Spontaneous Skyrmion Ground States in Magnetic Metals

A.N. Bogdanov (IFW Dresden); C. Pfleiderer (TU Munich); U.K. Roessler

REVIEW

Spontaneous Skyrmion Ground States in Magnetic Metals

U.K. Roessler, A.N. Bogdanov (IFW Dresden), C. Pfleiderer (TU Munich)

Statistical Mechanics · Submitted 5 Mar 2006 · 42 references

Pith's verdict: A physically motivated and historically important paper whose central stabilization mechanism holds up, but whose 'rigorously establishes' claim and printed Euler equations need correction.

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CitationA.N. Bogdanov (IFW Dresden), C. Pfleiderer (TU Munich), U.K. Roessler. "Spontaneous Skyrmion Ground States in Magnetic Metals." arXiv:cond-mat/0603103 (2006). https://pith.science/paper/EWRKRIAW

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Editor's note

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Short version: this is the paper that introduced η<1 longitudinal softening as a stabilization mechanism for spontaneous skyrmion phases. That is a real step beyond Wright–Mermin and beyond the authors' own earlier field-induced skyrmion work. The mechanism is physically transparent, and the numerical solutions in Figs. 2 and 3 support it for the parameters shown. I would not desk-reject it.

What is genuinely new: a minimal two-gradient model with separate longitudinal stiffness η, and the demonstration that for η<1 isolated cylindrical skyrmions become local minima with a finite temperature window where they beat the helix in a circular-cell energy comparison. The η=1 anchor is an honest external check: the model reproduces the Wright–Mermin no-skyrmion result. The phase diagram with TS, Tt, and metastable helix is useful. The MnSi discussion is suggestive rather than overclaimed.

Where the soft spots are: the word 'rigorously establishes' overstates the evidence. The lower phase boundary Tt comes from the approximate equal-average-energy condition for a circular skyrmion cell versus the helix, not from a full periodic calculation. The one full 2D lattice solution (Fig. 4) is never energetically compared with the helix or the paramagnetic state, and its staggered 2×2 structure is not the same state as the m(R)=0 cell used for Tt. So thermodynamic stability of the skyrmion lattice over the whole claimed interval is not actually demonstrated. The stress-test note is right: Eq. (7) as printed is not the Euler–Lagrange equation of Eq. (5) — the G-term coefficient and the f0 term do not match — and Eq. (8) looks similarly unreproducible. That needs a corrigendum. The η values for EuS, Ni, and MnSi come from a separate supplement [20] not included in this submission, so the applicability claim rests on an external document I cannot check. No code is provided, but for a 2006 cond-mat paper that is a minor issue, not fatal.

Who this is for: anyone working on chiral magnets, skyrmions, MnSi, or thin-film magnetism. The citation pattern is appropriate: Wright–Mermin, Derrick–Hobart, and the prior Bogdanov–Hubert work are all credited. It deserves a serious referee; the referee should ask for corrected equations, the supplement with the η fits, and an energy comparison of Fig. 4 against competing states.

Referee report

Significance

If the central claim is correct, this overturns the long-standing belief that skyrmions in chiral magnets require external fields or topological-defect proliferation, and it identifies a new spontaneous condensed-matter phase with clear experimental signatures. The model is physically motivated and the paper provides numerically computed equilibrium profiles and a candidate lattice state. The paper also gives a concrete route to test the prediction via neutron scattering and real-space imaging. However, the thermodynamic stability of the skyrmion phase is not fully established by the calculations as presented: the phase boundary is obtained from an approximate cell energy comparison, and the one full lattice solution is never compared energetically with the competing helix or paramagnetic states.

Major objections (4)

[Methods, Eq. (7)] The printed Euler–Lagrange equation for m is inconsistent with the energy density in Eq. (5). Functional differentiation of fs with respect to m yields η(m''+m'/ρ) − G m − (1/(2Aq0²))∂f0/∂m = 0 after factoring out the common factors, not the printed − A G m − q0⁻² ∂f0/∂m. As printed, an independent reader cannot reproduce the numerical skyrmion profiles from the equations given, so this typo/error must be corrected.
[Fig. 2 and accompanying text] The statement "This analysis establishes rigorously that stable cylindrical skyrmions form spontaneously in a finite temperature interval Tt<T<TS" overstates the evidence. The boundary Tt is obtained from the approximate condition that the average energy of a circular skyrmion cell equals that of the helix, which is a cell approximation, not a solution of the periodic problem. The manuscript demonstrates local stability of an isolated cylindrical skyrmion, but not that a skyrmion lattice is the global free-energy minimum over the claimed temperature interval.
[Fig. 4] The full two-dimensional skyrmion lattice solution (η=0.4, b=0.05, T−TC=0.225) is presented as a minimum-energy solution, but its energy is not compared with that of the helix or the paramagnetic state. Without such a comparison, the conclusion that this lattice is a ground state rather than a metastable state is not supported by the data shown.
[Main text, paragraph on η values] The values η=0.925, 0.65, and 0.4 for EuS, Ni, and MnSi are asserted from an analysis of published longitudinal susceptibility data that is relegated to the separate supplementary information [20]. Since the existence of the skyrmion phase relies on η<1 and the quantitative predictions for MnSi depend on these numbers, the paper should state the essential steps or provide the relevant data so that the estimates can be checked without access to the supplementary.

Minor objections (4)

[Abstract and main text] There are several typographical errors, e.g., "hirachy" for "hierarchy" in the main text and "Fermi-lqiud" in reference [33]; these should be corrected.
[Methods] The word "uniaxal" should be "uniaxial" where it appears in the description of Cnv symmetry magnets.
[Discussion of MnSi] The claim that the proposed skyrmion phase "resolves all of the existing contradictions" in MnSi is stronger than the evidence presented; the argument is based on qualitative compatibility and an indirect estimate of TD−TC from [20], so the wording should be moderated.
[Fig. 1 caption] The phrase "when the fingers on a hand follow the moments rolling outward the thumb defines a winding sense" is grammatically unclear and should be rephrased for clarity.

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The paper

The paper claims

In chiral magnets with soft magnetization, skyrmions can form spontaneously as ground states, with no external field or defect proliferation needed.

The reading

The paper argues that the long-standing belief that skyrmions cannot form spontaneous ground states in magnets is wrong. For chiral ferromagnets with a softened longitudinal stiffness of the magnetization (a condition metallic magnets naturally satisfy), cylindrical skyrmions are thermodynamically stable in a finite temperature interval between the paramagnetic and helical phases. This means skyrmion lattices or amorphous skyrmion condensates may be intrinsic equilibrium states in a broad class of noncentrosymmetric magnetic materials, including thin films and bulk compounds like MnSi. The result connects magnetic skyrmion textures to Abrikosov vortex lattices and to the blue phases of liquid crystals, but with a distinct stabilization mechanism.

What carries the argument

The key object is the longitudinal stiffness parameter η entering the two-gradient micromagnetic free energy functional. It expresses the ratio of the energy cost for changing the magnetization amplitude to the cost for rotating the magnetization direction. The argument is carried by the radial energy density comparison: for fixed-moment skyrmions the energy gain from double-twisting near the core is outweighed by excess energy near the cell boundary, but for η < 1 the modulus m(ρ) decreases to zero at the boundary, removing that excess and making the localized cylindrical skyrmion a minimum of the free energy. The Euler equations for m(ρ) and θ(ρ) determine the skyrmion profiles and the phase boundaries T_t and T_S, and full two-dimensional solutions of the extended model confirm that a skyrmion lattice is a genuine ground state.

Extended reading notes

Core claim

The central claim is that for a chiral magnet described by the free energy density f = $Am^{2}$ Σ(∂_i n_j)^2 + ηA(∇m)^2 + D m·rot m + f_0(m), the dimensionless parameter η (the ratio of longitudinal to transverse magnetic stiffness) controls whether skyrmions can be stable. When η = 1 (fixed magnetization amplitude, as in classical Heisenberg models), double-twisting costs too much energy at large distances and skyrmions are unstable, as Wright and Mermin showed. When η < 1, the magnetization amplitude can shrink toward zero at the skyrmion cell boundary, eliminating the energy penalty of double-twist at large radii. Numerical minimization of the Euler equations (7)–(8) then establishes a stable skyrmion phase in the finite temperature window T_t < T < T_S, bounded below by a first-order transition into the helix and above by a continuous transition into the paramagnetic state. The paper further predicts that extended skyrmion condensates form a regular lattice under favorable conditions (analogous to Abrikosov lattices), or a liquid-like/amorphous arrangement in three dimensions, and that such a state explains the partial magnetic order and non-Fermi-liquid behavior observed in pressurized MnSi.

Load-bearing premise

The spontaneous skyrmion ground state exists only if the magnetization amplitude in real chiral magnets is genuinely softer than its stiffness in classical local-moment magnets (η < 1), and the paper's estimates of η for EuS, Ni, and MnSi come from a separate supplementary analysis that is not reproduced in the paper and could be inaccurate.

Editorial extensions

If this is right

Spontaneous skyrmion lattices should exist in noncentrosymmetric ferromagnets with η < 1, both in thin films with broken inversion symmetry and in bulk compounds with chirality, without any applied magnetic field.
The skyrmion phase appears in a narrow temperature window just above the helical ordering temperature; materials with prominent longitudinal softening, such as MnSi (η ≈ 0.4) and Ni (η ≈ 0.65), are the most promising candidates.
The transition from the skyrmion phase to the helix is first-order because the two textures have different topology, while the transition into the paramagnet at T_S is continuous with diverging skyrmion radius and vanishing core magnetization.
The predicted skyrmion condensate resolves the puzzle of partial magnetic order in MnSi: an amorphous arrangement of skyrmion tubes yields neutron scattering intensity concentrated on great circles that intersect along ⟨110⟩, matching observed broad maxima, and gives a T^(3/2) resistivity in an ultrapure, non-Fermi-liquid system.
The mechanism extends to antiferromagnets by replacing the magnetization with a staggered order parameter, and to any chiral ordered state where exchange dominates, chiral interactions are subleading, and anisotropy is weak.

Reading between the lines

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

A direct experimental test of the paper's premise is to measure the longitudinal susceptibility of a chiral ferromagnet like MnSi near T_C and extract η; an observed η ≥ 1 would falsify the predicted spontaneous skyrmion phase, whereas η clearly below 1 would strengthen its likelihood.
The analogy with Abrikosov lattices suggests that impurities and pinning should distort the skyrmion lattice into a glassy or amorphous condensate, so the same mechanism may explain the broad, history-dependent scattering patterns seen in MnSi rather than sharp Bragg peaks.
The paper leaves open the behavior under applied fields: if spontaneous skyrmion lattices exist, moderate fields should reorient and eventually melt them, producing a field-temperature phase diagram with features that should be distinguishable from the field-induced skyrmion phase that appears only under field.
The role of dipolar interactions is treated qualitatively in a footnote; a quantitative extension of the model including dipolar anisotropy would test whether the skyrmion phase survives in real materials where magnetostatic energies are appreciable, especially in thin films.

Original abstract

Since the 1950s Heisenberg and others have attempted to explain the appearance of countable particles in quantum field theory in terms of stable localized field configurations. As an exception Skyrme's model succeeded to describe nuclear particles as localized states, so-called 'skyrmions', within a non-linear field theory. Skyrmions are a characteristic of non-linear continuum models ranging from microscopic to cosmological scales. Skyrmionic states have been found under non-equilibrium conditions, or when stabilised by external fields or the proliferation of topological defects. Examples are Turing patterns in classical liquids, spin textures in quantum Hall magnets, or the blue phases in liquid crystals, respectively. However, it is believed that skyrmions cannot form spontaneous ground states like ferromagnetic or antiferromagnetic order in magnetic materials. Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed matter systems with chiral interactions without the assistance of external fields or the proliferation of defects. We show this within a phenomenological continuum model, that is based on a few material-specific parameters that may be determined from experiment. As a new condition not considered before, we allow for softened amplitude variations of the magnetisation - a key property of, for instance, metallic magnets. Our model implies that spontaneous skyrmion lattice ground states may exist quite generally in a large number of materials, notably at surfaces and in thin films as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Since the 1950s Heisenberg and others have attempted to explain the appearance of countable particles in quantum field theory in terms of stable localized field configurations. As an exception Skyrme's model succeeded to describe nuclear particles as localized states, so-called 'skyrmions', within a non-linear field theory. Skyrmions are a characteristic of non-linear continuum models ranging from microscopic to cosmological scales. Skyrmionic states have been found under non-equilibrium conditions, or when stabilised by external fields or the proliferation of topological defects. Examples are Turing patterns in classical liquids, spin textures in quantum Hall magnets, or the blue phases in liquid crystals, respectively. However, it is believed that skyrmions cannot form spontaneous ground states like ferromagnetic or antiferromagnetic order in magnetic materials. Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed matter systems with chiral interactions without the assistance of external fields or the proliferation of defects. We show this within a phenomenological continuum model, that is based on a few material-specific parameters that may be determined from experiment. As a new condition not considered before, we allow for softened amplitude variations of the magnetisation - a key property of, for instance, metallic magnets. Our model implies that spontaneous skyrmion lattice ground states may exist quite generally in a large number of materials, notably at surfaces and in thin films as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Since the 1950s Heisenberg and others have attempted to explain the appearance of countable particles in quantum field theory in terms of stable localized field configurations.	background 1.00	0.02	0.01	asserted 0.96
s1	As an exception Skyrme's model succeeded to describe nuclear particles as localized states, so-called 'skyrmions', within a non-linear field theory.	background 1.00	0.03	0.02	asserted 1.00
s2	Skyrmions are a characteristic of non-linear continuum models ranging from microscopic to cosmological scales.	background 1.00	0.05	0.02	asserted 1.00
s3	Skyrmionic states have been found under non-equilibrium conditions, or when stabilised by external fields or the proliferation of topological defects.	background 0.99	0.05	0.02	asserted 1.00
s4	Examples are Turing patterns in classical liquids, spin textures in quantum Hall magnets, or the blue phases in liquid crystals, respectively.	background 1.00	0.03	0.01	asserted 0.98
s5	However, it is believed that skyrmions cannot form spontaneous ground states like ferromagnetic or antiferromagnetic order in magnetic materials.	gap 0.54	0.03	0.03	hedged possibility 0.57
s6	Here, we show theoretically that this assumption is wrong and that skyrmion textures may form spontaneously in condensed matter systems with chiral interactions without the assistance of external fields or the proliferation of defects.	contribution 0.55	0.95	0.96	asserted 0.79
s7	We show this within a phenomenological continuum model, that is based on a few material-specific parameters that may be determined from experiment.	method 0.98	0.90	0.05	asserted 0.99
s8	As a new condition not considered before, we allow for softened amplitude variations of the magnetisation - a key property of, for instance, metallic magnets.	method 0.55	0.85	0.04	asserted 0.94
s9	Our model implies that spontaneous skyrmion lattice ground states may exist quite generally in a large number of materials, notably at surfaces and in thin films as well as in bulk compounds, where a lack of space inversion symmetry leads to chiral interactions.	interpretation 0.93	0.83	0.52	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.81.

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Figures

**Figure 1.** FIG. 1: Chiral modulated structures for noncentrosymmetri

Figures reproduced from arXiv:cond-mat/0603103 by the authors.

References

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Note that dipolar interactions do not commute with ferr omagnetic and antiferromagnetic order. Further, dipolar interactions are larger the larger the ordered magnetic moment. This suggests that η < 1 even in the limit q → 0. While classical dipolar interactions may suppress 16 the formation of certain skyrmions they favour the skyrmion s relevant for iso...

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M. Uchida, Y. Onose, Y. Matsui, Y. Tokura, Real-Space Ob servation of Helical Spin Order. Science 311, 359-361 (2006). c ρ b ρ a x -0.16 -0.12 -0.08 -0.04 0 0 0.2 0.4 0.6 0.8 1 energy density w( ρ) radius ρ / R helix skyrmion with fixed m skyrmion with varying m η=0.4 T - TC = 0.22 b=0.05 Fig. 1 18 FIG. 1: Chiral modulated structures for noncentrosymmetri...

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How the literature uses it

6 arXiv papers cite this in 6 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Cites it as background 4 papers

The form of the solitonic Skyrmion solutions stabilized by the Skyrme and Dzyaloshinskii–Moriya (DM) interactions has been investigated extensively [ref] , as has their organization into Skyrmion lattices [ref] , see Ref. [ref] for a recent review.

Revisiting fermion bound states in baby Skyrme background with Dzyaloshinskii Moriya interaction · Introduction · 0.97
It was observed for the first time in 2D magnetic systems [ref] involving Dzyaloshinskii-Moriya interactions (DMI) [ref] , known as magnetic skyrmions.

The relation between the radii and the densities of magnetic skyrmions · Introduction · 0.97
Helimagnets are a special class of materials that realize noncollinear long-range-ordered magnetic structures in the form of proper-screw spin helices [ref] , and fall into two general categories based on their underlying crystal structures and correspondingly the mechanism of stabilization of the magnetic structure: (1) helimagnets in compounds having noncentrosymmetric crystal structures or in which Dzyaloshinkii-Moriya (DM) interactions are possible (Dzyaloshinskii-type helimagnets) [ref] ; and (2) helimagnets in compounds having centrosymmetric crystal structures with inversion symmetry at the midpoint between consecutive magnetic atoms (Yoshimori-type helimagnets) [ref] .

Rich Magnetic Phase Diagram of Putative Helimagnet Sr$_3$Fe$_2$O$_7$ · Introduction · 1.00

Discussion

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Note that dipolar interactions do not commute with ferr omagnetic and antiferromagnetic order. Further, dipolar interactions are larger the larger the ordered magnetic moment. This suggests that η < 1 even in the limit q → 0. While classical dipolar interactions may suppress 16 the formation of certain skyrmions they favour the skyrmion s relevant for iso...

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