shaped as necklace-like modes, with the number of \textquotedblleft beads" (multipole's order) $n=2S$ , up to $S=6$ . The stability area of the multipoles shrinks with the increase of $S$ . For higher values of $S$ the centrifugal effect associated with the phase winding destabilizes the annular density and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy and peak density on the norm (number of particles) and $S$ is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the non-rotating configurations.We investigate the formation, stability, and dynamics of 3D ring-shaped and multipole vortical quantum droplets (QDs) in non-rotating dipolar Bose-Einstein condensates held in a toroidal trapping potential. The QD dynamics are investigated in the framework of the extended Gross-Pitaevskii equation, which includes long-range dipole-dipole interactions (DDI) and the beyond-mean-field Lee-Huang-Yang (LHY) term, revealing the emergence of self-bound states. Stable stationary solutions for multipole QDs with different values of the topological charge (vorticity $S$) are shaped as necklace-like modes, with the number of \textquotedblleft beads" (multipole's order) $n=2S$, up to $S=6$. The stability area of the multipoles shrinks with the increase of $S$. For higher values of $S$ the centrifugal effect associated with the phase winding destabilizes the annular density and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy and peak density on the norm (number of particles) and $S$ is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the non-rotating configurations.

arXiv:2602.09683Feb 10, 20260 citations

Two-dimensional vortex quantum droplets in Bose-Einstein condensates

Hao Huang

Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.Bose-Einstein condensation(BEC) has been a research hotspot in cold atom physics for the past few decades. Recently, experiments on BECs have reported the observation of liquid-like quantum droplets, attracting significant interest. The physics for forming quantum droplets can be explained by zero-point quantum fluctuation which effectively prohibit the collapse caused by the mean-field interactions between the atoms. The stabilization mechanism beyond the mean-field theory, induced by quantum fluctuations, can be described by the Lee–Huang–Yang(LHY) correction. Normally, forming stable 2D or 3D solitons in free space is a challenging problem. Additionally, an intriguing question is how to generate 2D or 3D solitons with embedded vorticity. The LHY stability mechanism offers a new approach to producing stable solitons and vortex solitons in BECs. In this thesis, two-dimensional vortex quantum droplets in Bose-Einstein condensates, including those with and without external potential, are examined. The study of solitons and vortex solitons in BECs under mean-field interactions and external potential has yielded significant results. This thesis expands this field by introducing the LHY correction to study stable quantum droplets and vortex droplets in BECs. The work chapters begin with studying binary quantum droplets in free space, considering the case where interspecies interactions are weakly attractive and intraspecies interactions are weakly repulsive. The zero-vorticity, semi-vortex, and binary vortex quantum droplets are studied in Chapter 3. Stable semi-vortex quantum droplets with embedded vorticity up to 8 are found in this system. This result gives a way to form stable vortex droplets with high topological charges in free space by controlling one component with zero vorticity and the other component with embedded vorticity. The stability and properties of these droplets are systematically studied. Chapter 4 explores the vortex quantum droplets and hidden vortex droplets confined within a radial lattice trap. It is found that the existence curves of the droplets confined within this optical trap could violate the Vakhitov-Kolokolov(VK) criterion, which is a necessary condition to form a stable soliton. A remarkable finding is that the vortex and hidden vortex quantum droplets in this radial lattice remain stable when they carry the embedded vorticity up to 11. The double-ring vortex and hidden vortex quantum droplets confined within this radial lattice are also discussed. The work expands from non-magnetic atoms to magnetic atoms in Chapter 5, which needs to consider the dipole-dipole interactions(DDIs). The anisotropic dipolar vortex quantum droplets trapped in an annular potential are studied. Under the effect of the ring shape potential, the double peak ring-shaped vortex droplets remain stable when they carry the embedded vorticity up to 11. It is the first time stable droplets with high topological charges have been formed in a dipolar system. Additionally, the shape transformation of dipolar vortex droplets induced by the strength of the mean-field interaction is examined. Finally, the behavior of dipolar vortex droplets confined within a weak-depth annular potential is also discussed. The results of this thesis may prove highly useful in creating stable vortex quantum droplets in Bose-Einstein condensates and advancing the broader field of ultracold atoms.

Feb 7, 20260 citations

S

and winding number

M

of the intrinsic twist, whose product determines the \textit{Hopf number},

Q_{H}=MS

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

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}

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

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.

S

embedded into each component. Because of the effect of the logarithmic factor, the QDs feature a flat-top shape, which expands with the increase of

S

and norm

N

. An essential finding, produced by a systematic numerical investigation and analytical estimates, is that the vortical QDs are stable (which is a critical issue for vortex solitons in nonlinear models) up to

S=5

, for

N

exceeding a certain threshold value, which is predicted to scale as ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{4}$ for large

S

(for three-dimensional QDs, the scaling is ${N}_{\mathrm{th}}\ensuremath{\sim}{S}^{6}$). The prediction is corroborated by numerical findings. Pivots of QDs with $S\ensuremath{\ge}2$ are subject to structural instability, as specially selected perturbations can split the single pivot in a set of

S

S+2

pivots corresponding to unitary vortices; however, the structural instability remains virtually invisible, as it occurs in a broad central ``hole'' of the vortex soliton, where values of fields are very small, and it does not cause any dynamical instability. In the condensate of

^{39}\mathrm{K}

atoms, in which QDs with

S=0

and a quasi-2D shape were created recently, the vortical droplets may have radial size $\ensuremath{\lesssim}30\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{m}$, with the number of atoms in the range of

{10}^{4}--{10}^{5}

. The role of three-body losses is considered too, demonstrating that they do not prevent the creation of the vortex droplets but may produce a noteworthy effect, leading to sudden splitting of ``light'' droplets. In addition, hidden-vorticity (HV) states in QDs, with topological charges ${S}_{+}=\ensuremath{-}{S}_{\ensuremath{-}}=1$ in their components, which are prone to strong instability in other settings, have their stability region too. Unstable HV states tend to spontaneously merge into zero-vorticity solitons. Collisions of QDs, which may lead to their merger, and dynamics of elliptically deformed QDs (which form rotating elongated patterns or ones with oscillations of the eccentricity) are briefly considered too.

{m}_{1}={m}_{2}=1

{m}_{1}={m}_{2}=2

in their components, are constructed by means of numerical and approximate analytical methods. The analysis reveals stability regions for the vortex droplets (in broad and relatively narrow parameter regions for

{m}_{1,2}=1

and

{m}_{1,2}=2

, respectively). The results provide a scenario for the creation of stable 3D self-trapped states with the double vorticity (

{m}_{1,2}=2

). The stable modes are shaped as flat-top ones, with the space between the inner hole, induced by the vorticity, and the outer boundary filled by a nearly constant density. On the other hand, all modes with hidden vorticity, i.e., topological charges of the two components ${m}_{1}=\ensuremath{-}{m}_{2}=1$, are unstable. The stability of the droplets with

{m}_{1,2}=1

against splitting (which is the main scenario of possible instability) is explained by estimating analytically the energy of the split and unsplit states. The predicted results may be implemented, exploiting dilute quantum droplets in mixtures of Bose-Einstein condensates.

Toroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar

^{164}

Dy Quantum Droplets

S

) are shaped as necklace-like modes, with the number of \textquotedblleft beads" (multipole's order)

n=2S

, up to

S=6

. The stability area of the multipoles shrinks with the increase of

S

. For higher values of

S

the centrifugal effect associated with the phase winding destabilizes the annular density and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy and peak density on the norm (number of particles) and

S

is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the non-rotating configurations.