Stabilizing hopfion

[Neeti Keswani]

Dear All,

Pl guide how should i stabilize an hopfion in a nanodisc.

Can somebody share parameters? 

Thanking you.

Regards

Your sincerely,

 Neeti Keswani

[WW]

Dear keswani
,

Two ingredients are needed together: a suitable initial state, and a perpendicular boundary condition on the top and bottom surfaces. Boundary conditions alone are not enough, since the hopfion is metastable and separated from the competing states by an energy barrier, so relaxation from a random or uniform state will almost never reach it. Seed with a hopfion ansatz, or with a skyrmion tube spanning the disc, which often relaxes into Q_H = 1.

For the boundary condition I have used three approaches, all of which enforce m close to (0,0,1) on the top and bottom:

1. Freeze the spins in the top and bottom layers to (0,0,1) using the mumax3 API (FrozenSpins mask together with m.SetRegion). This is non physical, but it is the most robust of the three and it is convenient for parameter scans.

2. Define top and bottom regions with a large PMA and anisU = (0,0,1). This is the approach of Liu, Lake and Zang, and it is the closest to a real multilayer sample.

3. Apply a static perpendicular field. This is the least reliable, because the working window is narrow and depends on the material parameters. In my simulations it only works if Msat is reduced in the top and bottom layers. Otherwise the in plane shape anisotropy of those layers wins and the boundary condition is never established.

Parameters. The standard reference set is Liu, Lake and Zang, Binding a hopfion in a chiral magnet nanodisk, Phys. Rev. B 98, 174437 (2018):

  Aex  = 2.195e-12 J/m
  D    = 0.395e-3 J/m^2 (bulk DMI)
  Msat = 384e3 A/m
  Ku1  = 0.8e6 J/m^3, applied only in the top and bottom layers, anisU = (0,0,1), 10 nm thick each
  helical period L_D = 4*pi*Aex/D = 70 nm
  disc radius r = 100 nm = 1.43 L_D, chiral layer thickness t = 70 nm = 1.0 L_D
  cell size 2 nm (convergence verified down to 0.5 nm)

The geometry is what decides stability, so it is better to think in units of L_D than in nanometres. The hopfion survives only in a limited window, roughly t = 1 L_D. If you change A or D, rescale r and t so that r/L_D and t/L_D stay inside that window.

Best regards,

waleed

[Neeti Keswani]

Thank you Waleed for the detailed information.

Also can you guide me how I should check if I am actually able to stabilize the hopfion?

Or paraphrasing if the structure achieved is a hopfion?

Thanks! 

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Thanking you.

Regards

Your sincerely,

Dr. Neeti Keswani

[Electricity & Magnetism]

Three checks are usually sufficient, and all three should be done on the relaxed configuration rather than the seed.

1. Preimages. The Hopf invariant is the linking number between two arbitrary closed-loop preimages. Plot the preimages of s = (1,0,0) and s = (-1,0,0). For a hopfion they form two closed loops linked once. Avoid s = (0,0,±1), since that is the background direction and its preimage will not close inside the volume. A monopole-antimonopole pair gives no closed loops at all; its preimages instead join at the monopole and antimonopole. See Liu, Lake and Zang, Phys. Rev. B 98, 174437 (2018), Fig. 1(g,h).

2. Emergent field. Q_H is gauge-invariant only if the emergent field B is free of singularities. Plot B in a cross-section: for a hopfion it flows smoothly, whereas a monopole-antimonopole state shows two opposite-charge monopoles near the surfaces, and Q_H is ill-defined there. Same reference, Fig. 2.

3. Numerical Hopf number. Compute Q_H independently rather than inferring it from the preimages. Solving for A in momentum space in the Coulomb gauge is the standard route. Sweep the grid number and report the convergence curve; a value slightly below the integer is expected from finite size and open boundaries (0.96 in the reference above, Fig. 3). Implementations and a comparison of methods are given in Knapman et al., arXiv:2410.22058, including MuMax3 extensions.

Best regards,

w.w

To view this discussion visit https://groups.google.com/d/msgid/mumax2/CABBCV7hegKVu49pHTgyxRZgeT7DTxuJ%3DN0YuWEFemKNDkw%3DLPA%40mail.gmail.com.

[Electricity & Magnetism]

Three checks are usually sufficient, and all three should be done on the relaxed configuration rather than the seed.

1. Preimages. The Hopf invariant is the linking number between two arbitrary closed-loop preimages. Plot the preimages of s = (1,0,0) and s = (-1,0,0). For a hopfion they form two closed loops linked once. Avoid s = (0,0,±1), since that is the background direction and its preimage will not close inside the volume. A monopole-antimonopole pair gives no closed loops at all; its preimages instead join at the monopole and antimonopole. See Liu, Lake and Zang, Phys. Rev. B 98, 174437 (2018), Fig. 1(g,h).

2. Emergent field. Q_H is gauge-invariant only if the emergent field B is free of singularities. Plot B in a cross-section: for a hopfion it flows smoothly, whereas a monopole-antimonopole state shows two opposite-charge monopoles near the surfaces, and Q_H is ill-defined there. Same reference, Fig. 2.

3. Numerical Hopf number. Compute Q_H independently rather than inferring it from the preimages. Solving for A in momentum space in the Coulomb gauge is the standard route. Sweep the grid number and report the convergence curve; a value slightly below the integer is expected from finite size and open boundaries (0.96 in the reference above, Fig. 3). Implementations and a comparison of methods are given in Knapman et al., arXiv:2410.22058, including MuMax3 extensions.

Best regards,

w.w

On Tue, Aug 25, 2026 at 11:42 AM Neeti Keswani <neetike...@gmail.com> wrote:

To view this discussion visit https://groups.google.com/d/msgid/mumax2/CABBCV7hegKVu49pHTgyxRZgeT7DTxuJ%3DN0YuWEFemKNDkw%3DLPA%40mail.gmail.com.

[Neeti Keswani]

Thanks for the reply. 

It seems really tough though . How it shud be executed? 

Will these paper give better outlook 

Thanking you.

Regards

Your sincerely,

Dr. Neeti Keswani

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