The behavior of any physical system from cosmological objects in field theories1 through living organisms in biology2 up to synthetic objects in condensed3 and soft4 matter is determined by the order parameter that lives in spacetime. The spatial distribution of the order parameter is governed by the geometry of the physical space of the object, in particular its dimensionality and curvature5,6. As the order parameter and fields are local quantities, it is natural to approach the description of curvilinear-geometry-induced effects based on theories, which involve local interactions only. For many systems, if not for all, this local description is incomplete as nonlocal interactions naturally appear in gravity theory7, quantum information science (quantum nonlocality and entanglement)8, response theory in electrodynamics9 and magnetism10,11. The interplay of the order parameter of magnetically ordered materials (magnetization) with the curvilinear geometry is still described at the level when local and nonlocal interactions are treated separately12. This makes the description of the systems inherently incomplete as the theory fails to describe the impact of curvature effects on the statics and dynamics of micromagnetic textures like magnetic domains, skyrmion bubbles and vortices.

Here, we present a micromagnetic framework of curvilinear magnetism. The theory describes the impact of curvature-induced effects, driven by both local13 and nonlocal14,15 interactions, on the statics and dynamics of magnetic textures in extended curved thin shells. Fundamentally, we report nonlocal chiral symmetry-breaking effect revealing itself even in statics, which does not exist in flat thin films (compare Fig. 1a–h). To understand the nonlocal physics in curved thin shells, we go beyond the established physical picture relying on surface and volume magnetostatic charges (Fig. 1i). We demonstrate that the interactions in geometrically curved thin shells unveil themselves in full by involving three fictitious mathematical constructions in the description. In the spirit of the seminal work by Brown10, we name them magnetostatic charges now consisting of a surface charge and two charges originated from the divergence of magnetization, which is classically referred to as a volume charge. According to their geometrical properties, we term them tangential charge and geometrical charge (compare Fig. 1i, j). Being determined by the tangential magnetization components only, the tangential charge coincides with the volume charge in flat films. The geometrical charge is determined by local characteristics of the curved surface and properties of the magnetic texture. In contrast to surface and tangential charges, this newcomer resembles the coupling of the magnetic subsystem and geometrical curvature. We demonstrate that in geometrically curved systems, all three charges do interact pairwise with nontrivial contribution to the total energy (compare Fig. 1k–m). The nonlocal chiral symmetry-breaking effect in static magnetic textures is not only unexplored, it is not even expected in thin flat geometries. We are convinced that its analysis will stimulate the use of curved films to realize chiral magnets for fundamentally appealing and technologically relevant skyrmionic spin–orbitronic systems. The impact of this work goes well beyond the magnetism community. The presented approach can be extended for studying the evolution of generic vector fields on curved shells in different models of condensed (graphene16 and superconductors17) and soft (nematics18 and cholesterics19) matter.

Fig. 1: Micromagnetism of static magnetization texture in flat and curved thin shells.
figure 1

Local spin–orbit interaction and symmetry breaking at the interfaces lead to the appearance of anisotropy (a)54, and chiral terms (b)55, in the energy functional of a magnetic thin film. Geometrically broken symmetry in curved magnetic shells brings about emergent anisotropy (e)56, and chiral (f)57, terms stemming from local exchange interaction. Nonlocal magnetostatic interaction can be reduced to a shape anisotropy in the case of thin flat (c)58, and curved (g)59, magnetic shells. Chiral symmetry-breaking effects stemming from nonlocal magnetostatics do not exist in statics in the case of flat thin films. d We report nonlocal chiral symmetry-breaking effect in curvilinear magnetic shells: h representative example, where nonlocal chiral symmetry breaking leads to the selection of the handedness of the Neel domain wall in a complex 3D curved out-of-plane magnetized shell. i Nonlocal magnetostatic interactions in the case of flat thin films can be described in the framework of Brown10 using two fictitious charges: surface (pink shaded disks) and volume (red-shaded body). j The nonlocal physics in curved shells can be understood relying on three chargers: in addition to surface charge, it is insightful to introduce tangential (red-shaded body) and geometrical charges (green-shaded disks); in flat films tangential charge coincides with the volume charge. k, l All three charges do interact pairwise with a nontrivial contribution to the energy. m In flat films, the interaction between surface and volume charges vanishes. Panel (a) reprinted by permission from Springer Nature, Nature Materials54; (b) reprinted by permission from Springer Nature, Nature Materials55; (c) reprinted by permission from Springer Nature, Nature58; (e) reprinted with permission from ref. 56 copyright (2011) by the American Physical Society; (f) reprinted with permission from ref. 57 copyright (2018) by the American Physical Society; (g) reprinted with permission from ref. 59 copyright (2012) by the American Physical Society.


Magnetostatics of a curvilinear shell

We consider a curved ferromagnetic shell of thickness h. Its shape is locally described by Gaussian \({\mathcal{K}}({\bf{r}})\) and mean \({\mathcal{H}}({\bf{r}})\) curvatures. Shells are described as an extrusion of a surface ς(r) by a constant value h along the vector \(\hat{{\bf{n}}}=\hat{{\bf{n}}}({\bf{r}})\) normal to the surface. We assume that the magnetization does not depend on the thickness coordinate along \(\hat{{\bf{n}}}\). Our model includes exchange, anisotropy, and magnetostatic interactions in the energy functional. To separate the explicit curvature effects from the spurious effects of the curvilinear reference frame, it is imperative to choose the Darboux reference frame along the principal axes (see Supplementary Note 1 for details). This allows to adapt the language of covariant derivatives used in general relativity to the case of thin-shell condensed matter systems. Accounting for local intrinsic interactions only (exchange and anisotropy) allows to identify emergent curvature-induced interactions, i.e., anisotropy and Dzyaloshinskii–Moriya interaction (DMI)13,20. These interactions are essentially local and already widely explored for the prediction of chiral effects in curved wires and shells12,21,22.

Following the same formalism but taking into consideration the nonlocal magnetostatic interaction, we identify the first striking property: in contrast to a flat thin film, the transformation in a curvilinear frame requires that the divergence of magnetization m is determined not only by the spatial derivatives of the magnetization, which determine the tangential charge ρ, but also by a new geometrical magnetostatic charge g, representing coupling of the magnetic texture and curvature

$$\begin{array}{l}-{\boldsymbol{\nabla }}\cdot {\bf{m}}=\rho +g,\\ \rho ({\bf{r}})=-{\eth}_{i}{m}_{i}\equiv -{\eth}_{\alpha }{m}_{\alpha },\qquad g({\bf{r}})={\mathcal{H}}({\bf{r}}){m}_{n}({\bf{r}}).\end{array}$$

The symbol \(\eth\) defines the modified tangential derivatives of magnetization in curvilinear reference frame, \({\bf{m}}={m}_{i}{{\bf{e}}}_{i}\equiv {m}_{\alpha }{{\bf{e}}}_{\alpha }+{m}_{n}\hat{{\bf{n}}}\), with tangential components mα and the normal component mn; \({\mathcal{H}}\) is the mean curvature, which describes locally the shape of the curved shell. The geometrical charge vanishes for flat films and minimal surfaces (with \({\mathcal{H}}\equiv 0\)) or for tangential magnetic texture with mn = 0.

The geometrical charge leads to the appearance of physical effects that are intrinsically nonlocal and reveal themselves in nonlocal symmetry breaking. The geometrically broken symmetry due to the curvature results in the reorganization of the magnetostatic energy terms in the form, adapted to the geometry

$${E}_{{\rm{d}}}={M}_{{\rm{s}}}^{2}\int {\rm{d}}S{w}_{\sigma -\sigma }+{M}_{{\rm{s}}}^{2}\int {\rm{d}}{\bf{r}}{w}_{\rho -\rho }$$
$$+{M}_{{\rm{s}}}^{2}\int {\rm{d}}{\bf{r}}\left({w}_{g-\sigma }+{w}_{g-g}+{w}_{g-\rho }+{w}_{\sigma -\rho }\right),$$

where Ms is the saturation magnetization. The first two terms (2A) are similar to the planar case

$${w}_{\sigma -\sigma }=\frac{\sigma ({\bf{r}})}{2}\int \frac{\sigma ({\bf{r}}^{\prime} ){\rm{d}}S^{\prime} }{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|},\ {w}_{\rho -\rho }=\frac{\rho ({\bf{r}})}{2}\int {\rm{d}}{\bf{r}}^{\prime} \frac{\rho ({\bf{r}}^{\prime} )}{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|}.$$

Here, wσσ describes the interaction between the surfaces charges σ± = m n± at the top/bottom surfaces of the shell. In the main order on the shell thickness h, this surface term is local, \({w}_{\sigma -\sigma }=(h/2){({\bf{m}}\cdot \hat{{\bf{n}}})}^{2}+{\scriptstyle \mathcal{O}}(h)\)23,24, and typically leads to the renormalization of anisotropy coefficients. It is the only term, linear in h stemming from the magnetostatic interaction. All other contributions to the magnetostatic interaction are essentially nonlocal23,24,25. In the thin-shell limit, they scale as \({\scriptstyle \mathcal{O}}(h)\).

The magnetostatic terms wgσ and wgg describe the interaction between geometrical and surface charges, and the interaction between geometrical charges, respectively

$${w}_{g-\sigma }=g({\bf{r}})\int\frac{\sigma ({\bf{r}}^{\prime} ){\rm{d}}S^{\prime} }{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|},\ {w}_{g-g}=\frac{g({\bf{r}})}{2}\int\frac{g({\bf{r}}^{\prime} ){\rm{d}}{\bf{r}}^{\prime} }{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|}.$$

Both contributions provide nonlocal coupling between normal magnetization components. The interaction between the geometrical and tangential charges

$${w}_{g-\rho }=\rho ({\bf{r}})\int\frac{g({\bf{r}}^{\prime} ){\rm{d}}{\bf{r}}^{\prime} }{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|},$$

favors the coupling between the normal magnetization mn and spatial derivatives of the in-surface components mα. Another magnetostatic term of similar symmetry

$${w}_{\sigma -\rho }=\rho ({\bf{r}})\int\frac{\sigma ({\bf{r}}^{\prime} ){\rm{d}}S^{\prime} }{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|},$$

is caused by the interaction between magnetostatic surface and tangential charges. We note that this term does not contribute to the total energy of extended flat thin films (Fig. 1m, wσρ ≡ 0). Among others, wσρ naturally appears in cylindrical and spherical shells due to the difference in the area of the inner and outer surfaces. Both interactions wgρ and wσρ vanish for any homogeneous magnetic texture in a curved shell.

In addition to the established curvature-induced local anisotropy and chiral terms, driven by the exchange interaction13,20, the nonlocal symmetry breaking of magnetic textures discussed in the following section, stems from the magnetostatic interaction. Understanding of nonlocal physics in curvilinear shells naturally involves three magnetostatic charges: surface, tangential, and geometrical charges (Fig. 1k). The latter one is determined by the mean curvature \({\mathcal{H}}\) coupled to the magnetic texture. In flat thin films, the interaction between volume ρ and surface σ charges does not contribute to the energy: taking into account that magnetization does not depend on the normal coordinate, surface charges on opposite surfaces of any flat thin film will compensate each other; the term (Eq. 2F) gives necessarily zero contribution to the energy density of a thin film, \({w}_{\sigma -\rho }^{{\rm{flat}}}\equiv 0\) (Fig. 1m). In contrast, in curved magnetic shells, these symmetry operations (i.e., mirror symmetry about the curved surface) are absent, hence wσρ ≠ 0 for inhomogeneous textures (Fig. 1k, l).

Symmetry-breaking effects

The pair interactions between three charges, ρ, σ, and g, result in symmetry-breaking effects. Different energy contributions (Eqs. 2DF) have different spatial and magnetic symmetries. First, the magnetic symmetry of wgσ and wgg preserves the mirror symmetry of the magnetic texture about the curved surface, and therefore cannot lead to chiral effects.

Two other terms, wgρ and wσρ, favor a coupling between the normal magnetization component and the direction of the gradient of the in-surface magnetization components. For instance, due to the wgρ term, and for the case mn ≠ 0 and ðαmα > 0, the magnetization favors tilt outward the surface for positive mean curvatures, \({\mathcal{H}}\,> \,0\), and inward for \({\mathcal{H}}\, <\, 0\). The symmetry of these terms supports chiral symmetry breaking yet in specific cases. The sufficient conditions to admit the nonlocal chiral symmetry breaking about the x1 axis can be formulated as follows: (i) existence of a symmetry about a flat mirror in Cartesian frame about one of the principal axis x1 for a curved shell, (ii) the magnetic texture needs to have both out-of-surface components mn and spatially inhomogeneous in-surface magnetic texture at least along x1, and (iii) the magnetization components mn and mα should be of different parity with respect to the reflection x1 → −x1. These conditions can be satisfied for noncollinear magnetic textures like domain walls not only for a class of developable surfaces (Fig. 2), e.g., a generalized cylinder with generatrix x1, but also for a complex curvilinear shell as an elliptic cylinder shell with a bottleneck shown in Fig. 1h. Such surfaces provide preferential direction x1, which can act as a possible chiral axis for the magnetization texture. Depending on the geometry and material parameters, one of the possible equilibrium states is the normally magnetized shell. We emphasize that this is an assumed ground state that is a subject of further modifications due to local and nonlocal curvilinear effects. With respect to the modifications of this assumed state due to the local interaction (see Fig. 2): local anisotropy together with local chiral effects will lead to the appearance of the azimuthal component m2(x2). In accordance with the above analysis, there will be a nonlocal chiral symmetry breaking for the Neel domain wall: two domain walls (along and opposite x1) are energetically not equivalent, see Fig. 1h. In this respect, the nonlocal chiral symmetry breaking is similar to its local counterpart, the DMI: the sign of DMI vector determines whether right-handed or left-handed canting between the two spins is the lower energy configuration. This is how the local DMI as well as the nonlocal chiral symmetry breaking can favor homochiral spin textures with one handedness over the other26. This is the first prediction of chiral symmetry breaking stemming from nonlocal interactions in static magnetic textures.

Fig. 2: Interplay between symmetry of magnetic interactions, curvature, and magnetic texture.
figure 2

Effects of curvature-induced interactions on assumed equilibrium state \(\widetilde{{\bf{m}}}\) given by the anisotropy for different symmetries. Here, each of the curvatures is considered either zero or nonconstant function. We consider that the respective energy term will impact the texture if the term is nonzero. The central and right-hand parts of the figure contain consequences of the input given on the left. A direction ea is one of the unit vectors or their linear combination.

Although there are striking similarities with intrinsic DMI and curvature-induced exchange-driven DMI (both are local interactions), the nonlocal chiral symmetry breaking possesses essential fundamental differences. Namely, the symmetry of local chiral interaction terms is dictated by Lifshitz invariants, possessing properties similar to the antisymmetric curl operator. Therefore, there appear twists on the magnetic texture in Dzyaloshinskii spirals in 1D wires27, chiral28 and transverse29 domain walls in flat nanostripes with intrinsic DMI, or twists in transverse30 and periodic31 domain walls in quasi-1D magnetic helices. Instead, the nonlocal chiral symmetry breaking originates from the pair interaction of tangential, surface, and geometrical charges. The differential properties, caused by the tangential charges, are similar to the surface divergence operator, which does not support twists on the magnetic texture.


The developed micromagnetism of curvilinear shells allows to make generic predictions on how the properties of magnetic textures will change, depending on the type of curved surface they live in. Considering magnetic shells with strong uniaxial anisotropy, we can readily perform analysis of textures, which do not deviate significantly from the assumed equilibrium state stabilized by the magnetocrystalline anisotropy. We can assess if the resulting magnetic texture will be modified due to the presence of a curvature-induced anisotropy and if the texture is chiral. In this respect, we analyze the energy contributions from local (anisotropy \({w}_{{\rm{A}}}^{{\rm{x}}}\) and DMI \({w}_{{\rm{D}}1}^{{\rm{x}}}\) and \({w}_{{\rm{D}}2}^{{\rm{x}}}\)) and nonlocal terms (Fig. 2) accounting for local curvatures and the direction of the easy axis in the vicinity of a given point. To illustrate the above approach, we consider a developable surface with vanishing Gaussian curvature, \({\mathcal{K}}=0\), and nonconstant second principal curvature (κ1 = 0, κ2 ≠ 0), e.g., elliptic cylinder or ripple, and assume that magnetic easy axis is pointing normally to the surface. Then, the magnetic texture m = m(x2) is influenced by \({w}_{{\rm{D}}2}^{{\rm{x}}}\) (local chiral term). Nonlocal magnetostatic terms, wgσ, wgg, and wgρ, are also present since the mean curvature is nonzero. The term wσρ, which is responsible for the interaction between the surface and tangential charges, can appear due to inequivalence of top and bottom surfaces of the shell for inhomogeneous magnetization texture with nonvanishing tangential magnetostatic charges. Based on these considerations, the assumed equilibrium state \(\widetilde{{\bf{m}}}\) (e.g., normally magnetized elliptic cylinder) will be modified due to local and nonlocal curvature effects as follows: (i) the state will be with a break of symmetry (and, possibly, chiral), i.e., deviation from the \(\hat{{\bf{n}}}\) is linear with respect to κ2(x2), and (ii) effective easy–normal anisotropy will be inhomogeneously changed. As a result of this consideration, the initially assumed strictly normal magnetization distribution is modified by the appearance of the m2 component: m = {0, m2(x2), mn(x2)}. Further details are described in “Methods”.

The proposed theory can be used to describe not only complex static magnetic textures but also dynamic excitations. As a characteristic example, the dynamic responses of tubular and spherical architectures are discussed in Supplementary Note 1.

The magnetism in curved geometries encompasses a range of fascinating geometry-induced effects ranging from emergent anisotropy and DMI to the possibility to create new magnetoelectric responses12,32. Here, we put forth a platform for theoretical analysis of micromagnetic textures in curvilinear ferromagnetic shells of different geometries. The developed generalized micromagnetic framework of curvilinear ferromagnetic shells treats local (exchange and anisotropy) and nonlocal (magnetostatics) interactions on equal footing.

A direct consequence of accounting for nonlocality is the appearance of three magnetostatic charges, which do interact with each other pairwise. This interaction brings about nonlocal chiral symmetry-breaking effect. We note that this finding is not the result of adding new energy terms to the energy functional of a flat thin film. Furthermore, there is no counterpart of this mechanism in any flat system possessing the same energy functional. Similar to the established spin–orbit-induced DMI and curvature-induced local chiral interactions, the nonlocal chiral symmetry breaking leads to the stabilization of magnetic textures with preferred handedness. Still, the physical consequences of this effect are distinct from local chiral effects. While local intrinsic DMI in helimagnets causes twists in magnetic textures (as spiralization27), nonlocal magnetostatics results in handedness but not in the appearance of a twist. In this respect, nonlocal chiral symmetry breaking is different even from the curvature-induced DMI originating from the exchange interaction: the local exchange-driven DMI in chiral curvilinear geometries (helix31,33, Möbius34) results in the appearance of a twist in a magnetic texture similar to the intrinsic DMI.

Magnetostatics influences on the magnetic textures of a thin curved shell because of the presence of shell surfaces: both face and edge ones. In the current study, we focus on the curvature effects caused by face surfaces, which is applicable for extended shells. Hence, we do not consider possible boundary effects. For instance, complex geometry of the boundary (e.g., boundary notches and curvature) can be the source of other effects, which are already widely used to control the domain wall propagation in flat stripes with complex boundaries35,36,37, and formation of vortices in magnetically soft disks38. The reason is that boundary effects result in complementary but not the same ways to the symmetry breaking in magnetic textures. Here we discuss only the effects, related to point geometrical characteristics, fundamentally important for curvilinear shells.

The nonlocal chiral symmetry breaking is a mechanism toward tuning material responses relying on the choice of the geometry in contrast to the conventional route of the optimization of material properties. The possibility to stabilize the preferred handedness of a nonlocal magnetic texture, e.g., the domain wall in the intrinsically not chiral magnets even in statics is technologically relevant for current-driven domain wall motion in racetrack memory39,40 and domain wall logic devices41. The broken time-reversal and space-inversion symmetries in curved magnetic shells are fundamentally appealing for the realization of exotic materials possessing ferrotoroidic order42 and design of magnetoelectric responses in geometrically curved but still conventional ferromagnetic thin films. These appealing predictions are yet to be explored theoretically and in experiments.

The impact of this work goes beyond the magnetism community. The approach toward the description of a generic vector field can be applied in various emergent fields of studies of curvature effects in condensed and soft matter. Among the prospective application, there are modification of local electronic properties of a twisted graphene bilayer induced by a strain and a high curvature16, the evolution of the electron spin orientation d vector in curved superconducting nanostructures43, domain texture, described by the polarization vector in flexible ferroelectric inorganic44 and organic45 crystals, and manipulation of molecule alignment in liquid crystals18, including the effect of curvature on defects and assembly46.


Magnetic interactions in curvilinear geometries

We consider a model of ferromagnetic shell with magnetic texture controlled by the exchange, anisotropy, and magnetostatic interactions. The total energy has the following form:

$$E= \, {E}^{{\rm{x}}}+{E}^{{\rm{an}}}+{E}^{{\rm{d}}},\\ {E}^{{\rm{x}}}= \, -A\int {\rm{d}}{\bf{r}}\ {\bf{m}}\cdot {{\boldsymbol{\nabla }}}^{2}{\bf{m}},\\ {E}^{{\rm{an}}}= \, \frac{K}{2}\int {\rm{d}}{\bf{r}}{\left({\bf{m}}\cdot {{\bf{e}}}_{{\rm{a}}}\right)}^{2},\\ {E}^{{\rm{d}}}= \, \frac{{M}_{{\rm{s}}}^{2}}{2}\int {\rm{d}}{\bf{r}}\int\ {\rm{d}}{\bf{r}}^{\prime} \left({\bf{m}}({\bf{r}})\cdot {\boldsymbol{\nabla }}\right)\left({\bf{m}}({\bf{r}}^{\prime} )\cdot {\boldsymbol{\nabla }}^{\prime} \right)\frac{1}{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|},$$

where A is the exchange stiffness and K is the constant of uniaxial anisotropy. We suppose that the anisotropy direction ea is determined by the surface geometry, it corresponds to one of the principal directions or their linear combination, and mn is the normal component of magnetization, see also Supplementary Note 2. By choosing the curvilinear reference frame, adapted to the geometry of an object, anisotropy obtains its usual spatially invariant form.

We generalize the description of magnetic curvilinear shells by introducing tangential derivatives of unit magnetization vector m = M/Ms defined as the following:

$${\eth}_{\alpha }{m}_{\beta }:= \, \frac{{\partial }_{\alpha }{m}_{\beta }+{\epsilon }_{\beta \gamma }\left({{\bf{e}}}_{1}\cdot {\partial }_{\alpha }{{\bf{e}}}_{2}\right){m}_{\gamma }}{\sqrt{{{\mathfrak{g}}}_{\alpha \alpha }}},\\ {\eth }_{\alpha }{m}_{n}:= \, \frac{{\partial }_{\alpha }{m}_{n}}{\sqrt{{{\mathfrak{g}}}_{\alpha \alpha }}},\quad \alpha,\beta =1,2.$$

Here, \({{\mathfrak{g}}}_{\alpha \beta }\) is metric tensor and ϵβγ is totally antisymmetric tensor, mi is an ith magnetization component in the curvilinear orthonormal Darboux three-frame \(\left\{{{\bf{e}}}_{1},{{\bf{e}}}_{2},\hat{{\bf{n}}}\right\}\) on the surface ς, where e1 and e2 are unit vectors corresponding to the principal directions, see Supplementary Note 2 for details. We use Greek letters αβ, … = 1, 2 to denote indices restricted to the shell surface. To indicate all three components of any vector, we use Latin indices ij, … = 1, 2, n. The Einstein summation convention is also used.

The main purpose of using this language is to separate two effects: a system-specific curvature effect and spurious effect of the curvilinear reference frame. Because of the geometry-broken symmetry, it is natural to restructure all magnetic energy terms containing spatial derivatives. A characteristic example is an exchange interaction: being isotropic in the Cartesian reference frame, it contains three components of different symmetries in curvilinear coordinates, \({E}^{{\rm{x}}}={E}_{0}^{{\rm{x}}}+{E}_{{\rm{A}}}^{{\rm{x}}}+{E}_{{\rm{D}}}^{{\rm{x}}}\)13,47. Here, \({E}_{0}^{{\rm{x}}}\) is a “common,” regular isotropic part of exchange interaction, which has the form similar to the one in a planar film

$${E}_{0}^{{\rm{x}}}=Ah\int \left({\eth }_{\alpha }{m}_{i}\right)\left({\eth }_{\alpha }{m}_{i}\right){\rm{d}}S.$$

We emphasize that all effects that stem from the choice of the reference frame are properly and unambiguously assigned to \({E}_{0}^{{\rm{x}}}\). This represents the major advantage of the approach based on tangential derivatives.

The second term in the exchange energy reads

$${E}_{{\rm{A}}}^{{\rm{x}}}=Ah\int {w}_{{\rm{A}}}^{{\rm{x}}}{\rm{d}}S,\quad {w}_{{\rm{A}}}^{{\rm{x}}}={{\mathscr{K}}_{\!\!ij}}{m}_{i}{m}_{j}.$$

In general, this energy term describes the curvature-induced biaxial anisotropy, \(\left\Vert {{\mathscr{K}}}_{\!\!ij}\right\Vert ={\rm{diag}}\left({\kappa }_{1}^{2},{\kappa }_{2}^{2},{\kappa }_{1}^{2}+{\kappa }_{2}^{2}\right)\) with κ1 and κ2 being local values of principal curvatures related to Gaussian and mean curvature as \({\mathcal{K}}={\kappa }_{1}{\kappa }_{2}\) and \({\mathcal{H}}={\kappa }_{1}+{\kappa }_{2}\), respectively. A striking manifestation of the curvature-induced anisotropy is shape-induced patterning, for a review see ref. 12. The last term in the exchange energy is a curvature-induced chiral interaction (extrinsic DMI)13

$${E}_{{\rm{D}}}^{{\rm{x}}}=Ah\int {\rm{d}}S\left({w}_{{\rm{D}}1}^{{\rm{x}}}+{w}_{{\rm{D}}2}^{{\rm{x}}}\right),\quad {w}_{{\rm{D}}\alpha }^{{\rm{x}}}=2{\kappa }_{\alpha }{{\mathscr{L}}}_{\alpha n}^{(\alpha )},$$

where no summation over α in Eq. (5C) is applied. This term is determined by the curvilinear-geometry analog of Lifshitz invariants

$${{\mathscr{L}}}_{ij}^{(\alpha )}={m}_{i}{\partial }_{\alpha }{m}_{j}-{m}_{j}{\partial }_{\alpha }{m}_{i}.$$

The two Lifshitz invariants in (5C) are determined by principal curvatures κ1 and κ2. The curvature-induced DMI is a reason for a local chiral symmetry breaking, i.e., magnetochiral effects15, for a review see ref. 12.

Classification of curvature effects by shell type

A direct analysis of all magnetostatic energy contributions (Eq. 2A) is complicated by the nonlocal integration kernels. For this reason, we apply a symmetry analysis to the energy of a ferromagnetic shell to distinguish sources of possible effects of curvature on the magnetic texture.

In the following, we consider the case of strong anisotropies, which allows us to study a magnetic texture, which does not deviate significantly from the assumed equilibrium state \(\widetilde{{\bf{m}}}\) given by the anisotropy. We are interested in how local properties, i.e., local curvatures of the surface and local orientation of the magnetic easy axis, impact the resulting global magnetic state.

For our discussion, we consider uniaxial magnets with special types of anisotropy along one of the principal directions for the following distinct cases of surfaces:

  1. (i)

    A class of developable surfaces of zero Gaussian curvature, \({\mathcal{K}}({\bf{r}})=0\), includes cylinders, cones, and tangent surfaces48. They can be locally developed into a plane without stretching. Since cones and tangent surfaces are singular ones49, here, we consider generalized cylindrical surfaces only.

  2. (ii)

    Minimal surfaces with vanishing mean curvature, \({\mathcal{H}}({\bf{r}})=0\), have principal curvatures of opposite signs, and in the vicinity of each point, they are saddle-shaped. Minimal surfaces provide the minimal surface area enclosed by a given boundary.

  3. (iii)

    General case with nonvanishing \({\mathcal{H}}\) and \({\mathcal{K}}\) and arbitrary local surface elements, including convex and saddle ones.

The impact of the geometry on a magnetic texture is summarized in Fig. 2. It is given by the interplay of the curvature-induced energy terms and the type of anisotropy and orientation of the anisotropy axis. We refer to the curvature-related energy terms as the following:

$$E= \ {E}_{0}+Ah\int {\rm{d}}S\left({w}_{{\rm{A}}}^{{\rm{x}}}+{w}_{{\rm{D}}1}^{{\rm{x}}}+{w}_{{\rm{D}}2}^{{\rm{x}}}\right)\\ +\ {M}_{{\rm{s}}}^{2}\int {\rm{d}}{\bf{r}}\left({w}_{g-\sigma }+{w}_{g-g}+{w}_{g-\rho }+{w}_{\sigma -\rho }\right).$$

Here, \({E}_{0}={E}_{0}^{{\rm{x}}}+{E}_{0}^{{\rm{an}}}\) comprises terms that have the form similar to flat systems; they contain derivatives of magnetization components, and can result in symmetry breaking and chiral effects even in a purely planar case due to chiral magnetic texture, the so-called pattern-induced chirality breaking12. These effects are well studied for magnons on the background of solitons50, vortices51, and skyrmions52.

The curvature-induced exchange terms \({w}_{{\rm{A}}}^{{\rm{x}}}\), \({w}_{{\rm{D}}1}^{{\rm{x}}}\), and \({w}_{{\rm{D}}2}^{{\rm{x}}}\) scale linearly with the shell thickness. Magnetostatic terms wgσ, wgg, and wgρ are present only for curved shells with a nonzero mean curvature, \({\mathcal{H}}\, \ne \, 0\); for the infinitesimally thin shells, they scale quadratically with thickness. These magnetostatic terms are absent for minimal surfaces, e.g., for catenoids and helicoids. The term wσρ is present only for the inhomogeneous magnetization texture if top and bottom surfaces of the shell are not equivalent. It appears due to the coupling between surface and tangential magnetostatic charges σ± and ρ (we do not discuss here the specific picture of interaction between edge-surface charges with volume ones, which is available also in flat systems53).

It is important to stress that such an approach cannot be considered as a sufficient condition of existence and moreover stability of the corresponding magnetization states. No other criteria are applied with assembly in Fig. 2. Here, we discuss possible statical states by applying only symmetrical arguments to the energy functional irrelevant whether they are in local minimum of energy or not. Investigation of equilibrium magnetization texture for the concrete geometry should be a purpose of a separate work. For example, \({w}_{{\rm{D}}1}^{{\rm{x}}}\) does not impact magnetic textures (it vanishes) for the following cases: either κ1 ≡ 0, or magnetic texture does not vary along x1, or mn ≡ 0. A possible magnetic texture is assumed based on the sample symmetry and interplay between intrinsic and curvature-induced anisotropies.

Minimal surfaces do not exhibit effects, which explicitly depend on the curvature due to \({\mathcal{H}}\equiv 0\). At the same time, all geometry-induced exchange-driven terms are present for any texture symmetry, except easy-surface anisotropy or easy-axis anisotropy along e2. The symmetry-broken magnetostatics-driven term wσρ is always present for inhomogeneous textures with nonzero magnetostatic charge if the top and bottom surfaces of a shell are not equivalent. As in the previous case, the initially assumed strictly normal magnetic texture is modified by the appearance of the m1 component: m = {m1(x1), 0, mn(x1)} for a catenoid.

For the general case of \({\mathcal{H}}\, \ne \, 0\) and \({\mathcal{K}}\, \ne \, 0\), all effects stemming from both local exchange and nonlocal magnetostatics interaction are possible. Note that Fig. 2 is also valid for nonlinear excitations of the equilibrium state like domain walls if their symmetry corresponds to the function given in the “Texture symmetry” column.