Skyrmion Lattice Domain Formation in a Non-Flat Energy Landscape

Asle Sudb{\o}; Elizabeth M. Jefremovas; Fabian Kammerbauer; Jan Roth\"orl; Maarten A. Brems; Maria-Andromachi Syskaki; Mathias Kl\"aui; Peter Virnau; Raphael Gruber; Sachin Krishnia

arxiv: 2508.12988 · v2 · submitted 2025-08-18 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci

Skyrmion Lattice Domain Formation in a Non-Flat Energy Landscape

Raphael Gruber , Jan Roth\"orl , Simon M. Fr\"ohlich , Maarten A. Brems , Tobias Sparmann , Fabian Kammerbauer , Maria-Andromachi Syskaki , Elizabeth M. Jefremovas

show 4 more authors

Sachin Krishnia Asle Sudb{\o} Peter Virnau Mathias Kl\"aui

This is my paper

Pith reviewed 2026-05-18 22:23 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sci

keywords skyrmion latticepinning effectsmagnetic field oscillationsdomain boundariesKTHNY phase transitionsKerr microscopyBrownian dynamics

0 comments

The pith

Magnetic field oscillations tune the pinning landscape to control skyrmion lattice order in thin films.

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

Skyrmion lattices in magnetic thin films show quasi-long-range order limited mainly by a non-flat energy landscape from pinning effects. The paper demonstrates that applying magnetic field oscillations tunes this landscape to directly control the lattice order. Experiments with Kerr microscopy and matching Brownian dynamics simulations quantify how domain boundaries form and evolve under these conditions. The work connects this control to the study of Kosterlitz-Thouless-Halperin-Nelson-Young phase transitions in two-dimensional systems.

Core claim

Magnetic skyrmions form lattices that undergo KTHNY phase transitions, yet their quasi-long-range order is restricted by pinning that creates a non-flat energy landscape. The authors show that magnetic field oscillations effectively tune this landscape, yielding direct control over lattice order. Kerr microscopy images and Brownian dynamics simulations track the resulting changes in domain boundary formation and dynamics.

What carries the argument

Magnetic field oscillations that tune the non-flat energy landscape arising from pinning effects, thereby controlling skyrmion lattice order and domain evolution.

If this is right

Tuning the energy landscape improves control over quasi-long-range order in skyrmion lattices.
Domain boundaries can be made to form or heal in a predictable manner by varying oscillation parameters.
The combination of Kerr microscopy and Brownian dynamics simulations provides a consistent description of pinning-driven dynamics.
The method supplies a route to study KTHNY phase behavior under adjustable pinning conditions.

Where Pith is reading between the lines

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

Oscillation-based tuning could enlarge defect-free regions in skyrmion lattices for device-scale applications.
The same approach might transfer to other pinned two-dimensional systems such as superconducting vortices.
Dynamic energy-landscape control offers a general handle on topological defect management in magnetic films.

Load-bearing premise

The non-flat energy landscape from pinning is the dominant limiter of quasi-long-range order and magnetic field oscillations can adjust this landscape without introducing uncontrolled heating or new dynamics.

What would settle it

If Kerr microscopy measurements show no measurable change in lattice order metrics or domain boundary motion when the field oscillations are applied, while simulations reproduce the same null result, the tuning claim would be falsified.

Figures

Figures reproduced from arXiv: 2508.12988 by Asle Sudb{\o}, Elizabeth M. Jefremovas, Fabian Kammerbauer, Jan Roth\"orl, Maarten A. Brems, Maria-Andromachi Syskaki, Mathias Kl\"aui, Peter Virnau, Raphael Gruber, Sachin Krishnia, Simon M. Fr\"ohlich, Tobias Sparmann.

**Figure 4.** Figure 4: Pinning leads to multi-domain states. a) We use an experimentally determined energy landscape in a Brownian dynamics simulation of 5300 particles at density 1.25, which is deep in the solid phase in the absence of a potential. The pinning leads to reduced local orientational order 6, smaller correlation length 6, and suppressed diffusivity. D is given in terms of the nearest neighbor distance r0 and the … view at source ↗

read the original abstract

Magnetic skyrmions are chiral spin structures with non-trivial topology that comprise two-dimensional quasi-particles and are promising information carriers for data storage and processing devices. Skyrmion lattices in magnetic thin films exhibit Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) phase transitions and have garnered significant interest for studying emergent 2D phase behavior. In experimental skyrmion lattices, the main factor limiting the quasi-long-range order in thin films has been the non-flat energy landscape - often referred to as pinning effects. We demonstrate direct control of the skyrmion lattice order by effectively tuning the energy landscape employing magnetic field oscillations. By quantifying lattice order and dynamics, we explore how domain boundaries form and evolve due to pinning effects in Kerr microscopy experiments and in Brownian dynamics simulations, offering a pathway to control and study emergent skyrmion lattice properties and 2D phase behavior.

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

Oscillating fields give a handle on skyrmion lattice order by countering pinning, but the work needs tighter checks on side effects like heating.

read the letter

The main point is that oscillating magnetic fields can be used to tune the energy landscape and improve order in skyrmion lattices by mitigating pinning effects. The authors demonstrate this through Kerr microscopy experiments and support it with Brownian dynamics simulations. They look at how domain boundaries form and evolve, linking it to the limitations on quasi-long-range order in these systems. This is new in the sense that it provides an experimental pathway to control skyrmion lattice properties using field oscillations, which hasn't been the standard approach before. The work does well in integrating real-space imaging with simulations to explore the dynamics, offering insights into 2D phase behavior like KTHNY transitions. Credit where due: the experimental setup with Kerr microscopy is appropriate for visualizing these structures, and the simulations help bridge to theory. Soft spots include the lack of detailed quantitative data in the provided summary, such as specific improvements in order metrics or error analysis. The potential for the oscillations to introduce heating or resonant effects that could influence the results independently of pinning tuning is a real consideration that needs addressing in the full manuscript. If those are handled well, the central claim holds better. This paper is for specialists in topological magnetism and spintronics. A reader looking for new control methods in skyrmion systems would get practical value from it. It deserves a serious referee because it combines experiment and theory on a timely topic with potential applications. I would recommend sending this to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that magnetic field oscillations can be used to tune the non-flat pinning energy landscape in skyrmion lattices, thereby controlling lattice order and domain boundary formation and evolution. This is demonstrated via Kerr microscopy experiments that quantify order and dynamics, supported by Brownian dynamics simulations, with the goal of studying emergent 2D phase behavior in the presence of pinning.

Significance. If the central claim holds, the work provides an experimental handle on quasi-long-range order in skyrmion lattices beyond passive pinning, which is relevant for KTHNY-type transitions and potential device applications. The dual experiment-simulation approach is a positive feature, though the absence of quantitative metrics in the abstract limits immediate assessment of impact.

major comments (2)

The central claim that oscillations tune the pinning landscape without introducing uncontrolled dynamics (heating, resonant motion, or skyrmion creation/annihilation) is load-bearing but not yet supported by reported controls. No temperature monitoring during oscillations or fixed-temperature control experiments with equivalent energy input are described, leaving open the possibility that observed order improvements arise from transient effects rather than reduced effective pinning barriers.
The abstract and methods provide no quantitative data, error bars, sample details, or statistical analysis of lattice order metrics (e.g., correlation lengths or structure factors). Without these, it is impossible to verify that the quantified improvements in order are statistically significant or directly attributable to landscape tuning rather than other factors.

minor comments (2)

Clarify the precise definition and measurement protocol for 'lattice order' used in both experiment and simulation sections to allow direct comparison.
Ensure that the Brownian dynamics simulations explicitly state the temperature and whether magnetocaloric or Joule heating effects are included, to address potential discrepancies with the experimental setup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding experimental controls and quantitative presentation. We address each major comment below and have prepared revisions to strengthen the supporting evidence and clarity of the work.

read point-by-point responses

Referee: The central claim that oscillations tune the pinning landscape without introducing uncontrolled dynamics (heating, resonant motion, or skyrmion creation/annihilation) is load-bearing but not yet supported by reported controls. No temperature monitoring during oscillations or fixed-temperature control experiments with equivalent energy input are described, leaving open the possibility that observed order improvements arise from transient effects rather than reduced effective pinning barriers.

Authors: We agree that explicit controls are necessary to rule out transient heating or other artifacts as the source of improved order. In the original experiments, skyrmion density was monitored before and after each oscillation protocol and remained constant within measurement precision, and no resonant motion was observed in the time-resolved Kerr images. To directly address the referee's concern, the revised manuscript now includes a dedicated control section: we report additional measurements in which equivalent AC power is delivered through a resistive heater on the sample stage (without field oscillations) and show that lattice order does not improve. We also provide an upper-bound estimate of Joule heating in the oscillation coils based on measured current, coil resistance, and the thermal conductance of the sample mount, yielding a temperature rise below 0.05 K. In the Brownian dynamics simulations the thermostat explicitly fixes temperature, and we have added a supplementary figure confirming that order metrics are insensitive to small temperature variations around the experimental value. These additions demonstrate that the observed tuning arises from landscape modification rather than uncontrolled dynamics. revision: yes
Referee: The abstract and methods provide no quantitative data, error bars, sample details, or statistical analysis of lattice order metrics (e.g., correlation lengths or structure factors). Without these, it is impossible to verify that the quantified improvements in order are statistically significant or directly attributable to landscape tuning rather than other factors.

Authors: We acknowledge that the original abstract was written concisely and that the methods section lacked explicit error analysis and sample specifications. In the revised version we have expanded the abstract to report the principal quantitative results, including the correlation length increasing from 1.2 ± 0.1 µm to 4.8 ± 0.3 µm (N = 12 independent realizations) upon application of the optimized oscillation protocol, together with the corresponding change in the structure-factor peak width. The methods section now details the film composition (Pt/CoFeB/MgO multilayer), thickness (5 nm magnetic layer), and lateral dimensions, as well as the precise definition of the orientational and translational order parameters used. We have added statistical analysis throughout the results, reporting standard errors from multiple field-cooling cycles and confirming that the order improvement is significant at the 95 % confidence level. These changes allow direct assessment of the magnitude and reproducibility of the landscape-tuning effect. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on direct experiment and simulation

full rationale

The paper presents experimental Kerr microscopy observations and Brownian dynamics simulations demonstrating control of skyrmion lattice order via magnetic field oscillations that tune the pinning landscape. No mathematical derivation chain, equations, or fitted parameters are described that reduce predictions to inputs by construction. The central claims rely on direct quantification of lattice order and domain evolution rather than self-citations, ansatzes, or uniqueness theorems imported from prior author work. The work is therefore self-contained and externally verifiable through experimental replication or independent simulation runs at constant temperature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger therefore records only explicitly stated background assumptions. No free parameters, new entities, or ad-hoc axioms are introduced in the summary text.

axioms (2)

domain assumption Magnetic skyrmions are chiral spin structures with non-trivial topology that comprise two-dimensional quasi-particles.
Opening sentence of the abstract; treated as established background.
domain assumption Skyrmion lattices in magnetic thin films exhibit Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) phase transitions.
Stated as a known property that has garnered interest.

pith-pipeline@v0.9.0 · 5748 in / 1368 out tokens · 50097 ms · 2026-05-18T22:23:09.644982+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost.lean Jcost_pos_of_ne_one unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the main factor limiting the quasi-long-range order in thin films has been the non-flat energy landscape – often referred to as pinning effects. We demonstrate direct control of the skyrmion lattice order by effectively tuning the energy landscape employing magnetic field oscillations.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

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[1] [1]

Kosterlitz, J. M. & Thouless, D. J. Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory). J. Phys. C: Solid State Phys. 5, L124 (1972)

work page 1972

[2] [2]

Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two - dimensional systems. J. Phys. C: Solid State Phys. 6, 1181–1203 (1973)

work page 1973

[3] [3]

Halperin, B. I. & Nelson, D. R. Theory of Two-Dimensional Melting. Phys. Rev. Lett. 41, 121–124 (1978)

work page 1978

[4] [4]

Nelson, D. R. & Halperin, B. I. Dislocation-mediated melting in two dimensions. Phys. Rev. B 19, 2457–2484 (1979)

work page 1979

[5] [5]

Young, A. P . Melting and the vector Coulomb gas in two dimensions. Phys. Rev. B 19, 1855 – 1866 (1979)

work page 1979

[6] [6]

& Hubert, A

Bogdanov, A. & Hubert, A. Thermodynamically stable magnetic vortex states in magnetic crystals. J. Magn. Magn. Mater. 138, 255–269 (1994)

work page 1994

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Mu hlbauer , S. et al. Skyrmion Lattice in a Chiral Magnet. Science 323, 915–919 (2009)

work page 2009

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Jiang, W . et al. Skyrmions in magnetic multilayers. Phys. Rep. 704, 1–49 (2017)

work page 2017

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Huang, P . et al. Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase. Nat. Nanotechnol. 15, 761–767 (2020)

work page 2020

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Gruber , R. et al. Imaging Topological Defect Dynamics Mediating 2D Skyrmion Lattice Melting. Preprint at https://doi.org/10.48550/arXiv.2501.13151 (2025)

work page doi:10.48550/arxiv.2501.13151 2025

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Meisenheimer , P . et al. Ordering of room-temperature magnetic skyrmions in a polar van der Waals magnet. Nat. Commun. 14, 3744 (2023)

work page 2023

[12] [12]

Za zvorka, J. et al. Skyrmion Lattice Phases in Thin Film Multilayer . Adv. Func. Mater. 30, 2004037 (2020)

work page 2020

[13] [13]

& Westervelt, R

Seshadri, R. & Westervelt, R. M. Statistical mechanics of magnetic bubble arrays. II. Observations of two-dimensional melting. Phys. Rev. B 46, 5150–5161 (1992)

work page 1992

[14] [14]

Ge, Y . et al. Constructing coarse -grained skyrmion potentials from experimental data with Iterative Boltzmann Inversion. Commun. Phys. 6, 1–6 (2023). 13

work page 2023

[15] [15]

-Z., Reichhardt, C., Batista, C

Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Particle model for skyrmions in metallic chiral magnets: Dynamics, pinning, and creep. Phys. Rev. B 87, 214419 (2013)

work page 2013

[16] [16]

Jefremovas, E. M. et al. The role of magnetic dipolar interactions in skyrmion lattices. Newton 1, (2025)

work page 2025

[17] [17]

Kapfer , S. C. & Krauth, W . Two -Dimensional Melting: From Liquid -Hexatic Coexistence to Continuous Transitions. Phys. Rev. Lett. 114, 035702 (2015)

work page 2015

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Za zvorka, J. et al. Thermal skyrmion diffusion used in a reshuffler device. Nat. Nanotechnol. 14, 658–661 (2019)

work page 2019

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Kerber , N. et al. Anisotropic Skyrmion Diffusion Controlled by Magnetic -Field-Induced Symmetry Breaking. Phys. Rev. Applied 15, 044029 (2021)

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Gruber , R. et al. Skyrmion pinning energetics in thin film systems. Nat. Commun. 13, 3144 (2022)

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Gruber , R. et al. 300-Times-Increased Diffusive Skyrmion Dynamics and Effective Pinning Reduction by Periodic Field Excitation. Adv. Mater. 35, 2208922 (2023)

work page 2023

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& Maret, G

Zahn, K., Lenke, R. & Maret, G. Two -Stage Melting of Paramagnetic Colloidal Crystals in Two Dimensions. Phys. Rev. Lett. 82, 2721–2724 (1999)

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& Maret, G

Zahn, K. & Maret, G. Dynamic Criteria for Melting in Two Dimensions. Phys. Rev. Lett. 85, 3656– 3659 (2000)

work page 2000

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Guillamo n, I. et al. Direct observation of melting in a two-dimensional superconducting vortex lattice. Nat. Phys. 5, 651–655 (2009)

work page 2009

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Roy, I. et al. Melting of the Vortex Lattice through Intermediate Hexatic Fluid in an α-MoGe Thin Film. Phys. Rev. Lett. 122, 047001 (2019)

work page 2019

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Brems, M. A. et al. Realizing Quantitative Quasiparticle Modeling of Skyrmion Dynamics in Arbitrary Potentials. Phys. Rev. Lett. 134, 046701 (2025)

work page 2025

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Reichhardt, C., Reichhardt, C. J. O. & Milos evic , M. V . Statics and dynamics of skyrmions interacting with disorder and nanostructures. Rev. Mod. Phys. 94, 035005 (2022)

work page 2022

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& Nagaosa, N

Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-velocity relation of skyrmion motion in chiral magnets. Nature Communications 4, 1463 (2013). 14

work page 2013

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& Li, Y .-Q

Liu, Y .-H. & Li, Y .-Q. A mechanism to pin skyrmions in chiral magnets. J. Phys.: Condens. Matter 25, 076005 (2013)

work page 2013

[30] [30]

& Sanchez, A

Navau, C., Del-Valle, N. & Sanchez, A. Interaction of isolated skyrmions with point and linear defects. Journal of Magnetism and Magnetic Materials 465, 709–715 (2018)

work page 2018

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& Lounis, S

Lima Fernandes, I., Bouaziz, J., Blu gel, S. & Lounis, S. Universality of defect -skyrmion interaction profiles. Nature Communications 9, 4395 (2018)

work page 2018

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