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REVIEW 2 major objections 5 minor 52 references

Quantum Hopfion rings in the cluster mean-field approximation

T0 review · 2 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read Cluster mean-field calculations show hopfion rings host localized drops in magnetization length that classical micromagnetics cannot describe, so a regularized Landau-Lifshitz equation is required for their zero-mode motion.

desk verdict Solid, usable CMFA maps of |m|<1 on hopfion rings and kπ-skyrmions; the 3-D localization claim is a lower-bound result that still motivates regularized dynamics, even if full 3-D entanglement remains unchecked. read the letter →

arxiv 2607.09140 v1 pith:2S22OQIQ submitted 2026-07-10 cond-mat.mes-hall cond-mat.str-el

Vladyslav M. Kuchkin , Thomas L. Schmidt This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-el
keywords clustermean-fieldapproximationhopfionringskπ-skyrmionsquantumspintexturesmagnetizationlengthregularizedLandau-Lifshitzequationdensitymatrixrenormalizationgroupchiralmagnets
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

The paper develops a cluster mean-field method that uses the cylindrical symmetry of skyrmions and hopfion rings: radial spin chains are treated fully quantum-mechanically with density-matrix renormalization group, while azimuthal and axial couplings stay at the mean-field level. This combination lets the authors compute quantum fluctuations in large, metastable topological textures that standard methods cannot reach. The resulting maps show finite regions where the magnetization length falls below one, strongest on the hopfion rings themselves and roughly three times larger than on a pure skyrmion string. Those length variations lie outside ordinary micromagnetic theory, which forces unit length, and they also block the usual Landau-Lifshitz description of free sliding along the string. A recently proposed regularized continuum equation that allows the length to change restores the zero mode, giving both a practical quantum route to three-dimensional topological magnets and a concrete reason to extend classical continuum models.

What carries the argument

Cluster mean-field approximation with cylindrical symmetrization: one-dimensional radial chains are solved quantum-mechanically by DMRG while inter-cluster (azimuthal and z) interactions are replaced by self-consistent mean-field expectation values that respect axial rotation symmetry, reducing dimensionality and granting access to metastable states.

What would settle it

A full three-dimensional tensor-network calculation on a modest hopfion-ring geometry that finds either no localized |m|<1 regions on the rings or a qualitatively different radial entanglement pattern would falsify the claim that the cluster approach captures the essential quantum features.

Watch

Extended reading notes

Core claim

Within the cluster mean-field approximation the spatial profiles of magnetization length for hopfion rings exhibit finite-size regions where |m| < 1, concentrated on the rings and roughly three times stronger than the fluctuations of a pure skyrmion string. These features cannot be described by the classical micromagnetic model and show that an extended continuum description is required; the regularized Landau-Lifshitz equation based on an S^{3}-valued order parameter is compatible with the translational zero mode of such quantum hopfion rings.

Load-bearing premise

The method assumes entanglement is strongest along the radial direction, so azimuthal and axial couplings can be replaced by classical mean fields without losing the dominant quantum fluctuations.

Editorial extensions

If this is right

  • Quantum fluctuations in hopfion rings are localized on the rings and stronger than in pure skyrmion strings.
  • Classical micromagnetic models that enforce unit magnetization length miss essential spatial structure of these textures.
  • The regularized Landau-Lifshitz equation can describe zero-mode motion of quantum hopfions while the standard equation cannot.
  • Cluster mean-field supplies a systematic upper bound on energies (and lower bound on fluctuations) for large topological spin textures.
  • Metastable kπ-skyrmions and hopfion rings become computationally accessible for quantum study.

Reading between the lines

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

  • Full three-dimensional quantum simulations would likely show still larger magnetization-length reductions, so the present results already set a useful floor for experimental detection.
  • Materials hosting hopfion rings may exhibit local suppressions of magnetic moment detectable by high-resolution magnetometry.
  • The same radial-cluster construction can be applied to other axially symmetric textures such as skyrmion bags.
  • If the regularized dynamics is confirmed, field-pulse experiments could test continuum predictions of quantum hopfion sliding.
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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

2 major / 5 minor

Summary. The manuscript develops a cluster mean-field approximation (CMFA) that exploits cylindrical symmetry of kπ-skyrmions and hopfion rings: radial chains are treated fully quantum-mechanically with DMRG while azimuthal (and, in 3D, z) couplings are replaced by self-consistent mean fields obtained from a rotation of the local spin expectation values. Benchmarks against full 2D DMRG on square and disk lattices recover consistent magnetization profiles and the expected reduction of fluctuations under CMFA; the same framework is then applied to metastable kπ-skyrmions and to hopfion rings threaded on skyrmion strings. Spatial maps of 1−|m| show finite regions of reduced magnetization length that peak at the skyrmion radius or at the hopfion rings themselves. The authors argue that these regions render the ordinary Landau-Lifshitz equation incompatible with the translational zero mode of a hopfion ring, while the recently proposed regularized S^{3}-valued dynamics remains compatible.

Significance. The work supplies a practical route to quantum fluctuations in large, topologically nontrivial metastable textures that lie beyond the reach of exact diagonalization or unrestricted DMRG. The 2D benchmarks, bond-dimension and variance convergence documented in the Supplemental Material, and the public ITensor-based code constitute solid methodological contributions. The observation that CMFA already produces |m|<1 hotspots, together with the explicit compatibility check against the regularized micromagnetic equation, gives a concrete, falsifiable motivation for extending classical continuum models. Even if the quantitative localization of fluctuations in 3D remains approximate, the lower-bound character of CMFA and the qualitative necessity of |m| dynamics are robust.

major comments (2)
  1. In the section “Quantum Hopfion rings” and Supplemental Eqs. (11)–(12), both azimuthal and z-direction bonds are replaced by mean-field averages, so the clusters remain strictly one-dimensional radial chains. While axial symmetry and earlier 2D skyrmion studies justify this for planar textures, a hopfion is a genuinely three-dimensional linked object; missing inter-layer entanglement can in principle shift or delocalize the 1−|m| maxima shown in Fig. 3(f,g). Those spatial maps are the sole quantitative support for the claim that ordinary LL dynamics is incompatible with the zero mode while the regularized equation (5) is compatible. A controlled test (e.g., a small fully quantum 3D cluster, or a comparison with two-dimensional radial–z clusters) or an explicit statement that only the existence—not the precise location—of |m|<1 regions is claimed would strengthen the dynamical argument.
  2. The energy inequalities Eq ≤ ECMFA ≤ EMFA are asserted without a derivation or reference that covers the present non-ground-state, finite-temperature-free setting. Because the hopfion rings are metastable, a short justification (or citation) that the variational character survives the self-consistent mean-field loop and the imposed classical boundary spins would remove any ambiguity about the claimed lower bound on fluctuations.
minor comments (5)
  1. Typographical inconsistency: the abbreviation appears as “CFM”, “CFMA” and “CMFA” in the main text and figure captions; standardize to CMFA.
  2. Fig. 1(f) caption and main text refer to “panels (f) and (g)” while only (f) is present; renumber or restore the missing panel.
  3. In the hopfion section the phrase “approximately three times larger” is given without error bars or a precise definition of the averaging region; a short quantitative statement (peak value or integrated 1−|m|) would improve reproducibility.
  4. Supplemental Material III shows variance and energy versus bond dimension only for the 2D skyrmion; a corresponding plot for the 1D CMFA chains used for hopfions would complete the convergence documentation.
  5. Equation (5) introduces the phenomenological parameter ε without stating the range explored or whether the qualitative zero-mode motion is independent of its value; a one-sentence clarification would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: CMFA |m| profiles are independent DMRG computations; regularized LL is applied only as a compatibility check, not as a self-defining prediction.

full rationale

The paper’s load-bearing results are the self-consistent CMFA magnetization-length maps for kπ-skyrmions and hopfion rings (Figs. 1–3). Those maps are obtained by (i) writing the chiral Heisenberg model in cylindrical coordinates, (ii) retaining radial bonds as quantum operators while replacing azimuthal and z bonds by mean-field averages that respect axial symmetry, and (iii) iterating DMRG until the expectation values converge. This is the ordinary CMFA fixed-point loop; the target observable 1-|m| is not inserted by definition or by a fit. The subsequent dynamical argument is a one-line compatibility check: the classical LL equation conserves |m| by construction (∂t|m|²=0), so it cannot transport the |m|<1 regions found by CMFA, whereas the regularized S³ dynamics of Refs. [50,51] (already published) allows ∂t|m|≠0 when the phenomenological parameter ε is nonzero. No parameter is fitted to data and then re-labeled a prediction; no uniqueness theorem is imported to forbid alternatives; the radial-cluster ansatz is an explicit approximation justified by symmetry and earlier 2-D studies, not a circular reduction. Self-citations of the authors’ prior regularized-micromagnetics papers are sequential research, not load-bearing circularity. Score 0 is therefore the correct assessment.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central numerical claims rest on the standard Heisenberg + bulk DMI Hamiltonian, the CMFA factorization, and the assumption that radial entanglement dominates. Free parameters are the usual material constants (h, u) and discretization choices; no new particles or forces are postulated. The regularized LL equation is imported from earlier work of overlapping authors but is used only for qualitative compatibility.

free parameters (3)
  • external field h and anisotropy u = h=0.4–0.85, u=0–0.3
    Chosen by hand (h=0.85 for skyrmions, h=0.65 for kπ-skyrmions, h=0.4/u=0.3 for hopfion rings) to stabilize the desired metastable textures; results are therefore parameter-specific.
  • radial/azimuthal/z discretizations δr, δϕ, δz = δr=1/16 etc.
    Finite-difference spacings that control continuum limit; set to 1/16 in the hopfion runs.
  • DMRG bond dimension = up to 256
    Truncation parameter whose convergence is checked but still finite.
assumptions (4)
  • domain assumption CMFA factorization: inter-cluster operators replaced by products of operators and expectation values
    Standard cluster mean-field approximation; invoked throughout Sec. 'Skyrmion in the CMFA' and for the 3-D hopfion clusters.
  • domain assumption Axial symmetry implies that neighboring clusters are related by a simple rotation matrix R(±δϕ)
    Used to close the mean-field equations (Eq. 4); justified by classical continuum limit and earlier 2-D results.
  • ad hoc to paper Entanglement is strongest along the radial direction
    Explicit modeling choice that reduces the problem to 1-D chains; supported by citations to 2-D skyrmion studies but not independently verified for hopfion rings.
  • standard math Energy ordering Eq ≤ E_CMFA ≤ E_MFA holds for the studied spin-1/2 systems
    Follows from variational arguments once entanglement is neglected; used to interpret CMFA as a bound.

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Pith. "Pith review of Quantum Hopfion rings in the cluster mean-field approximation." pith.science (2026). https://pith.science/paper/2S22OQIQ

@misc{pith2026260709140,
  author       = {Pith},
  title        = {Pith review of: Quantum Hopfion rings in the cluster mean-field approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2S22OQIQ}},
  note         = {Machine review of arXiv:2607.09140}
}
abstract

We study the quantum properties of two- and three-dimensional spin textures -- $k\pi$-skyrmions and hopfion rings -- within the cluster mean-field approximation (CMFA). By combining the CMFA with a symmetrization procedure, we achieve two key advances: the accurate computation of quantum fluctuations in large spin textures and reliable access to metastable states. These challenges are generally insurmountable using standard methods, which are severely limited by the curse of dimensionality and typically restricted to ground-state properties. Exploiting the cylindrical symmetry of the studied magnetic configurations, we construct one-dimensional chain-like clusters that can be efficiently simulated using the density matrix renormalization group method, while inter-cluster interactions are treated at the mean-field level. The resulting spatial profiles of quantum features such as the local variation of the magnetization length in hopfion rings reveal limitations of the classical micromagnetic model and indicate the necessity of its extension. We demonstrate that the recently proposed regularized micromagnetic equation provides a suitable framework for this purpose.

Figures

Figures reproduced from arXiv: 2607.09140 by the authors.

Figure 1
Figure 1. FIG. 1. Results of 2D DMRG and CMFA simulations for a quantum skyrmion. The positions of quantum (red) and classical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. (f), (g)] must move as well. However, the recently introduced regularized LL equa￾tion [50] for the motion of Bloch points can describe the zero mode for quantum hopfion rings. Following the ap￾proach in Ref. [51], we introduce an S 3 -order parameter ν, with |ν| = 1, which generalizes the magnetization vec- [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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Works this paper leans on

52 extracted references · 23 canonical work pages

  1. [1]

    Tokura and N

    Y. Tokura and N. Kanazawa, Magnetic skyrmion mate- rials, Chemical Reviews121, 2857–2897 (2020)

    2020

  2. [2]

    K. M. Song, J.-S. Jeong, B. Pan, X. Zhang, J. Xia, S. Cha, T.-E. Park, K. Kim, S. Finizio, J. Raabe, J. Chang, Y. Zhou, W. Zhao, W. Kang, H. Ju, and S. Woo, Skyrmion-based artificial synapses for neuromor- phic computing, Nature Electronics3, 148–155 (2020)

    2020

  3. [3]

    Psaroudaki and C

    C. Psaroudaki and C. Panagopoulos, Skyrmion qubits: A new class of quantum logic elements based on nanoscale magnetization, Physical Review Letters127, 10.1103/physrevlett.127.067201 (2021)

    work page doi:10.1103/physrevlett.127.067201 2021

  4. [4]

    J. Xia, X. Zhang, X. Liu, Y. Zhou, and M. Ezawa, Univer- sal quantum computation based on nanoscale skyrmion helicity qubits in frustrated magnets, Physical Review Letters130, 10.1103/physrevlett.130.106701 (2023)

    work page doi:10.1103/physrevlett.130.106701 2023

  5. [5]

    A. P. Petrovi´ c, C. Psaroudaki, P. Fischer, M. Garst, and C. Panagopoulos, Colloquium : Quantum properties and functionalities of magnetic skyrmions, Reviews of Mod- ern Physics97, 10.1103/revmodphys.97.031001 (2025)

    doi:10.1103/revmodphys.97.031001 2025

  6. [6]

    E. M. Chudnovsky and D. A. Garanin, Magnetic skyrmion as Schr¨ odinger’s cat, Europhysics Letters (2025)

    2025

  7. [7]

    Juge, S.-G

    R. Juge, S.-G. Je, D. d. S. Chaves, L. D. Buda-Prejbeanu, J. Pe˜ na-Garcia, J. Nath, I. M. Miron, K. G. Rana, L. Aballe, M. Foerster, F. Genuzio, T. O. Mente¸ s, A. Lo- catelli, F. Maccherozzi, S. S. Dhesi, M. Belmeguenai, Y. Roussign´ e, S. Auffret, S. Pizzini, G. Gaudin, J. Vo- gel, and O. Boulle, Current-driven skyrmion dynamics and drive-dependent sky...

    doi:10.1103/physrevap- 2019

  8. [8]

    Tengdin, B

    P. Tengdin, B. Truc, A. Sapozhnik, L. Kong, N. del Ser, S. Gargiulo, I. Madan, T. Sch¨ onenberger, P. R. Baral, P. Che, A. Magrez, D. Grundler, H. M. Rønnow, T. La- grange, J. Zang, A. Rosch, and F. Carbone, Imaging the ultrafast coherent control of a skyrmion crystal, Physical Review X12, 10.1103/physrevx.12.041030 (2022)

    work page doi:10.1103/physrevx.12.041030 2022

Show all 52 references
  1. [9]

    Hirosawa, A

    T. Hirosawa, A. Mook, J. Klinovaja, and D. Loss, Mag- netoelectric cavity magnonics in skyrmion crystals, PRX Quantum3, 10.1103/prxquantum.3.040321 (2022)

    2022 doi

  2. [10]

    Lemesh, K

    I. Lemesh, K. Litzius, M. B¨ ottcher, P. Bassirian, N. Kerber, D. Heinze, J. Z´ azvorka, F. B¨ uttner, L. Caretta, M. Mann, M. Weigand, S. Finizio, J. Raabe, M. Im, H. Stoll, G. Sch¨ utz, B. Dup´ e, M. Kl¨ aui, and G. S. D. Beach, Current-induced skyrmion generation through mo...

    2018 doi

  3. [11]

    Vi˜ nas Bostr¨ om, A

    E. Vi˜ nas Bostr¨ om, A. Rubio, and C. Verdozzi, Micro- scopic theory of light-induced ultrafast skyrmion excita- tion in transition metal films, npj Computational Mate- rials8, 10.1038/s41524-022-00735-5 (2022)

    2022 doi

  4. [12]

    Feldtkeller, Mikromagnetisch stetige und unstetige magnetisierungskonfigurationen, Zeitschrift f¨ ur Ange- wandte Physik19, 530 (1965)

    E. Feldtkeller, Mikromagnetisch stetige und unstetige magnetisierungskonfigurationen, Zeitschrift f¨ ur Ange- wandte Physik19, 530 (1965)

    1965

  5. [13]

    D¨ oring, Point singularities in micromagnetism, Jour- nal of Applied Physics39, 1006 (1968)

    W. D¨ oring, Point singularities in micromagnetism, Jour- nal of Applied Physics39, 1006 (1968)

    1968

  6. [14]

    X. Chen, D. Yang, Z. Li, J. Guo, H. Wang, Y. Hu, V. M. Kuchkin, A. S. Savchenko, H. Zhang, B. Ding, Z. Hou, W. Shi, F. N. Rybakov, O. Eriksson, S. Bl¨ ugel, Y. Han, R. E. Dunin-Borkowski, N. S. Kiselev, X. Fu, and F. Zheng, Laser-induced nucleation of magnetic hopfions, Nature...

    2026

  7. [15]

    Foster, C

    D. Foster, C. Kind, P. J. Ackerman, J.-S. B. Tai, M. R. Dennis, and I. I. Smalyukh, Two-dimensional skyrmion bags in liquid crystals and ferromagnets, Nature Physics 6 15, 655–659 (2019)

    2019

  8. [16]

    L. Kern, V. M. Kuchkin, V. Deinhart, C. Klose, T. Sidiropoulos, M. Auer, S. Gaebel, K. Gerlinger, R. Battistelli, S. Wittrock, T. Karaman, M. Schneider, C. M. G¨ unther, D. Engel, I. Will, S. Wintz, M. Weigand, F. B¨ uttner, K. H¨ oflich, S. Eisebitt, and B. Pfau, Con- trolled...

    2025 doi

  9. [17]

    Zheng, N

    F. Zheng, N. S. Kiselev, F. N. Rybakov, L. Yang, W. Shi, S. Bl¨ ugel, and R. E. Dunin-Borkowski, Hopfion rings in a cubic chiral magnet, Nature623, 718–723 (2023)

    2023

  10. [18]

    X. Chen, D. Song, F. N. Rybakov, N. S. Kiselev, L. Li, W. Shi, R. Wu, X. Fu, O. Eriksson, S. Bl¨ ugel, R. E. Dunin-Borkowski, H. Du, and F. Zheng, Electric-current- assisted nucleation of zero-field hopfion rings, Advanced Materials38, 10.1002/adma.202523417 (2026)

    2026 doi

  11. [19]

    M¨ uhlbauer, B

    S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B¨ oni, Skyrmion lattice in a chiral magnet, Science323, 915–919 (2009)

    2009

  12. [20]

    Lancaster, F

    T. Lancaster, F. Xiao, Z. Salman, I. O. Thomas, S. J. Blundell, F. L. Pratt, S. J. Clark, T. Prokscha, A. Suter, S. L. Zhang, A. A. Baker, and T. Hesjedal, Transverse field muon-spin rotation measurement of the topological anomaly in a thin film of mnsi, Physical Review B93, 1...

    2016 doi

  13. [21]

    vortices

    A. N. Bogdanov and D. A. Yablonskii, Thermodynami- cally stable “vortices” in magnetically ordered crystals. the mixed state of magnets, Sov. Phys. JETP68(1989)

    1989

  14. [22]

    Bogdanov and A

    A. Bogdanov and A. Hubert, The properties of isolated magnetic vortices, physica status solidi (b)186, 527 (1994)

    1994

  15. [23]

    F. N. Rybakov and N. S. Kiselev, Chiral magnetic skyrmions with arbitrary topological charge, Physical Re- view B99, 10.1103/physrevb.99.064437 (2019)

    2019 doi

  16. [24]

    V. M. Kuchkin, B. Barton-Singer, F. N. Rybakov, S. Bl¨ ugel, B. J. Schroers, and N. S. Kiselev, Mag- netic skyrmions, chiral kinks, and holomorphic functions, Physical Review B102, 10.1103/physrevb.102.144422 (2020)

    2020 doi

  17. [26]

    V. M. Kuchkin, N. S. Kiselev, F. N. Rybakov, I. S. Lobanov, S. Bl¨ ugel, and V. M. Uzdin, Heliknoton in a film of cubic chiral magnet, Frontiers in Physics11, 10.3389/fphy.2023.1201018 (2023)

    2023 doi

  18. [27]

    V. M. Kuchkin and N. S. Kiselev, Homotopy tran- sitions and 3d magnetic solitons, APL Materials10, 10.1063/5.0097559 (2022)

    2022 doi

  19. [28]

    Yamamoto, Correlated cluster mean-field theory for spin systems, Physical Review B79, 10.1103/phys- revb.79.144427 (2009)

    D. Yamamoto, Correlated cluster mean-field theory for spin systems, Physical Review B79, 10.1103/phys- revb.79.144427 (2009)

    2009 doi

  20. [29]

    H. A. Bethe, Statistical theory of superlattices, Proceed- ings of the Royal Society of London. Series A - Mathe- matical and Physical Sciences150, 552–575 (1935)

    1935

  21. [30]

    Peierls, On ising’s model of ferromagnetism, Mathe- matical Proceedings of the Cambridge Philosophical So- ciety32, 477–481 (1936)

    R. Peierls, On ising’s model of ferromagnetism, Mathe- matical Proceedings of the Cambridge Philosophical So- ciety32, 477–481 (1936)

    1936

  22. [31]

    P. R. Weiss, The application of the bethe-peierls method to ferromagnetism, Physical Review74, 1493–1504 (1948)

    1948

  23. [32]

    O. M. Sotnikov, V. V. Mazurenko, J. Colbois, F. Mila, M. I. Katsnelson, and E. A. Stepanov, Probing the topol- ogy of the quantum analog of a classical skyrmion, Physi- cal Review B103, 10.1103/physrevb.103.l060404 (2021)

    2021 doi

  24. [33]

    Haller, S

    A. Haller, S. Groenendijk, A. Habibi, A. Michels, and T. L. Schmidt, Quantum skyrmion lattices in heisenberg ferromagnets, Physical Review Research4, 10.1103/physrevresearch.4.043113 (2022)

    2022 doi

  25. [34]

    V. V. Mazurenko, I. A. Iakovlev, O. M. Sotnikov, and M. I. Katsnelson, Estimating patterns of classical and quantum skyrmion states, Journal of the Physical Society of Japan92, 10.7566/jpsj.92.081004 (2023)

    2023 doi

  26. [35]

    Salvati, M

    F. Salvati, M. I. Katsnelson, A. A. Bagrov, and T. West- erhout, Stability of a quantum skyrmion: Projective mea- surements and the quantum zeno effect, Physical Review B109, 10.1103/physrevb.109.064409 (2024)

    2024 doi

  27. [36]

    Liscak, A

    S. Liscak, A. Haller, A. Michels, T. L. Schmidt, and V. M. Kuchkin, Skyrmionic schr¨ odinger cat states in monoaxial chiral magnets, Physical Review B113, 10.1103/842q- ghyt (2026)

    2026 doi

  28. [37]

    S. R. White, Density matrix formulation for quantum renormalization groups, Physical Review Letters69, 2863–2866 (1992)

    1992

  29. [38]

    R´ ozsa, D

    L. R´ ozsa, D. Wuhrer, S. A. D´ ıaz, U. Nowak, and W. Belzig, Hidden quantum correlations in the ground states of quasiclassical spin systems, Physical Review B 111, 10.1103/physrevb.111.174441 (2025)

    2025 doi

  30. [39]

    Nishimura and R

    K. Nishimura and R. Yoshii, Inhomogeneous entangle- ment structure in monoaxial chiral ferromagnetic quan- tum spin chain (2025)

    2025

  31. [40]

    Millard and H

    K. Millard and H. S. Leff, Infinite-spin limit of the quan- tum heisenberg model, Journal of Mathematical Physics 12, 1000–1005 (1971)

    1971

  32. [41]

    E. H. Lieb, The classical limit of quantum spin systems, Communications in Mathematical Physics31, 327–340 (1973)

    1973

  33. [42]

    Donahue and R

    M. Donahue and R. McMichael, Exchange energy repre- sentations in computational micromagnetics, Physica B: Condensed Matter233, 272–278 (1997)

    1997

  34. [43]

    Romming, C

    N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Writing and deleting single magnetic skyrmions, Science341, 636–639 (2013)

    2013

  35. [44]

    Takashima, H

    R. Takashima, H. Ishizuka, and L. Balents, Quantum skyrmions in two-dimensional chiral magnets, Physical Review B94, 10.1103/physrevb.94.134415 (2016)

    2016 doi

  36. [45]

    A. S. Savchenko, V. M. Kuchkin, F. N. Rybakov, S. Bl¨ ugel, and N. S. Kiselev, Chiral standing spin waves in skyrmion lattice, APL Materials10, 10.1063/5.0097651 (2022)

    2022 doi

  37. [46]

    Sallermann, B

    M. Sallermann, B. Zimmermann, F. Lux, and S. Bl¨ ugel, The Skyrmion Radius Calculator (SKM 2021, Virtual (Germany), 27 Sep 2021 - 1 Oct 2021, 2021)

    2021

  38. [47]

    Komineas, C

    S. Komineas, C. Melcher, and S. Venakides, Chiral mag- netic skyrmions across length scales, New Journal of Physics25, 023013 (2023)

    2023

  39. [48]

    Azhar, S

    M. Azhar, S. C. Shaju, R. Knapman, A. Pignedoli, and K. Everschor-Sitte, 3d magnetic textures with mixed topology: Unlocking the tunable hopf index (2024)

    2024

  40. [49]

    V. M. Kuchkin, B. Barton-Singer, P. F. Bessarab, and N. S. Kiselev, Symmetry-governed dynamics of magnetic skyrmions under field pulses, Communications Physics8, 10.1038/s42005-024-01913-1 (2025)

    2025 doi

  41. [50]

    V. M. Kuchkin, A. Haller, A. Michels, T. L. Schmidt, and N. S. Kiselev, Regularized micromagnetic the- 7 ory for bloch points, Communications Physics9, 10.1038/s42005-026-02565-z (2026)

    2026 doi

  42. [51]

    V. M. Kuchkin, A. Haller, S. Liscak, M. P. Adams, V. Rai, E. P. Sinaga, A. Michels, and T. L. Schmidt, Quantum and classical magnetic bloch points, Physical Review Research7, 10.1103/physrevresearch.7.013195 (2025)

    2025 doi

  43. [52]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    2022

  44. [53]

    Quantum Hopfion rings in the cluster mean-field approximation

    M. B. Hastings, An area law for one-dimensional quan- tum systems, Journal of Statistical Mechanics: Theory and Experiment2007, P08024–P08024 (2007). 8 Supplemental Material for “Quantum Hopfion rings in the cluster mean-field approximation” I. HAMIL TONIAN DIMENSIONALIZA TION...

    2007

Pith tools

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