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Stable hopfions in trapped quantum droplets

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Pith's one-line read Stable hopfion quantum droplets with Hopf numbers 0 through 7 can exist in toroidally trapped binary Bose gases when the Lee-Huang-Yang quantum-fluctuation term is present.

desk verdict Promising scalar-model study of LHY-stabilized hopfions; the binary-condensate stability claim outruns the numerics. read the letter →

arxiv 2507.10910 v1 pith:W45DHS37 submitted 2025-07-15 cond-mat.quant-gas

Zibin Zhao , Guilong Li , Huanbo Luo , Bin Liu , Guihua Chen , Boris A. Malomed , Yongyao Li This is my paper · ORCID
classification cond-mat.quant-gas PACS 03.75.Lm67.85.-d
keywords hopfionsquantumdropletsBose-EinsteincondensatesLee-Huang-YangtermtoroidaltrapGross-PitaevskiiequationstopologicalsolitonsHopfnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that hopfions—three-dimensional knot-like solitons built as twisted vortex tori—can exist as stable quantum droplets in a toroidally trapped binary Bose–Einstein condensate. Working from the coupled Gross–Pitaevskii equations with cubic mean-field self-attraction, quartic Lee–Huang–Yang repulsion, and a harmonic toroidal trap, the authors report stable families with azimuthal winding $M=0$, twist $S=1$ (Hopf number $Q_H=0$) once the trap's inner radius exceeds $R_0 \simeq 1.1\,\mu$m, and partly stable families of true hopfions with $M=1$ through $7$ and $S=1$ ($Q_H=1$ through $7$). The Lee–Huang–Yang term is the load-bearing ingredient: with only the cubic mean-field nonlinearity every $Q_H \neq 0$ hopfion is unstable, while a purely LHY nonlinearity still supports partly stable hopfions. If correct, the results give concrete, experimentally accessible signatures—double-ring density patterns and petal-like preimage knots—for observing hopfions in ultracold atomic gases.

What carries the argument

The carrying object is the twisted-torus ansatz $\varphi = (r')^S \exp[-(r')^2/A + iM\theta + iS\phi]$, in which $M$ winds around the vertical axis and $S$ winds in the poloidal $(r,z)$ plane, so the Hopf number is $Q_H = MS$. Stationary solutions are found by Newton–conjugate-gradient iteration of the dimensionless scalar GPE $-\frac{1}{2}\nabla^2\psi + g|\psi|^2\psi + \gamma|\psi|^3\psi + \frac{1}{2}\omega[(r-r_0)^2+z^2]\psi = \mu\psi$, and stability is checked by the negative-slope criterion $dN/d\mu < 0$ together with direct simulation of perturbed evolution. The geometric preimage construction—constant values of $(\mathrm{Re}\,\varphi,\mathrm{Im}\,\varphi)$ mapped back to real space through the Hopf map and stereographic projection—turns the abstract Hopf number into visible linking and petal counts.

What would settle it

Initialize the coupled two-component GPEs (2)–(3) with a stable symmetric hopfion and add a small antisymmetric perturbation, $\Psi_1=\Psi/\sqrt{2}+\epsilon$ and $\Psi_2=\Psi/\sqrt{2}-\epsilon$; if the relative mode grows and breaks the torus on the reported timescales, the binary-mixture stability claim fails even though the scalar numerics are correct.

Watch

Extended reading notes

Core claim

The central discovery is that the scalar Gross–Pitaevskii equation obtained from the binary system by the symmetric reduction $\Psi_1=\Psi_2=\Psi/\sqrt{2}$ possesses stable toroidal soliton solutions with two independent winding numbers: the azimuthal winding $M$ around the vertical axis and the twist $S$ around the torus core, whose product is the Hopf number $Q_H=MS$. Numerically, with parameters for $^{39}$K, the $Q_H=0$ family ($M=0$, $S=1$) is stable for inner torus radii $R_0 \gtrsim 1.1\,\mu$m and ceases to exist below $R_0 \approx 0.4\,\mu$m; the true hopfions with $S=1$, $M=1$–$7$ have shrinking stability regions as $Q_H$ grows. In the absence of the LHY term ($\gamma=0$) the negative-slope condition on $N(\mu)$ is violated and these hopfions are unstable, whereas in the LHY-only case ($g=0$) at least the $Q_H=1$ family is mostly stable. The paper also establishes an elementary geometric reading of the Hopf number: curves of constant $(\mathrm{Re}\,\varphi,\mathrm{Im}\,\varphi)$ are non-intersecting concentric circles for $Q_H=0$, and for $Q_H \ge 1$ they intersect in $Q_H$ points forming a $Q_H$-petal knot in the $Z=0$ plane.

Load-bearing premise

The load-bearing premise is that setting the two components equal, $\Psi_1=\Psi_2=\Psi/\sqrt{2}$, faithfully represents the binary system; the paper never simulates the coupled two-component dynamics, so an antisymmetric perturbation between the components could in principle destroy the hopfions even though the scalar calculation is stable.

Editorial extensions

If this is right

  • Hopfion quantum droplets with Hopf numbers up to $Q_H=7$ should be realizable in toroidally trapped binary Bose gases with $^{39}$K parameters, provided the torus inner radius and atom number are above the reported thresholds.
  • The double-ring density profile in the $Z=0$ plane gives an experimental fingerprint that distinguishes hopfions from ordinary vortex droplets.
  • The LHY correction is not a minor quantitative shift but the qualitative stabilizer: removing it destabilizes all nonzero-Hopf families, while keeping only LHY restores partial stability.
  • Stability windows narrow as $Q_H$ grows, so the highest Hopf numbers will be hardest to observe and the lowest ones the best experimental targets.
  • Without the toroidal trap the hopfions decay on sub-millisecond timescales, so experimental realization must supply the toroidal confinement.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the stability analysis is confined to the symmetric subspace $\Psi_1=\Psi_2$, the binary-mixture claim is an extrapolation from a single-component reduction; the full two-component equations have not been tested for relative-density or relative-phase instabilities.
  • Since the preimage linking count equals $Q_H$, a phase- and density-resolved reconstruction could in principle read the Hopf number off the number of petals in the horizontal plane.
  • The sharp contrast between unstable mean-field-only hopfions and partly stable LHY-supported ones suggests the same stabilization mechanism may apply to other three-dimensional topological solitons in quantum droplets, such as vortex knots and skyrmions in binary mixtures.
  • A systematic scan of the stability boundary in the $(R_0, N, \delta a)$ parameter space would produce a concrete, experimentally testable phase diagram for hopfion droplets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript studies toroidally trapped binary Bose-Einstein condensates described by coupled Gross-Pitaevskii equations with cubic mean-field terms and the Lee-Huang-Yang quartic correction. Under the symmetric ansatz Ψ1=Ψ2=Ψ/√2 in Eq. (5), the two-component system is reduced to the scalar GPE in Eq. (9), and stationary hopfion solutions with S=1 and M=0,...,7 are computed by a Newton-conjugate-gradient method starting from the twisted toroidal ansatz in Eq. (14). Stability is assessed through the Vakhitov-Kolokolov criterion and finite-time perturbed evolution, leading to the claims that QH=0 states are stable for inner torus radius above a critical value, that QH=1,...,7 states are partly stable, that they are completely unstable without the LHY term, and that a purely LHY-driven ('LHY superfluid') regime supports at least partly stable hopfions. The paper also presents preimage-based geometric representations of the Hopf number and petal structures.

Significance. If the stability results survive a full two-component analysis, the paper would deliver a new class of three-dimensional topological solitons, hopfion quantum droplets with Hopf numbers 1 through 7, in a realistic binary BEC setup. The identification of the LHY term as the stabilizing nonlinearity and the proposed double-ring experimental signature are valuable. The systematic parameter scans in Figs. 2, 6, and 9 and the geometric illustrations in Figs. 4 and 8 are useful and clearly presented. However, the central claim for a binary gas is conditional on the unanalyzed component-antisymmetric sector, and the topological charge is not independently computed for the converged numerical states, so the significance is currently prospective rather than fully established.

major comments (4)
  1. [Sec. II, Eq. (5); Sec. III] All stationary solutions and all reported stability evolutions solve the scalar GPE (9), but the physical model is the binary system (2)-(3). The manuscript never analyzes perturbations that break the condition Ψ1=Ψ2. The relative mode η=δΨ1-δΨ2 obeys a linearized equation that contains an anomalous term proportional to (G-G')Ψ²η*, which is absent from Eq. (9); for the main parameters a=100a0 and a'=-110a0 this coupling is large. Consequently, the Vakhitov-Kolokolov criterion and the perturbed evolutions in Figs. 2, 6, and 9 certify stability only in the symmetric subspace. A coupled two-component Bogoliubov-de Gennes spectrum, or direct binary-GPE simulations seeded with antisymmetric perturbations, is required before the claim of stable hopfions in binary atomic gases is supported.
  2. [Sec. III, Eq. (14); Figs. 4 and 8] The ansatz (14) explicitly builds in the winding numbers e^{iMθ} and e^{iSϕ}, so the value QH=MS is effectively imposed by construction. The paper does not compute a Hopf invariant of the converged numerical solution, for example through an integral of the Hopf density or through the linking number of preimage curves obtained from an explicitly defined normalized map to S2. The preimage and petal plots therefore illustrate the labels put in by the ansatz rather than independently establish the topological charge of the numerical state. Please define the map Φ:R3→S2, including the boundary condition at the edge of the trap, and report QH computed directly for each converged solution.
  3. [Secs. III A and III B; Figs. 2, 6, 9] The stability labels are based on the Vakhitov-Kolokolov criterion, which the paper itself notes is only a necessary condition, and on finite-time perturbed evolution lasting roughly 10-40 ms. Weak instabilities with small growth rates can be missed on these timescales, especially for the higher-QH families that are divided into stable and unstable segments. A Bogoliubov-de Gennes spectrum for representative points in each stable and unstable segment would make the labels much more robust. At minimum, the perturbation amplitudes and a convergence check of the finite-time runs should be reported.
  4. [Sec. III B, Fig. 9; Abstract and Conclusion] The abstract states that true hopfions with QH=1,...,7 form partly stable families 'including the case of the LHY superfluid', and the conclusion states that when only the LHY term is present, hopfions with QH≠0 can stably exist. However, the only LHY-superfluid results shown in Fig. 9 are for QH=1. No data are presented for QH=2,...,7 in the g=0 regime, so the broader statements in the abstract and conclusion are not supported by the reported numerics. Please either provide those families or qualify the claim to QH=1.
minor comments (7)
  1. [Fig. 5 caption] The last panel labels read '92267 (f1-f3)', but they should refer to the g-column; please correct this typo.
  2. [Eq. (15)] The angle ϕ=arctan((r-r0)/z) is singular at r=r0 and z=0; it would be helpful to state how the numerical initial guess regularizes the vortex core in that region.
  3. [Fig. 5 caption] The symbol R in the caption should be R0 to match the notation introduced in Sec. II.
  4. [References] Reference [33] and reference [87] are the same work and should be merged or renumbered.
  5. [Sec. III] The manuscript does not report the numerical box size, grid resolution, or convergence tolerance for the Newton-conjugate-gradient method; these details are needed for reproducibility.
  6. [Abstract] In the sentence 'true hopfions with S=1, M=1~7, which correspond, accordingly, to QH=1~7', the dependence on S=1 is clear, but it would be safer to state explicitly that QH=MS with S=1.
  7. [Sec. III] The terms 'vorticity' and 'winding number' are used interchangeably for M; please define the terminology once at first use.

Circularity Check

1 steps flagged · score 2.0 of 10

Stability analysis is self-contained; only the topological QH/petal identification is fixed by the input ansatz, and the unreduced binary spin sector is a correctness caveat rather than circularity.

  1. self definitional [Sec. III (Stationary solutions), Eq. (14)-(15); with Eq. (1)]
    "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, Φ : R3 → S2 [2, 3, 82–86], therefore such states are named hopfions, which are characterized by the Hopf number (topological invariant) (Eq. 1)."

    The ansatz (14) explicitly contains the phase factors iMθ+iSϕ, and Eq. (1) defines QH=MS. The later geometric representation (Fig. 8) counts intersections of preimage contours and reports that the number of petals equals QH; that count is already fixed by the M and S inserted in the initial guess, so the Hopf number and the petal/linking structure are consequences of the input ansatz rather than an independently computed topological invariant. The stability calculation itself is not circular: it is obtained from the VK criterion and direct perturbed evolution of Eq. (9), with no fitted parameter renamed as a prediction.

full rationale

The central stability results are self-contained and do not reduce to their inputs. The model is the standard binary-GPE system (2)-(3), reduced under the explicitly stated symmetric condition (5) to the scalar GPE (9); the parameters (a=100a0, a'=-110a0, Ω=51000 Hz, R0 values) are physical inputs, not fits to the stability outcome. Stationary solutions are found by Newton-conjugate-gradient solution of Eq. (13), and stability is assessed by the Vakhitov-Kolokolov dN/dμ criterion plus direct simulations of perturbed evolution; none of these diagnostics is defined in terms of the conclusion. The only genuinely self-referential element is the topological labeling: since Eq. (14) starts from prescribed S and M, the correspondence QH=MS and the petal-count picture in Fig. 8 illustrate the input winding numbers rather than an independent measurement of the Hopf invariant. This is a minor presentational circularity and does not affect the stability claims. A separate, non-circular caveat is that all stability runs are performed in the Ψ1=Ψ2 subspace: Eq. (5) is imposed, and the antisymmetric (relative-density/phase) sector of the binary system is not analyzed, so the claim of stable hopfions in binary gases is conditional on that subspace being dynamically robust. That is a completeness/correctness risk, not a circular reduction. Self-citations in the reference list are contextual and not load-bearing for the numerical stability derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

This is a numerical study, so it contributes few fitted constants. The main hidden assumptions are the reduction to a single-component GPE and the use of finite-time dynamics plus the VK criterion as a stability certificate. The geometric hopfion identification is standard topology, not a new postulate.

free parameters (2)
  • Ansatz width A = not stated
    Eq. (14); this width controls the localization of the initial guess and may influence which stationary branch the Newton solver finds.
  • Numerical grid parameters
    Grid size, step sizes, and convergence tolerances are not reported; these choices can affect the apparent size of stability windows.
assumptions (4)
  • standard math Hopf fibration S3 to S2 has π3(S2)=Z, and the linking number of preimage circles equals the Hopf number.
    Used in Sec. III.A-B (around Eq. (16)) to identify the states as hopfions and to read the petal structure as the Hopf number.
  • domain assumption The symmetric substitution Ψ1=Ψ2=Ψ/√2 in Eq. (5) reduces the binary GPEs to a single scalar GPE and remains valid under perturbed evolution.
    Sec. II, Eq. (5); all numerical results in Sec. III are obtained from the scalar equation, and the binary relative mode is never simulated.
  • domain assumption Direct simulation of perturbed evolution for up to about 10-40 ms, together with the Vakhitov-Kolokolov criterion, is sufficient to classify a solution as stable.
    Secs. III.A-III.B; the VK criterion is necessary but not sufficient, and no Bogoliubov-de Gennes spectra are presented.
  • domain assumption The Lee-Huang-Yang coefficient Γ in Eq. (4) and its dimensionless form γ correctly capture quantum fluctuations in the binary mixture.
    Standard LHY theory from Ref. [61] is invoked in Sec. II and assumed valid for the 39K parameters used.

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Cite this review

Pith. "Pith review of Stable hopfions in trapped quantum droplets." pith.science (2026). https://pith.science/paper/W45DHS37

@misc{pith2026250710910,
  author       = {Pith},
  title        = {Pith review of: Stable hopfions in trapped quantum droplets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W45DHS37}},
  note         = {Machine review of arXiv:2507.10910}
}
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 $Q_{H}$.

Figures

Figures reproduced from arXiv: 2507.10910 by the authors.

Figure 1
Figure 1. FIG. 1. An example of a stable hopfion with vorticity [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The total atom number [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The unstable evolution of a hopfion with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Stable hopfions with [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The atom number [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The unstable evolution of hopfions with [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

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Reference graph

Works this paper leans on

93 extracted references · 80 canonical work pages · cited by 1 Pith paper

  1. [1]

    Thomson, II

    W. Thomson, II. On vortex atoms, London, Edinburgh, Dublin Philos. Mag. J. Sci. 34(227), 15-24 (1867)

  2. [2]

    The to- tal number of atoms in the system is N = ∫ ( |Ψ1|2 + |Ψ2|2 ) d3R = ∫ |Ψ|2d3R

    (4) For symmetric states with Ψ1 = Ψ 2 ≡ Ψ/ √ 2, (5) coupled GPEs (2) and (3) admit the reduction to a single equation, iℏ ∂ ∂T Ψ = − ℏ2 2m ∇2 XYZΨ + δG 2 |Ψ|2 Ψ + Γ |Ψ|3 Ψ + 1 2mΩ2 [ (R −R0)2 +Z 2 ] Ψ, (6) 3 where δG = ( 4πℏ2/m ) (a′ +a) ≡ ( 4πℏ2/m ) δa. The to- tal number of atoms in the system is N = ∫ ( |Ψ1|2 + |Ψ2|2 ) d3R = ∫ |Ψ|2d3R. (7) The set of ...

  3. [3]

    LHY superfluid

    Orange dashed lines designate positions of pivots (phase singularities) of the internal vortex rings, as observed in the Y = 0 plane. means of the Newton-conjugate-gradient method, start- ing with the ansatz (initial guess) φ (r,z ) = (r′)S exp ( − (r′)2 A +iMθ +iSϕ ) , (14) whereA> 0 is a real constant and the special coordinates are defined as (r′,θ,ϕ ) ...

  4. [4]

    Faddeev and A

    L. Faddeev and A. J. Niemi, Stable knot-like structures in classical field theory, Nature 387(6628), 58-61 (1997)

    1997

  5. [5]

    Gladikowski and M

    J. Gladikowski and M. Hellmund, Static solitons with nonzero Hopf number, Phys. Rev. D 56(8), 5194-5199 (1997)

    1997

  6. [6]

    R. A. Battye, Knots as Stable Soliton Solutions in a Three-Dimensional Classical Field Theory, Phys. Rev. Lett. 81(22), 4798-4801 (1998)

    1998

  7. [7]

    Aratyn, L

    H. Aratyn, L. A. Ferreira, and A. H. Zimerman, Toroidal solitons in 3+1 dimensional integrable theories, Phys. Lett. B 456(2), 162-170 (1999)

    1999

  8. [8]

    Hietarinta and P

    J. Hietarinta and P. Salo, Faddeev-Hopf knots: dynamics of linked un-knots, Phys. Lett. B 451(1), 60-67 (1999)

    1999

Show all 93 references
  1. [9]

    Aratyn, L

    H. Aratyn, L. A. Ferreira, and A. H. Zimerman, Exact Static Soliton Solutions of ( 3 + 1 ) -Dimensional Inte- grable Theory with Nonzero Hopf Numbers, Phys. Rev. Lett. 83(9), 1723-1726 (1999)

    1999

  2. [10]

    R. S. Ward, Skyrmions and Faddeev-Hopf solitons, Phys. Rev. D 70(6), 061701 (2004)

    2004

  3. [11]

    Sutcliffe, Knots in the Skyrme-Faddeev model, Proc

    P. Sutcliffe, Knots in the Skyrme-Faddeev model, Proc. R. Soc. A. 463(2087), 3001-3020 (2007)

    2007

  4. [12]

    Radu and M

    E. Radu and M. S. Volkov, Stationary ring solitons in field theory – Knots and vortons, Phys. Rep. 468(4), 101- 151 (2008)

    2008

  5. [13]

    G¨ obel, I

    B. G¨ obel, I. Mertig, and O. A. Tretiakov, Beyond skyrmions: Review and perspectives of alternative mag- netic quasiparticles, Phys. Rep. 895(12), 1-28 (2021)

    2021

  6. [14]

    Sallermann, H

    M. Sallermann, H. J´ onsson, and S. Bl¨ ugel, Stability of hopfions in bulk magnets with competing exchange in- teractions, Phys. Rev. B 107(10), 104404 (2023)

    2023

  7. [15]

    Raftrey and P

    D. Raftrey and P. Fischer, Field-Driven Dynamics of Magnetic Hopfions, Phys. Rev. Lett. 127(25), 257201 (2021)

    2021

  8. [16]

    Y. Liu, W. Hou, X. Han, et al., Three-Dimensional Dy- namics of a Magnetic Hopfion Driven by Spin Transfer Torque, Phys. Rev. Lett. 124(12), 127204 (2020)

    2020

  9. [17]

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

    2015

  10. [18]

    Y. Shen, Q. Zhang, P. Shi, et al., Optical skyrmions and other topological quasiparticles of light, Nat. Pho- ton. 18(1), 15-28 (2024)

    2024

  11. [19]

    Sugic, R

    D. Sugic, R. Droop, E. Otte, et al., Particle-like topolo- gies in light, Nat. Commun. 12(1), 6785 (2021)

    2021

  12. [20]

    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)

    2017

  13. [21]

    Luk’yanchuk, Y

    I. Luk’yanchuk, Y. Tikhonov, A. Razumnaya, et al., Hop- fions emerge in ferroelectrics, Nat. Commun. 11(1), 2433 (2020)

    2020

  14. [22]

    P. J. Ackerman, Diversity of Knot Solitons in Liquid Crystals Manifested by Linking of Preimages in Torons and Hopfions, Phys. Rev. X 7(1), 011006 (2017)

    2017

  15. [23]

    Kleckner and W

    D. Kleckner and W. T. M. Irvine, Creation and dynamics of knotted vortices, Nat. Phys. 9(4), 253-258 (2013)

    2013

  16. [24]

    Bolognesi, Hopf Skyrmion in QCD with adjoint quarks, Phys

    S. Bolognesi, Hopf Skyrmion in QCD with adjoint quarks, Phys. Rev. D 75(6), 065020 (2007)

    2007

  17. [25]

    Ezawa, Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon’s knot, trefoil knot, and other linked nodal va- rieties, Phys

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

    2017

  18. [26]

    Sutcliffe, Hopfions in chiral magnets, J

    P. Sutcliffe, Hopfions in chiral magnets, J. Phys. A: Math. Theor. 51(37), 375401 (2018)

    2018

  19. [27]

    Kobayashi and M

    M. Kobayashi and M. Nitta, Torus knots as Hopfions, Phys. Lett. B 728(20), 314-318 (2014)

    2014

  20. [28]

    X. S. Wang, A. Qaiumzadeh, and A. Brataas, Current- Driven Dynamics of Magnetic Hopfions, Phys. Rev. Lett. 123(14), 147203 (2019)

    2019

  21. [29]

    X. Yu, Y. Liu, K. V. Iakoubovskii, et al., Realization and Current-Driven Dynamics of Fractional Hopfions and Their Ensembles in a Helimagnet FeGe, Adv. Mater. 35(20), 2210646 (2023)

    2023

  22. [30]

    Ehrmanntraut, R

    D. Ehrmanntraut, R. Droop, D. Sugic, et al., Optical second-order skyrmionic hopfion, Optica, 10(6), 725-731 (2023)

    2023

  23. [31]

    Wang and S

    H. Wang and S. Fan, Photonic Spin Hopfions and Monopole Loops, Phys. Rev. Lett. 131(26), 263801 (2023)

    2023

  24. [32]

    F. N. Rybakov, N. S. Kiselev, A. B. Borisov, et al., Mag- netic hopfions in solids, APL Materials 10(11), 111113 (2022)

    2022

  25. [33]

    C. Wan, Y. Shen, A. Chong, et al., Scalar optical hop- fions, eLight 2(1), 22 (2022)

    2022

  26. [34]

    Y. Shen, B. Yu, H. Wu, et al., Topological transforma- tion and free-space transport of photonic hopfions, Adv. Photon. 5(1), 015001 (2023). 9

    2023

  27. [35]

    Z. Lyu, Y. Fang, and Y. Liu, Formation and Controlling of Optical Hopfions in High Harmonic Generation, Phys. Rev. Lett. 133(13), 133801 (2024)

    2024

  28. [36]

    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)

    2015

  29. [37]

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

    2020

  30. [38]

    J.-S. B. Tai and I. I. Smalyukh, Three-dimensional crys- tals of adaptive knots, Science 365(6460), 1449-1453 (2019)

    2019

  31. [39]

    J.-S. B. Tai, P. J. Ackerman, and I. I. Smalyukh, Topolog- ical transformations of Hopf solitons in chiral ferromag- nets and liquid crystals, Proc. Natl. Acad. Sci. U.S.A. 115(5), 921-926 (2018)

    2018

  32. [40]

    H. R. O. Sohn, P. J. Ackerman, T. J. Boyle, et al., Dy- namics of topological solitons, knotted streamlines, and transport of cargo in liquid crystals, Phys. Rev. E 97(5), 052701 (2018)

    2018

  33. [41]

    Smalyukh, H

    I. Smalyukh, H. Zhao, and B. Malomed, Topological solitonic macromolecules, Nature Commun. 14(1), 4581 (2023)

    2023

  34. [42]

    Vilenkin and E

    A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 1994)

    1994

  35. [43]

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

    2024

  36. [44]

    Foster, Massive hopfions, Phys

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

    2011

  37. [45]

    Thompson, A

    A. Thompson, A. Wickes, J. Swearngin, et al., Classifi- cation of electromagnetic and gravitational hopfions by algebraic type, J. Phys. A: Math. Theor. 48(20), 205202 (2015)

    2015

  38. [46]

    Smo/suppress lka and J

    T. Smo/suppress lka and J. Jezierski, Simple description of gener- alized electromagnetic and gravitational hopfions, Class. Quantum Grav. 35(24), 245010 (2018)

    2018

  39. [47]

    Proment, M

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

    2012

  40. [48]

    H.-B. Luo, L. Li, and W.-M. Liu, Three-Dimensional Skyrmions with Arbitrary Topological Number in a Fer- romagnetic Spin-1 Bose-Einstein Condensate, Sci. Rep. 9(1), 18804 (2019)

    2019

  41. [49]

    Zou, W.-K

    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)

    2021

  42. [50]

    Kawaguchi, M

    Y. Kawaguchi, M. Nitta, and M. Ueda, Knots in a Spinor Bose-Einstein Condensate, Phys. Rev. Lett. 100(18), 180403 (2008)

    2008

  43. [51]

    D. S. Hall, M. W. Ray, K. Tiurev, et al., Tying quantum knots, Nat. Phys. 12(5), 478-483 (2016)

    2016

  44. [52]

    R. N. Bisset, W. Wang, C. Ticknor, R. Carretero- Gonz´ alez, et al., Robust vortex lines, vortex rings, and hopfions in three-dimensional Bose-Einstein condensates, Phys. Rev. A 92(6), 063611 (2015)

    2015

  45. [53]

    Ollikainen, S

    T. Ollikainen, S. Masuda, M. M¨ ott¨ onen, et al., Quantum knots in Bose-Einstein condensates created by counter- diabatic control, Phys. Rev. A 96(6), 063609 (2017)

    2017

  46. [54]

    Liu, S.-J

    Y.-K. Liu, S.-J. Yang, G.-H. Yang, et al., Interlocked knot in spinor Bose-Einstein condensates, Chaos, Solitons & Fractals 140, 110209 (2020)

    2020

  47. [55]

    Jackson, J

    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)

    1999

  48. [56]

    W. Wang, R. N. Bisset, C. Ticknor, et al., Single and multiple vortex rings in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics, Phys. Rev. A 95(4), 043638 (2017)

    2017

  49. [57]

    Ticknor, W

    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)

    2018

  50. [58]

    V. P. Ruban, W. Wang, C. Ticknor, et al., Instabilities of a vortex-ring-bright soliton in trapped binary three- dimensional Bose-Einstein condensates, Phys. Rev. A 105(1), 013319 (2022)

    2022

  51. [59]

    M. Abad, M. Guilleumas, R. Mayol, et al., Vortex rings in toroidal Bose-Einstein condensates, Laser Phys. 18, 648-662 (2008)

    2008

  52. [60]

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

    2020

  53. [61]

    Y. M. Bidasyuk, A. V. Chumachenko, O. O. Prikhodko, et al., Stable Hopf solitons in rotating Bose-Einstein con- densates, Phys. Rev. A 92(5), 053603 (2015)

    2015

  54. [62]

    Y. V. Kartashov, B. A. Malomed, Y. Shnir, et al., Twisted Toroidal Vortex Solitons in Inhomogeneous Media with Repulsive Nonlinearity, Phys. Rev. Lett. 113(26), 264101 (2014)

    2014

  55. [63]

    D. S. Petrov, Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture, Phys. Rev. Lett. 115(15), 155302 (2015)

    2015

  56. [64]

    D. S. Petrov and G. E. Astrakharchik, Ultradilute Low- Dimensional Liquids, Phys. Rev. Lett. 117(10), 100401 (2016)

    2016

  57. [65]

    N. B. Jørgensen, G. M. Bruun, and J. J. Arlt, Dilute Fluid Governed by Quantum Fluctuations, Phys. Rev. Lett. 121(17), 173403 (2018)

    2018

  58. [66]

    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- 1145 (1957)

    1957

  59. [67]

    Z. Luo, W. Pang, B. Liu, et al., A new form of liquid matter: Quantum droplets, Front. Phys. 16, 1-21 (2021)

    2021

  60. [68]

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

    2021

  61. [69]

    Y. Li, Z. Chen, Z. Luo, et al., Two-dimensional vortex quantum droplets, Phys. Rev. A 98(6), 063602 (2018)

    2018

  62. [70]

    G. Li, Z. Zhao, X. Jiang, et al., Strongly Anisotropic Vortices in Dipolar Quantum Droplets, Phys. Rev. Lett. 133(5), 053804 (2024)

    2024

  63. [71]

    Zhang, X

    X. Zhang, X. Xu, Y. Zheng, et al., Semidiscrete Quantum Droplets and Vortices, Phys. Rev. Lett. 123(13), 133901 (2019)

    2019

  64. [72]

    Y. V. Kartashov, B. A. Malomed, L. Tarruell, et al., Three-dimensional droplets of swirling superfluids, Phys. Rev. A 98(1), 013612 (2018)

    2018

  65. [73]

    L. Dong, Y. V. Kartashov, Rotating Multidimensional Quantum Droplets, Phys. Rev. Lett. 126(24), 244101 (2021)

    2021

  66. [74]

    Z. Lin, X. Xu, Z. Chen, et al., Two-dimensional vortex quantum droplets get thick, Commun. Nonlinear Sci. Nu- mer. Simul. 93, 105536 (2021)

    2021

  67. [75]

    T. A. Yo˘ gurt, U. Tanyeri, A. Kele¸ s, et al., Vortex lat- tices in strongly confined quantum droplets, Phys. Rev. 10 A 108(3), 033315 (2023)

    2023

  68. [76]

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

    2018

  69. [77]

    D’Errico, A

    C. D’Errico, A. Burchianti, M. Prevedelli, et al., Obser- vation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Research 1(3), 033155 (2019)

    2019

  70. [78]

    Ferrier-Barbut, H

    I. Ferrier-Barbut, H. Kadau, M. Schmitt, et al., Obser- vation of Quantum Droplets in a Strongly Dipolar Bose Gas, Phys. Rev. Lett. 116(21) 215301 (2016)

    2016

  71. [79]

    R. N. Bisset, L. A. P. Ardila, L. Santos, Quantum Droplets of Dipolar Mixtures, Phys. Rev. Lett. 126(2), 025301 (2021)

    2021

  72. [80]

    L. Dong, M. Fan, B. A. Malomed, Three-dimensional vor- tex and multipole quantum droplets in a toroidal poten- tial, Chaos, Solitons & Fractals 188 115499 (2024)

    2024

  73. [81]

    Hopf, Selecta Heinz Hopf: Herausgegeben zu seinem

    H. Hopf, Selecta Heinz Hopf: Herausgegeben zu seinem

  74. [82]

    Geburtstag von der Eidgen¨ ossischen Technischen Hochschule Z¨ urich (Springer, Berlin, Heidelberg, 1964)

    1964

  75. [83]

    D. W. Lyons, An Elementary Introduction to the Hopf Fibration, Math. Mag. 76(2), 87-98 (2003)

    2003

  76. [84]

    J. H. C. Whitehead, An Expression of Hopf’s Invariant as an Integral, Proc. Natl. Acad. Sci. U.S.A. 33(5), 117-123 (1947)

    1947

  77. [85]

    Faddeev, A

    L. Faddeev, A. J. Niemi, Partially Dual Variables in SU(2) Yang-Mills Theory, Phys. Rev. Lett. 82(8), 1624 (1999)

    1999

  78. [86]

    J¨ aykk¨ a, J

    J. J¨ aykk¨ a, J. Hietarinta, P. Salo, Topologically nontrivial configurations associated with Hopf charges investigated in the two-component Ginzburg-Landau model, Phys. Rev. B 77(9), 094509 (2008)

    2008

  79. [87]

    Babaev, L

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

    2002

  80. [88]

    J. M. Speight, Supercurrent coupling in the Faddeev- Skyrme model, J. Geom. Phys. 60(4), 599-610 (2010)

    2010

  81. [89]

    J¨ aykk¨ a, J

    J. J¨ aykk¨ a, J. M. Speight, Supercurrent coupling destabi- lizes knot solitons, Phys. Rev. D 84(12), 125035 (2011)

    2011

  82. [90]

    Z. Lyu, Y. Fang, Y. Liu, Formation and Controlling of Optical Hopfions in High Harmonic Generation, Phys. Rev. Lett. 133(13), 133801 (2024)

    2024

  83. [91]

    N. G. Vakhitov, A. A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity sat- uration, Radiophys. Quantum Electron. 16(7), 783-789 (1973)

    1973

  84. [92]

    Berg´ e, Wave collapse in physics: principles and appli- cations to light and plasma waves, Phys

    L. Berg´ e, Wave collapse in physics: principles and appli- cations to light and plasma waves, Phys. Rep. 303(5-6), 259-370 (1998)

    1998

  85. [93]

    T. G. Skov, M. G. Skou, N. B. Jørgensen, et al., Ob- servation of a Lee-Huang-Yang Fluid, Phys. Rev. Lett. 126(23), 230404 (2021)

    2021

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