Antichiral Ferromagnetism

Filipp N. Rybakov∗1, Anastasiia Pervishko2, Olle Eriksson3, and Egor Babaev1

1Department of Physics, KTH Royal Institute of Technology, SE-10691 Stockholm, Sweden2Skolkovo Institute of Science and Technology, Moscow 121205, Russia

3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

Here by combining a symmetry-based analysis with numerical computations we predict a new kind ofmagnetic ordering – antichiral ferromagnetism. The relationship between chiral and antichiral magneticorder is conceptually similar to the relationship between ferromagnetic and antiferromagnetic order. With-out loss of generality, we focus our investigation on crystals with full tetrahedral symmetry where chiralinteraction terms – Lifshitz invariants – are forbidden by symmetry. However, we demonstrate that lead-ing chirality-related term leads to nontrivial smooth magnetic textures in the form of helix-like segmentsof alternating opposite chiralities. The unconventional order manifests itself beyond the ground state bystabilizing excitations such as domains and skyrmions in an antichiral form.

Chirality and chiral textures is a cornerstone con-cept in many fields of physics ranging from cosmol-ogy to nuclear and elementary particle physics [1]. Inmany materials, chiral crystal structure results in achiral ferromagnetic ordering [2, 3]. The chiral tex-tures in magnetism attracted renewed interest in re-cent decades thanks to their potential relevance totechnological applications, including alternative logicdevices and racetrack memory where the informationis encoded by virtue of magnetic textures [4–7]. In-terestingly, some of the concepts developed so farare intended for the use of materials that have fer-romagnetic ordering at the atomistic level, while atthe mesoscale the direction of magnetization m(r) ismodulated. The corresponding basic types of modu-lations are one-dimensional and conventionally calledspirals, helices, cycloids, screws, to name a few [8].

The emergence of modulated ferromagnetic order-ing may appear due to competing symmetric interac-tions between atoms in a lattice [9, 10]. The multi-spin exchange interactions, in particular four-spin in-teractions, also provide a possible mechanism for themodulated texture stabilization beyond pairwise sce-nario [11]. Most naturally, multi-spin exchange inter-actions can emerge in the form of two-site biquadraticor four-site spin interaction [12, 13]. The delicate in-terplay between competing pairs and/or accountingfor multi-spin interactions, leading to the stabilizationof different magnetic textures, has been investigatedboth theoretically and experimentally in Refs. [14–19].By virtue of symmetry of the above mentioned andother symmetric interactions, the reflection of modu-lated magnetic texture is defined by degenerate energystates with opposite chirality (handedness), where acertain chirality is a consequence of spontaneous sym-metry breaking.

An alternative mechanism of nucleation andstability of modulated ferromagnetic textures maybe attributed to the presence of the pairwise

Dzyaloshinskii-Moriya interaction (DMI), which is re-sponsible for an asymmetric nearest-neighbor spin ex-change, often discussed in magnetic systems whereinversion symmetry is absent. Along with the mag-netic ultrathin films and multilayers where the inver-sion symmetry is broken by natural means [5], the ef-fect of DMI is pronounced in cubic crystals with chiralpoint group symmetry T , such as B20-type FeGe andMnSi, in which a certain chirality of a magnetic he-lix is dictated by one of two possible enantiomorphicforms of the compound [20]. In this case, the corre-sponding magnetic Hamiltonian includes the Lifshitzinvariants [2, 21], and the chirality depends on thesign of their common factor.

In this paper, we predict a ferromagnetic orderingthat is fundamentally different from those discussedabove which we refer to as antichiral ferromagnetism.This term aims to reflect that spontaneous modula-tion of the magnetization direction appears in a waythat both types of chirality exist simultaneously, andalternate in space. Our analysis reveal that this mag-netic ordering naturally appears in a bulk ferromagnetwith the point group symmetry Td owing to achiralcrystal symmetry. This is a class of crystals in whichmany minerals are formed naturally [22, 23].

The macroscopic robustness of a magnetic config-uration is purely determined by its stability with re-spect to perturbations that violate spatial uniformity.If, for instance, inversion symmetry is broken the spinalignment might gain a certain chirality due to spin-orbit driven antisymmetric DMI that contributes tothe total energy with the terms linear with respectto the first spatial derivatives of magnetization. Ingeneral, the derivative linear contribution to the freeenergy of a ferromagnet can be casted in the form

H∇ =∫dr∑αβ

Ωαβ(m)∇αmβ , (1)

with the tensor Ωαβ(−m) = −Ωαβ(m) being odd∗f.n.rybakov@gmail.com

1

arX

iv:2

012.

0583

5v2

[co

nd-m

at.s

tr-e

l] 2

2 D

ec 2

020

under magnetization inversion [24], while α, β cor-respond to the spatial indexes. By expanding

Ωαβ(m) =∑γ Ω

(1)αβγmγ +

∑γδε Ω

(3)αβγδεmγmδmε in

powers of m and restricting to linear and cubic con-tribution only one arrives at

H∇ =1

2

∫dr∑αβγ

(DSαβγ∇α(mβmγ) +DAαβγL

(α)γβ

)+

∫dr∑αβγδ

Ω(3)αβγδεmγmδmε∇αmβ , (2)

where for convenience we have separated the termsarising due to Ω(1) into two parts, namely are symmet-ric terms, ∇α(mβmγ), and terms given in the form ofLifshitz invariants, L(α)βγ = mβ∇αmγ−mγ∇αmβ [25].Note that as the symmetric contribution can be ex-pressed in terms of surface integrals via Stokes’ theo-rem [24] its impact on the magnetic ordering in bulkcrystals can be discarded.

To proceed, we analyze (2) based on symmetrygrounds (see the Appendices). The solution for Tdpoint group symmetry is trivial with respect to Lif-shitz invariants, DAαβγ = 0, whereas in the case ofΩ(3) one can identify four independent componentswith the corresponding invariants given by∑

′∇α(m3βmγ +mβm3γ),

∑′∇α(m2αmβmγ), (3a)∑

′mαmβ∇γ(m2), (3b)

mxmymz∇ ·m, (3c)

where∑′ denote the sum over (α, β, γ) ∈

{(x, y, z), (y, z, x), (z, x, y)}. The invariant (3c) wasfirst reported by I. Ado et al. [26, 27], whereas we de-rive the complete set (3a)-(3c). After integrating byparts both terms (3a) can be discarded for the samereasons as the terms ∇α(mβmγ) in the above anal-ysis. Note that within the micromagnetic approachmagnetization is described in terms of a unit vector|m(r)| = 1 and thus the invariant (3b) vanishes aswell. Therefore we end up with (3c) as the only rel-evant term for Eq. (1).

We base our subsequent analysis on the minimalmodel of a tetrahedral ferromagnet that contains ex-change interaction and Ado interaction (3c):

H =∫dr(A|∇m|2 + Bmxmymz∇ ·m

), (4)

where A is the exchange stiffness. It worth notingthat in a somewhat more general case, the energydensity in (4) may be equipped with a magnetic-field-induced Zeeman term, cubic anisotropy ∝(m2xm

2y +m

2xm

2z +m

2ym

2z

), and extensions in the case

of exchange anisotropy. Without loss of generality, weassume B > 0, since the sign of this constant dependson the choice of the coordinates, see Fig. 1. To iden-tify the lowest energy state, as well as stable excitedconfigurations, we perform a numerical minimizationof the energy (see the Appendices).

x

𝓍

𝓎𝓏

+ℬmxmymzdiv(m)

−ℬm𝓍m𝓎m𝓏div(m)

Td crystal

[010]

[100]

[001]

y

z

Figure 1: A sketch of bulk crystal with achiralpoint group symmetry Td. By virtue of rotore-flection S4 invariants under the crystallographic pointgroup Td, the sign of the phenomenological constantB alternates at 90◦ rotations of the coordinate frame.

In Figs. 2(b)-(d) we provide relevant energy mini-mum configurations that emerge from an initial guesswith randomly oriented spins, as depicted in Fig. 2(a).We observe that the numerical solutions to (4) can beroughly classified into three different groups, namelyare multi-domain textures shown in Fig. 2(b), mod-ulated states with small inclusions such as edge dis-locations illustrated in Fig. 2(c), and defect-free pe-riodic modulations unveiled in Fig. 2(d). Interest-ingly, the magnetic structure in Fig. 2(d) was foundto be the energetically most favorable. Reducing thespatial dimension to two, while increasing the size ofthe modeling area, we also obtained that the type ofmodulations depicted in Fig. 2(d) is the most energet-ically favourable configuration. A closer inspection ofthe magnetic ordering in the structure is visualizedin Fig. 2(e). It can be clearly seen that locally inspace the magnetic spiral has a spontaneous distinctchirality. However the sign of the chirality varies inspace, resulting in a bichiral configuration that be-ing averaged over a modulation period produces zero.It is therefore not surprising that the relationship be-tween the local chirality as depicted in Fig. 2(e) is in away similar to the relationship between ferromagneticand antiferromagnetic ordering of individual atomicmoments. Here, alternating local moments are effec-tively replaced with alternating local chiralities. Onthis account we label this state as antichiral. Thepolarity demonstrated in Fig. 2(e) is twofold degener-ate. The spatial orientation of the texture is sixfolddegenerate, and the resulting twelvefold degeneracyshould be spontaneously broken in this model. In afinite sample, we expect that the significant net mag-netization and stray fields will favor the split into thedomains.

Having established the ground state of the sys-tem we proceed with the discussion on the possibil-ity of stable excited configurations, apart from thosefound by numerical energy minimization from the ran-dom state. Notably, several classes of magnets al-low particle-like topological excitations, such as mag-netic skyrmions [5, 7, 25]. In order to resolve numer-ically whether the present system has similar type

2

x

a b

minimizationenergy➪

random initial guess

color scheme

c d

m(r)

y

z

642.5A/ℬ

⟨m(r)⟩

{001}

polarity

handedness

e f g

h i50A/ℬ

m[110] m[110]m[001]

0.8

0.90.5

-0.5-0.54

0.54r[110]

Figure 2: Results of numerical simulations obtained by energy minimization. a, Initial guess withchaotically oriented magnetization for the cube under periodic boundary conditions. b – d, Typical solutionscorresponding to an energy minimum: multi-domain structure, magnetization modulations with edge disloca-tions, and perfect helix-like modulations, respectively. e, Antichiral magnetic texture that corresponds to theglobal energy minimum. The optimal period, L is found to be ≈ 75.72A/B, while the net magnetization is|〈m(r)〉| ≈ 0.854. The figure captures two periods, while the inserts at the bottom – one period. f – h, Stableantichiral excitations, including f, antiskyrmion, g, skyrmion, and h, a cluster composed by a pair of skyrmionand antiskyrmion. The approximate energy values of these states are 12.02, 12.02, 23.58, respectively. Theenergy is calculated relative to the global minimum in units of 2At, where t is the thickness in the directionperpendicular to the picture plane. i, Enlarged area of the skyrmion core.

excitations we use the vortex-like ansatz [25] as aninitial guess, superimposing modulation mimickingthe ground state discussed above. Our findings sug-gest that the model possesses stable states in shapeof skyrmions that form over antichiral background.Some of the solutions with distinct topology that havebeen discovered as a result of numerical energy min-imization are shown in Fig. 2(f)-(h). Interestingly,contrasting to their counterparts in chiral magnets,skyrmions that we obtain in this model do not have apredefined chirality.

We demonstrated that the magnetic ordering incrystals with Td (tetrahedral class) point group canbe of a novel type, that we refer to as antichiral ferro-magnetism. Among the methods that may be used toprobe the predicted antichiral ordering one can high-light small-angle neutron scattering, spin-polarizedscanning probe microscopy [28] as well as the electronmagnetic circular dichroism technique [29].

Our numerical findings reveal that antichiralground state has a set of stable topologically non-trivial excitations in the form of edge dislocationsand skyrmions. These excitations are antichiral incontrast to their chiral counterparts in systems withDzyaloshinskii-Moriya interaction. We strongly be-lieve their properties pose an intriguing question forfollow-up studies and will trigger experimental activ-ity in this field.

s3

[010]

[100]

[001]s4

s2

s1

h3

Figure 3: A possible structural building block,satisfying full tetrahedral symmetry. Magneticspins (arrows), and relevant magnetic interactionpaths (see explanation in the text) of the magneticunit cell.

Another interesting aspect of our numerical find-ings quests for the microscopic origin of the term (3c).Due to anisotropy of the crystal, shown in Fig. 1, themicroscopic Hamiltonian for classical spins si must beequipped with direction vectors. The possible termthat corresponds to such a contribution to the micro-

3

magnetic energy may be cast in the form

H = β

(4∑i=1

si · hi

)4∑i=1

(si · hi)3, (5)

where unit vectors hi are aligned with the bonds con-necting the center of a tetrahedron and the corre-sponding vertex, see Fig. 3, while β represents thestrength of the interaction. Note that multi-spin in-teractions whose Hamiltonian contains direction vec-tors were recently discovered in B20-type cubic chi-ral magnets [30]. Moreover, we expect that a certaininsight on the nature of pairwise and multi-spin in-teractions [31, 32] can be gained on the basis of first-principles calculations that could clarify the micro-scopic origin of the magnetic interactions leading toantichiral ferromagnetism in Td crystals and beyond.

Note added. Antichiral magnetic ordering pre-sented here should not be confused with the “helix”reported earlier in [27]. Right after the first versionof this manuscript [33], I. A. Ado et al. reported anapproximation of the ground state solution of Hamil-tonian (4) [34]. This approximation is close to oursolution and captures antichirality. Based on our nu-merical solution, we suggest a simple and very accu-rate analytical approximation for the ground state,

m[110] ≈ 0.54 sin(k r[110]), m[001] ≈ 0.5 cos(k r[110]),

m[1̄10] =√

1−m2[110] −m2[001],

k ≡ 2πL, r[110] ≡

x+ y√2, L ≈ 76A

B.

This expression gives average energy density〈ρH〉 ≈ −1.8676 · 10−3B2/A, while our original nu-merical result predicts

〈ρH〉 = −1.8696582776 · 10−3B2/A,L = 75.725021 A/B,|〈m(r)〉| = 0.853861965,

where all digits are expected to be accurate.The authors of [34] also proposed an alternative

to the version of the microscopic Hamiltonian usedhere (5).

Appendices

A1. Symmetry analysis method

Given a point group symmetry with generators R(µ),a tensorial structure is dictated by the symmetry re-lations

Ω(n)α1α2...αn = R(µ)α1β1R(µ)α2β2 . . .R

(µ)αnβn

Ω(n)β1β2...βn

, (8)

where R(µ)αβ are orthogonal matrices of three-dimensional irreducible representations for each ele-ment µ of the group. The corresponding matrices forthe tetrahedral point group Td can be found, e.g., inRefs. [35–37]. Reduction of the linear system (8) iden-tifies zero and non-zero components of the tensor Ω.

A2. Numerical energy minimization

The continuous model as yielded by Eq. (4) wasdiscretized using a rectangular grid under periodicboundary conditions. We implemented a discretiza-tion scheme giving eighth-order of accuracy by gener-alizing the approach of Donahue and McMichael [38].The typical number of grid points in each dimensionranged from 140 to 160. For minimization we used aGPU-parallelized nonlinear conjugate gradient algo-rithm [39], while the constraint |m| = 1 was satisfiedby means of the special use of stereographic projec-tions (see e.g. Supplemental Material in Ref. [40]).

Acknowledgments

The work of A.P. was supported by the Russian Sci-ence Foundation Project No. 20-72-00044. E.B.and F.N.R. were supported by the Swedish ResearchCouncil Grants No. 642-2013-7837, 2016-06122, 2018-03659, and Göran Gustafsson Foundation for Re-search in Natural Sciences. O.E. acknowledges sup-port from the Swedish Research Council, the Knutand Alice Wallenberg foundation, the Foundation forStrategic Research, the European Research Council,and eSSENCE.

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1 Appendices1.1 A1. Symmetry analysis method1.2 A2. Numerical energy minimization

2 Acknowledgments