Quadrature skyrmions in two-dimensionally arrayed parametric resonators

Daiki Hatanaka; Hiroshi Yamaguchi; Motoki Asano

arxiv: 2304.05018 · v2 · pith:MPEYZO4Lnew · submitted 2023-04-11 · ❄️ cond-mat.mes-hall · physics.app-ph

Quadrature skyrmions in two-dimensionally arrayed parametric resonators

Hiroshi Yamaguchi , Daiki Hatanaka , Motoki Asano This is my paper

Pith reviewed 2026-05-24 08:54 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.app-ph

keywords skyrmionparametric resonatorquadrature variablechiral exchangepiezoelectric membraneacoustic devicetopological solitonfinite-element simulation

0 comments

The pith

Parametric coupling of quadrature variables stabilizes skyrmions in resonator arrays

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

The paper proposes that quadrature variables in parametric resonators can supply the chiral exchange interaction required to stabilize skyrmions, which has been difficult to implement with vector variables in artificial arrays. Finite-element simulation of a piezoelectric membrane array shows that acoustic skyrmions form under this coupling. A sympathetic reader would care because the approach opens a route to topological solitons in acoustic, photonic, and electric resonator systems without needing conventional magnetic or spin interactions. The central object is the effective chiral exchange generated by the parametric term between the two quadrature amplitudes.

Core claim

Quadrature variables whose parametric coupling produces an effective chiral exchange interaction stabilize skyrmions in two-dimensionally arrayed parametric resonators; a finite-element simulation indicates that an acoustic skyrmion exists in a realistic piezoelectric membrane array.

What carries the argument

Parametric coupling between quadrature variables that generates an effective chiral exchange interaction

If this is right

Skyrmions become realizable and controllable in artificial acoustic resonator arrays.
The same quadrature mechanism applies to photonic and electric resonator systems.
Novel devices based on acoustic skyrmions can be designed using standard piezoelectric fabrication.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may allow skyrmion lattices or dynamics to be studied in purely mechanical systems without magnetic fields.
Tuning the parametric drive frequency and amplitude offers an experimental knob to switch skyrmion stability on and off.
Similar quadrature coupling could be tested in other two-dimensional resonator lattices to search for additional topological textures.

Load-bearing premise

The parametric coupling between quadrature variables can be realized and tuned to produce an effective chiral exchange interaction sufficient to stabilize skyrmions.

What would settle it

A finite-element simulation of the proposed piezoelectric membrane array that yields no skyrmion-like displacement pattern when the quadrature coupling strength is set to the reported value would falsify the existence claim.

read the original abstract

Skyrmions are topological solitons in two-dimensional systems and have been observed in various physical systems. Generating and controlling skyrmions in artificial resonator arrays lead to novel acoustic, photonic, and electric devices, but it is a challenge to implement a vector variable with the chiral exchange interaction. Here, we propose to use quadrature variables, where their parametric coupling enables skyrmions to be stabilized. A finite-element simulation indicates that a acoustic skyrmion would exist in a realistic structure consisting of a piezoelectric membrane array.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The simulation indicates skyrmions form under the assumed parametric coupling, but the paper does not derive the required chiral interaction from the resonator equations.

read the letter

The paper's main point is a proposal to stabilize skyrmions in 2D resonator arrays by treating the oscillators with quadrature variables whose parametric coupling supplies an effective chiral term. A finite-element simulation of a piezoelectric membrane array then shows an acoustic skyrmion appearing under those conditions. That combination is what is new: prior work on skyrmions in artificial systems had trouble getting both vector character and the right exchange interaction at once, and this route uses the parametric drive to address it directly. The simulation itself is concrete and tied to a realistic structure, which gives the claim some grounding beyond pure theory. The soft spot is exactly where the stress-test flagged it. The abstract introduces the parametric coupling as enabling the chiral interaction but supplies no derivation from the driven-oscillator equations to the effective Dzyaloshinskii-Moriya term. If the FEM model simply inserts a phenomenological coefficient rather than obtaining it from the drive, the result only shows that skyrmions exist once that term is present, not that the proposed mechanism produces it. That step is central, so the gap matters. The rest of the setup looks standard for the subfield with no obvious circularity or invented entities beyond the named quadrature skyrmion. This is for people working on topological solitons in acoustic or photonic resonator arrays who want a new stabilization route. A reader already thinking about parametric drives or membrane arrays would get the most from it. The work is coherent enough on its own terms to deserve peer review; the derivation issue is fixable in revision and the simulation provides a starting point worth checking.

Referee Report

2 major / 1 minor

Summary. The paper proposes stabilizing skyrmions in two-dimensional arrays of parametric resonators by employing quadrature variables, whose parametric coupling is claimed to generate an effective chiral exchange interaction. This is supported by a finite-element simulation indicating the existence of an acoustic skyrmion in a realistic piezoelectric membrane array.

Significance. If the mechanism is shown to follow from the resonator equations and the simulation is fully specified, the approach could enable topological solitons in acoustic and related resonator systems, offering a route to novel devices. The work is a simulation-based existence proof rather than a closed analytical derivation.

major comments (2)

[Proposal of quadrature variables and parametric coupling] The manuscript introduces parametric coupling of quadrature variables as enabling an effective chiral (Dzyaloshinskii-Moriya-like) interaction but provides no derivation of this term from the underlying driven-oscillator equations. This step is load-bearing for the central claim, as the FEM result then only confirms skyrmion stability under the inserted assumption rather than demonstrating that the proposed drive produces the required interaction.
[Finite-element simulation section] The FEM simulation parameters (including the value and origin of any chiral coefficient, mesh details, boundary conditions, and drive amplitudes), validation against limiting cases, and criteria used to identify the skyrmion are not described. This gap prevents assessment of whether the reported structure actually realizes the claimed stabilization.

minor comments (1)

[Abstract] Abstract contains the grammatical error 'a acoustic skyrmion' (should be 'an acoustic skyrmion').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important gaps in the presentation. We will revise the manuscript to include an explicit derivation of the effective chiral interaction from the driven-oscillator equations and to fully specify the finite-element simulation details. Point-by-point responses follow.

read point-by-point responses

Referee: The manuscript introduces parametric coupling of quadrature variables as enabling an effective chiral (Dzyaloshinskii-Moriya-like) interaction but provides no derivation of this term from the underlying driven-oscillator equations. This step is load-bearing for the central claim, as the FEM result then only confirms skyrmion stability under the inserted assumption rather than demonstrating that the proposed drive produces the required interaction.

Authors: We agree that the derivation is essential and was omitted. In the revised manuscript we will add a dedicated section deriving the effective chiral exchange term directly from the parametrically driven oscillator equations for the quadrature variables, showing how the parametric coupling generates the required interaction without additional assumptions. This will establish that the stabilization follows from the proposed drive. revision: yes
Referee: The FEM simulation parameters (including the value and origin of any chiral coefficient, mesh details, boundary conditions, and drive amplitudes), validation against limiting cases, and criteria used to identify the skyrmion are not described. This gap prevents assessment of whether the reported structure actually realizes the claimed stabilization.

Authors: We acknowledge the simulation section lacked sufficient detail. The revised manuscript will include a complete specification of all parameters (chiral coefficient and its origin, mesh, boundary conditions, drive amplitudes), validation against limiting cases, and the precise criteria used to identify the skyrmion structure. This will enable independent assessment of the results. revision: yes

Circularity Check

0 steps flagged

No circularity; simulation-based existence claim is independent of inputs

full rationale

The paper proposes quadrature variables whose parametric coupling stabilizes skyrmions and supports the claim via finite-element simulation of a piezoelectric membrane array. No derivation, equation, or self-citation is shown that reduces the simulated skyrmion to a fitted parameter, renamed input, or self-referential ansatz by construction. The simulation functions as an external numerical check under the stated model assumptions rather than a tautological prediction, making the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The proposal rests on the domain assumption that parametric coupling can substitute for chiral exchange, plus the standard mathematical definition of skyrmions as topological solitons; no free parameters or new entities with independent evidence are introduced beyond the simulation result.

axioms (1)

domain assumption Skyrmions require a chiral exchange interaction for stabilization in two-dimensional systems
Explicitly stated as the implementation challenge that the quadrature approach is meant to solve.

invented entities (1)

quadrature skyrmion no independent evidence
purpose: Topological soliton stabilized by parametric coupling of quadrature variables in resonator arrays
New concept introduced to enable skyrmions without traditional chiral materials; no independent falsifiable evidence supplied outside the simulation.

pith-pipeline@v0.9.0 · 5611 in / 1128 out tokens · 26056 ms · 2026-05-24T08:54:49.599145+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages

[1]

Manton and P

N. Manton and P. Sutcliffe, Topological Solitons (Cmbridge University Press, Cambridge, United Kingdom, 2004)

work page 2004
[2]

Dauxois and M

T. Dauxois and M. Peyrard, Physics of Solitons (Cmbridge University Press, Cambridge, United Kingdom, 2006)

work page 2006
[3]

Rubinstein, Sine‐Gordon Equation, J

J. Rubinstein, Sine‐Gordon Equation, J. Math. Phys. 11, 258 (1970)

work page 1970
[4]

T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962)

work page 1962
[5]

A. N. Bogdanov and U. K. Rossler, Chiral symmetry breaking in magnetic thin films and multilayers, Phys. Rev. Lett. 87, 037203 (2001)

work page 2001
[6]

U. K. Rossler, N. Bogdanov, and C. Pfleiderer, Spontaneous skyrmion ground states in magnetic materials, Nature 442, 797 (2006)

work page 2006
[7]

Muhlbauer et al., Skyrmion lattice in a chiral magnet, Science 323, 915 (2009)

S. Muhlbauer et al., Skyrmion lattice in a chiral magnet, Science 323, 915 (2009)

work page 2009
[8]

Nagaosa and Y

N. Nagaosa and Y . Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotech. 8, 899 (2013)

work page 2013
[9]

A. Fert, N. Reyren, and V . Cros, Magnetic skyrmions: advances in physics and potential applications, Nat. Rev. Mater. 2, 17031 (2017)

work page 2017
[10]

Gobel, I

B. Gobel, I. Mertig, O. A. Tretiakov, Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles, Physics Reports 895, 1 (2021)

work page 2021
[11]

Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J

I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958)

work page 1958
[12]

Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys

T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960)

work page 1960
[13]

Fert and P

A. Fert and P. M. Levy, Role of anisotropic exchange interactions in determining the properties of spin-glasses, Phys. Rev. Lett. 44, 1538 (1980). 11

work page 1980
[14]

Jonietz et al., Spin transfer torques in MnSi at ultralow current densities, Science 330, 1648 (2010)

F. Jonietz et al., Spin transfer torques in MnSi at ultralow current densities, Science 330, 1648 (2010)

work page 2010
[15]

X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose, and Y . Tokura, Skyrmion flow near room temperature in an ultralow current density, Nature Commun. 3, 988 (2012)

work page 2012
[16]

A. Fert, V . Cros, and J. Sampaio, Skyrmions on the track, Nat. Nanotechnol. 8, 152 (2013)

work page 2013
[17]

Tomasello, E

R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio, A strategy for the design of skyrmion racetrack memories, Sci. Rep. 4, 6784 (2014)

work page 2014
[18]

Zhang, M

X. Zhang, M. Ezawa, and Y . Zhou, Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions, Sci. Rep. 5, 9400 (2015)

work page 2015
[19]

Finocchio, M

G. Finocchio, M. Ricci, R. Tomasello, A. Giordano, M. Lanuzza, V . Puliafito, P . Burrascano, B. Azzerboni, and M. Carpentieri, Skyrmion based microwave detectors and harvesting, Appl. Phys. Lett. 107, 262401 (2015)

work page 2015
[20]

Garcia-Sanchez, N

F. Garcia-Sanchez, N. Reyren, J. Sampaio, V . Cros, and J.-V . Kim, Askyrmion- based spin-torque nano-oscillator, New J. Phys. 18, 075011 (2016)

work page 2016
[21]

K. Wang, Y . Huang, X. Zhang, and W. Zhao, Skyrmion-electronics: an overview and outlook, Proc. IEEE 140, 2040 (2016)

work page 2040
[22]

S. L. Sondhi, A. Karlhede, S.A. Kivelson, and E. H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47, 16419 (1993)

work page 1993
[23]

Fertig, L

H. Fertig, L. Brey, R. Cote, and A. H. MacDonald, Charged spin-texture excitations and the Hartree-Fock approximation in the quantum Hall effect, Phys. Rev. B 50, 11018 (1994)

work page 1994
[24]

S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Fillingν=1, Phys. Rev. Lett. 74, 5112 (1995)

work page 1995
[25]

Ho, Spinor Bose Condensates in Optical Traps, Phys

T.-L. Ho, Spinor Bose Condensates in Optical Traps, Phys. Rev. Lett. 81, 742 (1998)

work page 1998
[26]

Ohmi and K

T. Ohmi and K. Machida, Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases, J. Phys. Soc. Jan., 67, 1822 (1998)

work page 1998
[27]

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, Creation and Detection of Skyrmions in a Bose-Einstein Condensate, Phys. Rev. Lett. 103, 250401 (2009)

work page 2009
[28]

Ozawa et al., Topological photonics, Rev

T. Ozawa et al., Topological photonics, Rev. Mod. Phys. 91, 015006 (2019). 12

work page 2019
[29]

Y . Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y . Arakawa, and S. Iwamoto, Active topological photonics, Nanophotonics 9, 547 (2020)

work page 2020
[30]

Peano, M

V . Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, Topological phase transitions and chiral inelastic transport induced by the squeezing of light, Nature Commun. 7, 10779 (2016)

work page 2016
[31]

Z. Wang, Y . Chong, J. D. Joannopoulos, and M. Soljačić, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461, 772 (2009)

work page 2009
[32]

Peano, C

V . Peano, C. Brendel, M. Schmidt, and F. Marquardt, Topological phases of sound and light, Phys. Rev. X 5, 031011 (2015)

work page 2015
[33]

J. Cha, K. W. Kim, and C. Daraio, Experimental realization of on-chip topological nanoelectromechanical metamaterials, Nature 564, 229 (2018)

work page 2018
[34]

I. Kim, S. Iwamoto, and Y . Arakawa, Topologically protected elastic waves in one- dimensional phononic crystals of continuous media, Appl. Phys. Express 11, 017201 (2018)

work page 2018
[35]

Ge, X.-Y

H. Ge, X.-Y . Xu, L. Liu, R. Xu, Z.-K. Lin, S.-Y . Y u, M. Bao, J.-H. Jiang, M.-H. Lu, and Y .-F. Chen, Observation of Acoustic Skyrmions, Phys. Rev. Lett. 127, 144502 (2021)

work page 2021
[36]

L. Cao, S. Wan, Y. Zeng, Y. Zhu, and B. Assouar, Observation of phononic skyrmions based on hybrid spin of elastic waves, Sci. Adv. 9, eadf3652 (2023)

work page 2023
[37]

L. Du, A. Yang, A. V . Zayats, and X. Yuan, Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum, Nature Phys. 15, 650 (2019)

work page 2019
[38]

Zhang, Z

Q. Zhang, Z. Xie, L. Du, P. Shi, and X. Yuan, Bloch-type photonic skyrmions in optical chiral multilayers, Phys. Rev. Res. 3, 023109 (2021)

work page 2021
[39]

Tsesses, E

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science 361, 993 (2018)

work page 2018
[40]

Yamaguchi and S

H. Yamaguchi and S. Houri, Generation and propagation of topological solitons in a chain of coupled parametric-micromechanical-resonator arrays, Phys. Rev. Applied 15, 034091 (2021)

work page 2021
[41]

Rahav, I

S. Rahav, I. Gilary, and S. Fishman, Effective Hamiltonians for periodically driven systems, Phys. Rev. A 68, 013820 (2003)

work page 2003
[42]

Oka and S

T. Oka and S. Kitamura, Floquet Engineering of Quantum Materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019)

work page 2019
[43]

S. Yin, E. Galiffi, and A. Alù, Floquet metamaterials, eLight 2, 8 (2022). 13

work page 2022
[44]

Marthaler and M

M. Marthaler and M. I. Dykman, Switching via quantum activation: A parametrically modulated oscillator, Phys. Rev. A 73, 042108 (2006)

work page 2006
[45]

Bogdanov and A

A. Bogdanov and A. Hubert, J. Magn. Magn. Mater., 138, 255 (1994)

work page 1994
[46]

J. H. Han, J. Zang, Z. Yang, J.-H. Park, and N. Nagaosa, Skyrmion lattice in a two- dimensional chiral magnet, Phys. Rev. B 82, 094429 (2010)

work page 2010
[47]

Hatanaka, I

D. Hatanaka, I. Mahboob, K. Onomitsu, and H. Yamaguchi, Phonon waveguides for electromechanical circuits, Nature Nanotechnol. 9, 520 (2014)

work page 2014
[48]

Adachi, Physical Properties of III-V Semiconductor Compounds, Sec

S. Adachi, Physical Properties of III-V Semiconductor Compounds, Sec. 5.2 (John Wiley & Sons, Inc., New York, 1992)

work page 1992
[49]

Schmid, L

S. Schmid, L. G. Villanueva, and M. L. Roukes, Fundamentals of Nanomechanical Resonators, Sec. 4.5.1 (Springer, Switzerland 2016)

work page 2016
[50]

Okamoto, A

H. Okamoto, A. Gourgout, C.-Y . Chang, K. Onomitsu, I. Mahboob, E. Y . Chang, H. Yamaguchi, Nature Phys. 9, 480 (2013). 14 Supplemental materials

work page 2013
[51]

Derivation of parametric oscillation states We start from the single resonator Hamiltonian (2), 𝐻0 = 𝑝𝐴 2 + 𝑝𝐵 2 2 + 𝜔0 2 2 (𝑞𝐴 2 + 𝑞𝐵 2)+ 𝜔0 2Γcos2𝜔0𝑡(𝑞𝐴 2 − 𝑞𝐵 2) + 𝜔0 2𝛼 4 (𝑞𝐴 2 + 𝑞𝐵 2)2. Using the generator 𝐹 of the time-dependent canonical transformation for a rotating frame, 𝐹(𝑞𝐴,𝑄𝐴,𝑞𝐵,𝑄𝐵,𝑡)= ∑ ( 𝜔0𝑞𝑘2 2tan𝜔0𝑡 − √𝜔0𝑞𝑘𝑄𝑘 sin𝜔0𝑡 + 𝑄𝑘 2 2tan𝜔0𝑡) 𝑘=𝐴,𝐵 ...

work page
[52]

First, a solution providing the local minimum of ℎ0 is calculated

Stability test of skyrmion textures We perform a two-step calculation to investigate the stability of the skyrmion solution. First, a solution providing the local minimum of ℎ0 is calculated. Then , small perturbations are added to the quadratures, 𝑐(𝑖,𝑗)𝑘 → 𝑐(𝑖,𝑗)𝑘 + ∆𝑐(𝑖,𝑗)𝑘 𝑠(𝑖,𝑗)𝑘 → 𝑠(𝑖,𝑗)𝑘 + ∆𝑠(𝑖,𝑗)𝑘 16 at 𝑡 = 0, and their time evolution is calculate...

work page
[53]

Calculation of skyrmion number The skyrmion number of a continuous system is calculated using the formula, 𝑆 = ∫𝑛⃗ ∙ (𝜕𝑥𝑛⃗ × 𝜕𝑦𝑛⃗ )𝑑𝑥𝑑𝑦 4𝜋 Here, 𝑛⃗ is the unit vector of the moment. The discrete version is given by 𝑆 = ∑ 𝜂 (𝑖,𝑗)∙ (∆𝑥𝜂 (𝑖,𝑗)× ∆𝑦𝜂 (𝑖,𝑗))𝑖,𝑗 4𝜋 = ∑ 𝜂 (𝑖,𝑗)∙ (𝜂 (𝑖+1,𝑗)× 𝜂 (𝑖,𝑗+1))𝑖,𝑗 4𝜋 (S1) Here, 𝜂 (𝑖,𝑗) ≡ 𝑙⃑(𝑖,𝑗)/|𝑙⃑(𝑖,𝑗)| (in the case of 𝑙...

work page
[54]

Figure S4 shows the displacement distribution of x- and y-aligned linearly polarized modes in two coupled membrane resonators with different couplings 𝑔1 and 𝑔2

Full expression of interaction Hamiltonian First, let us start to derive the ferromagnetic interaction with a misaligned piezoelectric axis. Figure S4 shows the displacement distribution of x- and y-aligned linearly polarized modes in two coupled membrane resonators with different couplings 𝑔1 and 𝑔2. 19 Figure S4. Schematic drawing of the coupling of two...

work page
[55]

If we replace 𝑚⃗⃗ by 𝑙⃑, we obtain the quadrature DM interaction from as follows

Neel-type DM interactions 21 To generate a Neel-type skyrmion, 𝑛⃗ 12 = (0,1,0) for a resonator pair aligned to the x- axis, and ℎ𝐷𝑀 = 𝑔𝐷𝑀(𝑚1 𝑧𝑚2 𝑥 − 𝑚1 𝑥𝑚2 𝑧) and 𝑛⃗ 12 = (−1,0,0) for a resonator pair aligned to the y-axis, ℎ𝐷𝑀 = 𝑔𝐷𝑀(−𝑚1 𝑦𝑚2 𝑧 + 𝑚1 𝑧𝑚2 𝑦). If we replace 𝑚⃗⃗ by 𝑙⃑, we obtain the quadrature DM interaction from as follows. ℎ𝐷𝑀−𝑥 (𝐿) = 𝑔𝐷𝑀 √2...

work page
[56]

(8), we assume d that the parametric frequency shift has opposite sign s between two modes

Piezoelectric resonant frequency modulation calculated using a finite element method To obtain eq. (8), we assume d that the parametric frequency shift has opposite sign s between two modes. To confirm the feasibility of this assumption , we calculate d the flexural mode frequency as a function of d.c. gate voltage for a single membrane resonator using a ...

work page
[57]

Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations

Stability against the resonance frequency fluctuation We discuss the robustness of skyrmions against external fluctuations. Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations. Here, let us introduce a frequency detuning δ(𝑖,𝑗) = (𝜔0(𝑖,𝑗)− 𝜔0)/𝜔0 for each resonator obeying a Gaussian distribution. ...

work page
[58]

Transition textures In the parameter region between S-phase and SL-phase, we obtained highly isotropic texture shown in Fig. S7. In this parameter regions, the skyrmion deconfines and occupies the whole lattice in an extremely anisotropic shape. 24 Fig. S7. Calculated textures of 𝑙⃑(𝑖,𝑗) with an ideal DM interaction. 𝑔𝑠 = 0.02, Γ = 0.01, and 𝑔𝐷𝑀 = 0.11. W...

work page

[1] [1]

Manton and P

N. Manton and P. Sutcliffe, Topological Solitons (Cmbridge University Press, Cambridge, United Kingdom, 2004)

work page 2004

[2] [2]

Dauxois and M

T. Dauxois and M. Peyrard, Physics of Solitons (Cmbridge University Press, Cambridge, United Kingdom, 2006)

work page 2006

[3] [3]

Rubinstein, Sine‐Gordon Equation, J

J. Rubinstein, Sine‐Gordon Equation, J. Math. Phys. 11, 258 (1970)

work page 1970

[4] [4]

T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962)

work page 1962

[5] [5]

A. N. Bogdanov and U. K. Rossler, Chiral symmetry breaking in magnetic thin films and multilayers, Phys. Rev. Lett. 87, 037203 (2001)

work page 2001

[6] [6]

U. K. Rossler, N. Bogdanov, and C. Pfleiderer, Spontaneous skyrmion ground states in magnetic materials, Nature 442, 797 (2006)

work page 2006

[7] [7]

Muhlbauer et al., Skyrmion lattice in a chiral magnet, Science 323, 915 (2009)

S. Muhlbauer et al., Skyrmion lattice in a chiral magnet, Science 323, 915 (2009)

work page 2009

[8] [8]

Nagaosa and Y

N. Nagaosa and Y . Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotech. 8, 899 (2013)

work page 2013

[9] [9]

A. Fert, N. Reyren, and V . Cros, Magnetic skyrmions: advances in physics and potential applications, Nat. Rev. Mater. 2, 17031 (2017)

work page 2017

[10] [10]

Gobel, I

B. Gobel, I. Mertig, O. A. Tretiakov, Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles, Physics Reports 895, 1 (2021)

work page 2021

[11] [11]

Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J

I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, J. Phys. Chem. Solids 4, 241 (1958)

work page 1958

[12] [12]

Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys

T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Phys. Rev. 120, 91 (1960)

work page 1960

[13] [13]

Fert and P

A. Fert and P. M. Levy, Role of anisotropic exchange interactions in determining the properties of spin-glasses, Phys. Rev. Lett. 44, 1538 (1980). 11

work page 1980

[14] [14]

Jonietz et al., Spin transfer torques in MnSi at ultralow current densities, Science 330, 1648 (2010)

F. Jonietz et al., Spin transfer torques in MnSi at ultralow current densities, Science 330, 1648 (2010)

work page 2010

[15] [15]

X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose, and Y . Tokura, Skyrmion flow near room temperature in an ultralow current density, Nature Commun. 3, 988 (2012)

work page 2012

[16] [16]

A. Fert, V . Cros, and J. Sampaio, Skyrmions on the track, Nat. Nanotechnol. 8, 152 (2013)

work page 2013

[17] [17]

Tomasello, E

R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio, A strategy for the design of skyrmion racetrack memories, Sci. Rep. 4, 6784 (2014)

work page 2014

[18] [18]

Zhang, M

X. Zhang, M. Ezawa, and Y . Zhou, Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions, Sci. Rep. 5, 9400 (2015)

work page 2015

[19] [19]

Finocchio, M

G. Finocchio, M. Ricci, R. Tomasello, A. Giordano, M. Lanuzza, V . Puliafito, P . Burrascano, B. Azzerboni, and M. Carpentieri, Skyrmion based microwave detectors and harvesting, Appl. Phys. Lett. 107, 262401 (2015)

work page 2015

[20] [20]

Garcia-Sanchez, N

F. Garcia-Sanchez, N. Reyren, J. Sampaio, V . Cros, and J.-V . Kim, Askyrmion- based spin-torque nano-oscillator, New J. Phys. 18, 075011 (2016)

work page 2016

[21] [21]

K. Wang, Y . Huang, X. Zhang, and W. Zhao, Skyrmion-electronics: an overview and outlook, Proc. IEEE 140, 2040 (2016)

work page 2040

[22] [22]

S. L. Sondhi, A. Karlhede, S.A. Kivelson, and E. H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47, 16419 (1993)

work page 1993

[23] [23]

Fertig, L

H. Fertig, L. Brey, R. Cote, and A. H. MacDonald, Charged spin-texture excitations and the Hartree-Fock approximation in the quantum Hall effect, Phys. Rev. B 50, 11018 (1994)

work page 1994

[24] [24]

S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Fillingν=1, Phys. Rev. Lett. 74, 5112 (1995)

work page 1995

[25] [25]

Ho, Spinor Bose Condensates in Optical Traps, Phys

T.-L. Ho, Spinor Bose Condensates in Optical Traps, Phys. Rev. Lett. 81, 742 (1998)

work page 1998

[26] [26]

Ohmi and K

T. Ohmi and K. Machida, Bose-Einstein Condensation with Internal Degrees of Freedom in Alkali Atom Gases, J. Phys. Soc. Jan., 67, 1822 (1998)

work page 1998

[27] [27]

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, Creation and Detection of Skyrmions in a Bose-Einstein Condensate, Phys. Rev. Lett. 103, 250401 (2009)

work page 2009

[28] [28]

Ozawa et al., Topological photonics, Rev

T. Ozawa et al., Topological photonics, Rev. Mod. Phys. 91, 015006 (2019). 12

work page 2019

[29] [29]

Y . Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y . Arakawa, and S. Iwamoto, Active topological photonics, Nanophotonics 9, 547 (2020)

work page 2020

[30] [30]

Peano, M

V . Peano, M. Houde, C. Brendel, F. Marquardt, and A. A. Clerk, Topological phase transitions and chiral inelastic transport induced by the squeezing of light, Nature Commun. 7, 10779 (2016)

work page 2016

[31] [31]

Z. Wang, Y . Chong, J. D. Joannopoulos, and M. Soljačić, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461, 772 (2009)

work page 2009

[32] [32]

Peano, C

V . Peano, C. Brendel, M. Schmidt, and F. Marquardt, Topological phases of sound and light, Phys. Rev. X 5, 031011 (2015)

work page 2015

[33] [33]

J. Cha, K. W. Kim, and C. Daraio, Experimental realization of on-chip topological nanoelectromechanical metamaterials, Nature 564, 229 (2018)

work page 2018

[34] [34]

I. Kim, S. Iwamoto, and Y . Arakawa, Topologically protected elastic waves in one- dimensional phononic crystals of continuous media, Appl. Phys. Express 11, 017201 (2018)

work page 2018

[35] [35]

Ge, X.-Y

H. Ge, X.-Y . Xu, L. Liu, R. Xu, Z.-K. Lin, S.-Y . Y u, M. Bao, J.-H. Jiang, M.-H. Lu, and Y .-F. Chen, Observation of Acoustic Skyrmions, Phys. Rev. Lett. 127, 144502 (2021)

work page 2021

[36] [36]

L. Cao, S. Wan, Y. Zeng, Y. Zhu, and B. Assouar, Observation of phononic skyrmions based on hybrid spin of elastic waves, Sci. Adv. 9, eadf3652 (2023)

work page 2023

[37] [37]

L. Du, A. Yang, A. V . Zayats, and X. Yuan, Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum, Nature Phys. 15, 650 (2019)

work page 2019

[38] [38]

Zhang, Z

Q. Zhang, Z. Xie, L. Du, P. Shi, and X. Yuan, Bloch-type photonic skyrmions in optical chiral multilayers, Phys. Rev. Res. 3, 023109 (2021)

work page 2021

[39] [39]

Tsesses, E

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, G. Bartal, Optical skyrmion lattice in evanescent electromagnetic fields, Science 361, 993 (2018)

work page 2018

[40] [40]

Yamaguchi and S

H. Yamaguchi and S. Houri, Generation and propagation of topological solitons in a chain of coupled parametric-micromechanical-resonator arrays, Phys. Rev. Applied 15, 034091 (2021)

work page 2021

[41] [41]

Rahav, I

S. Rahav, I. Gilary, and S. Fishman, Effective Hamiltonians for periodically driven systems, Phys. Rev. A 68, 013820 (2003)

work page 2003

[42] [42]

Oka and S

T. Oka and S. Kitamura, Floquet Engineering of Quantum Materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019)

work page 2019

[43] [43]

S. Yin, E. Galiffi, and A. Alù, Floquet metamaterials, eLight 2, 8 (2022). 13

work page 2022

[44] [44]

Marthaler and M

M. Marthaler and M. I. Dykman, Switching via quantum activation: A parametrically modulated oscillator, Phys. Rev. A 73, 042108 (2006)

work page 2006

[45] [45]

Bogdanov and A

A. Bogdanov and A. Hubert, J. Magn. Magn. Mater., 138, 255 (1994)

work page 1994

[46] [46]

J. H. Han, J. Zang, Z. Yang, J.-H. Park, and N. Nagaosa, Skyrmion lattice in a two- dimensional chiral magnet, Phys. Rev. B 82, 094429 (2010)

work page 2010

[47] [47]

Hatanaka, I

D. Hatanaka, I. Mahboob, K. Onomitsu, and H. Yamaguchi, Phonon waveguides for electromechanical circuits, Nature Nanotechnol. 9, 520 (2014)

work page 2014

[48] [48]

Adachi, Physical Properties of III-V Semiconductor Compounds, Sec

S. Adachi, Physical Properties of III-V Semiconductor Compounds, Sec. 5.2 (John Wiley & Sons, Inc., New York, 1992)

work page 1992

[49] [49]

Schmid, L

S. Schmid, L. G. Villanueva, and M. L. Roukes, Fundamentals of Nanomechanical Resonators, Sec. 4.5.1 (Springer, Switzerland 2016)

work page 2016

[50] [50]

Okamoto, A

H. Okamoto, A. Gourgout, C.-Y . Chang, K. Onomitsu, I. Mahboob, E. Y . Chang, H. Yamaguchi, Nature Phys. 9, 480 (2013). 14 Supplemental materials

work page 2013

[51] [51]

Derivation of parametric oscillation states We start from the single resonator Hamiltonian (2), 𝐻0 = 𝑝𝐴 2 + 𝑝𝐵 2 2 + 𝜔0 2 2 (𝑞𝐴 2 + 𝑞𝐵 2)+ 𝜔0 2Γcos2𝜔0𝑡(𝑞𝐴 2 − 𝑞𝐵 2) + 𝜔0 2𝛼 4 (𝑞𝐴 2 + 𝑞𝐵 2)2. Using the generator 𝐹 of the time-dependent canonical transformation for a rotating frame, 𝐹(𝑞𝐴,𝑄𝐴,𝑞𝐵,𝑄𝐵,𝑡)= ∑ ( 𝜔0𝑞𝑘2 2tan𝜔0𝑡 − √𝜔0𝑞𝑘𝑄𝑘 sin𝜔0𝑡 + 𝑄𝑘 2 2tan𝜔0𝑡) 𝑘=𝐴,𝐵 ...

work page

[52] [52]

First, a solution providing the local minimum of ℎ0 is calculated

Stability test of skyrmion textures We perform a two-step calculation to investigate the stability of the skyrmion solution. First, a solution providing the local minimum of ℎ0 is calculated. Then , small perturbations are added to the quadratures, 𝑐(𝑖,𝑗)𝑘 → 𝑐(𝑖,𝑗)𝑘 + ∆𝑐(𝑖,𝑗)𝑘 𝑠(𝑖,𝑗)𝑘 → 𝑠(𝑖,𝑗)𝑘 + ∆𝑠(𝑖,𝑗)𝑘 16 at 𝑡 = 0, and their time evolution is calculate...

work page

[53] [53]

Calculation of skyrmion number The skyrmion number of a continuous system is calculated using the formula, 𝑆 = ∫𝑛⃗ ∙ (𝜕𝑥𝑛⃗ × 𝜕𝑦𝑛⃗ )𝑑𝑥𝑑𝑦 4𝜋 Here, 𝑛⃗ is the unit vector of the moment. The discrete version is given by 𝑆 = ∑ 𝜂 (𝑖,𝑗)∙ (∆𝑥𝜂 (𝑖,𝑗)× ∆𝑦𝜂 (𝑖,𝑗))𝑖,𝑗 4𝜋 = ∑ 𝜂 (𝑖,𝑗)∙ (𝜂 (𝑖+1,𝑗)× 𝜂 (𝑖,𝑗+1))𝑖,𝑗 4𝜋 (S1) Here, 𝜂 (𝑖,𝑗) ≡ 𝑙⃑(𝑖,𝑗)/|𝑙⃑(𝑖,𝑗)| (in the case of 𝑙...

work page

[54] [54]

Figure S4 shows the displacement distribution of x- and y-aligned linearly polarized modes in two coupled membrane resonators with different couplings 𝑔1 and 𝑔2

Full expression of interaction Hamiltonian First, let us start to derive the ferromagnetic interaction with a misaligned piezoelectric axis. Figure S4 shows the displacement distribution of x- and y-aligned linearly polarized modes in two coupled membrane resonators with different couplings 𝑔1 and 𝑔2. 19 Figure S4. Schematic drawing of the coupling of two...

work page

[55] [55]

If we replace 𝑚⃗⃗ by 𝑙⃑, we obtain the quadrature DM interaction from as follows

Neel-type DM interactions 21 To generate a Neel-type skyrmion, 𝑛⃗ 12 = (0,1,0) for a resonator pair aligned to the x- axis, and ℎ𝐷𝑀 = 𝑔𝐷𝑀(𝑚1 𝑧𝑚2 𝑥 − 𝑚1 𝑥𝑚2 𝑧) and 𝑛⃗ 12 = (−1,0,0) for a resonator pair aligned to the y-axis, ℎ𝐷𝑀 = 𝑔𝐷𝑀(−𝑚1 𝑦𝑚2 𝑧 + 𝑚1 𝑧𝑚2 𝑦). If we replace 𝑚⃗⃗ by 𝑙⃑, we obtain the quadrature DM interaction from as follows. ℎ𝐷𝑀−𝑥 (𝐿) = 𝑔𝐷𝑀 √2...

work page

[56] [56]

(8), we assume d that the parametric frequency shift has opposite sign s between two modes

Piezoelectric resonant frequency modulation calculated using a finite element method To obtain eq. (8), we assume d that the parametric frequency shift has opposite sign s between two modes. To confirm the feasibility of this assumption , we calculate d the flexural mode frequency as a function of d.c. gate voltage for a single membrane resonator using a ...

work page

[57] [57]

Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations

Stability against the resonance frequency fluctuation We discuss the robustness of skyrmions against external fluctuations. Fabrication inaccuracies affecting the resonance frequency are some of the most common causes of fluctuations. Here, let us introduce a frequency detuning δ(𝑖,𝑗) = (𝜔0(𝑖,𝑗)− 𝜔0)/𝜔0 for each resonator obeying a Gaussian distribution. ...

work page

[58] [58]

Transition textures In the parameter region between S-phase and SL-phase, we obtained highly isotropic texture shown in Fig. S7. In this parameter regions, the skyrmion deconfines and occupies the whole lattice in an extremely anisotropic shape. 24 Fig. S7. Calculated textures of 𝑙⃑(𝑖,𝑗) with an ideal DM interaction. 𝑔𝑠 = 0.02, Γ = 0.01, and 𝑔𝐷𝑀 = 0.11. W...

work page