A Derivation of the effective energy for the extended DW

# Effective description of domain wall strings

## Abstract

The analysis of domain wall dynamics is often simplified to one dimensional physics. For domain walls in thin films, more realistic approaches require the description as two dimensional objects. This includes the study of vortices and curvatures along the domain walls as well as the influence of boundary effects. Here we provide a theory in terms of soft modes that allows to analytically study the physics of extended domain walls and their stability. By considering irregular shaped skyrmions as closed domain walls, we analyze their plasticity and compare their dynamics with those of circular skyrmions. Our theory directly provides an analytical description of the the excitation modes of magnetic skyrmions, previously only accessible by sophisticated micromagnetic numerical calculations and spectral analysis. These analytical expressions provide the scaling behaviour of the different physics on parameters that experiments can test.

## I Introduction

The existence of distinguishable magnetic domains in ferromagnetic films Bloch (1932); Landau and Lifshits (1935) is the basis for magnetic memories devices. Parkin et al. (2008) The boundary between two opposite magnetic domains corresponds to a domain wall (DW). Slonczewski (1973); Hayashi et al. (2006); Tatara and Kohno (2004); Yamanouchi et al. (2004) DWs belong to a class of textures called topological textures. Analytically in the low-energy limit, DWs are often treated as one dimensional objects that can be described by two soft modes. Thiele (1973); Slonczewski (1973, 1972); Schryer and Walker (1974); Thiaville et al. (2004); Tatara et al. (2008); Beach et al. (2005); Atkinson et al. (2003) Experiments in thin films Jiang et al. (2015); Emori et al. (2014); Yamanouchi et al. (2004); Benitez et al. (2015); Boulle et al. (2016) reveal however much richer configurations including curved DWs and vortex DWs, LaBonte (1969) which in such a one dimensional picture are not captured. Therefore a two dimensional theory is needed.

In this paper, we consider DWs as strings described by a pair of fields of soft modes. We derive a theory that allows to treat the local dynamics of the DW and interactions with localized perturbations like impurities. Also the effects of external perturbations can be easily added to the formalism.

Beyond DWs, magnetic skyrmions are promissing candidates for applications in spintronics. Bogdanov and Yablonskii (1989); Mühlbauer et al. (2009); Heinze et al. (2011); Nagaosa and Tokura (2013a) Skyrmions have a quantized topological charge and respond more efficiently to applied currents.Jonietz et al. (2010); Schulz et al. (2012) Due to their reduced size and particle-like behavior they have attracted significant interest in spintronics Wolf et al. (2006) with applications as memory and logic devices. Parkin et al. (2008); Fert et al. (2013); Zhang et al. (2015); Allwood et al. (2005) Skyrmions are very much related to DWs in the sense at they can be viewed as closed DWs. Often, their theoretical description is based on a radially symmetric ansatz corresponding to a circular DW. Feldtkeller and Thomas (1965); Ezawa (2010); Kiselev et al. (2011); Rohart and Thiaville (2013) The radial symmetry, however, is not a requirement for their existence. Other shapes have been oberved, for example, close to a defectMüller and Rosch (2015); Romming et al. (2015) or by exciting their internal modes. Romming et al. (2015); Seki et al. (2012); Woo et al. (2016); Lin et al. (2014); Makhfudz et al. (2012) The plasticity of the skyrmion shape has several experimental advantages, including the ability to avoid impuritiesMüller and Rosch (2015); Romming et al. (2015) and increasing the response to external electrical currents. Iwasaki et al. (2013); Litzius et al. (2016) Here we describe skyrmions as closed DWs without imposing shape-related symmetries. Our formalism directly allows to study the dynamics of deformed magnetic skyrmions and their eigenmodes.

The paper is organized as follows. In Sec. II we present the effective description of a DW as a string. We describe the ansatz and obtain the effective energy for these topological textures. In Sec. III we obtain an action that describes the dynamics in terms of the soft mode fields. In Sec. IV we consider closed DWs and analyze the dynamics of deformed skyrmions. In particular, we consider three examples, we calculate the dynamics of deformations (bumps) along skyrmions and examine the rotational and the breathing mode dynamics of deformed skyrmions. In Sec. V we report our conclusions.

## Ii Description of Domain wall strings

We consider a ferromagnetic film with thickness , much smaller than the DW width, and with interfacial Dzyaloshinskii-Moriya interaction (DMI). Moriya (1960); Dzyaloshinsky (1958); Chen et al. (2013) The magnetization field is described as , where is a unit vector field and is the saturation magnetization. The micromagnetic Hamiltonian corresponds to

 H= d∫d2x(J2|∇^m|2+K(1−m2z) +D(mz∇⋅m⊥−m⊥⋅∇mz)), (1)

where is the vector of the in-plane components of the magnetization . The term proportional to is the ferromagnetic exchange interaction. It favors the alignment of the magnetization vectors. The term proportional to correspond to out-of-plane anisotropy interaction. It favors the magnetization pointing along the easy-axis, perpendicular to the plane of the film. The term proportional to is the interfacial DMI. For the ground state is given by a helix, Rohart and Thiaville (2013) which can be viewed as a periodic structure of DWs. In order to describe single DWs, we consider . We neglect explicit terms related to demagnetization fields as they may be interpreted as a contribution to an effective anisotropy in thin films. Draaisma and De Jonge (1988) The Hamiltonian from Eq. (II) has a characteristic length, the DW width , and a characteristic energy density scale, Hubert and Schafer (1998) These two parameters define the scale for the dynamics of topological textures.

We consider a DW in a thin film as an extended object described by a curve , see Fig. 1,

 X(s)=(x(s),y(s),0), (2)

where is a parameter along the curve. A local basis along the curve is given by the unitary longitudinal vector and the unitary normal vector

 ^el=X′|X′|,^en=^el×^ez, (3)

where for any function we define and . Note that , where the function is related to the local curvature via , see App. A. We assume that the radius of the local curvature at any point along the DW is much bigger than the DW width, i.e. .

We consider an ansatz for an extended DW described by a curve with a cross section along corresponding to a rigid one-dimensional DW, see Fig. 1 and App. A. Thus, each cross section is described by a pair of soft modes , the position of the cross section where the magnetization is in-plane, and , the angle of the in-plane magnetization with respect to the vector . The effective description of the DW string is in terms of the fields and .

In the case of finite thin films without periodic boundary conditions, DMI produces a twist of the magnetization at the boundaries. Rohart and Thiaville (2013); Meynell et al. (2014) This means that, for open DW curves in thin films, DMI twists the magnetic profile for cross sections close to the edge. We treat these edge twists as a perturbation to the state without edge twists and associate to them an effective potential. For DWs with periodic boundary conditions, there is no such twisting and, consequently, there is no potential associated to it. The effective energy obtained from the micromagnetic Hamiltonian given by Eq. (II) with periodic boundary conditions is in terms of the effective coordinates given by

 Heff=d∫ds( cEE|X′|+cκJ|X′|(Φ′−k)2 +D(π|X′|−cdk)cosΦ). (4)

Here, and are dimensionless constants that depend on the exact profile of the DW along the normal direction , see Appendix A. This effective Hamiltonian is one of the main results of this paper. It is invariant under reparametrization, where . The first term in Eq. (II) is proportional to the length of the DW in analogy to the energy of a rubber band. The solution that minimizes this term is a straight line. The second term describes the fact that bending the DW leads to a changes in and vice versa. As the azimuthal angle can be manipulated experimentally for example by local external magnetic fields or spin waves, Garcia-Sanchez et al. (2015); Wagner et al. (2016); Kläui et al. (2005) this provides a mechanism to introduce curvature in DWs. Note that without DMI the energy is invariant under global rotations of the azimuthal angle of the DW. As DMI breaks inversion symmetry, it directly couples the azimuthal angle with the curvature and the total length of the DW curve making the physics of the extended DW more complex.

To calculate the effective potential due to boundary conditions for open DWs, we assume the following two conditions: i) the longitudinal vector at the boundary is perpendicular to the edge; and ii) the length scale given by the DMI induced boundary twists is not much bigger than the DW width . The first condition implies that the magnetization configuration at the edge corresponds to a DW with the same width as in the bulk. Due to our assumption that the local curvature of the extended DW is larger than its width, the second condition grants that the DW is straight in the region close to the boundary. As a result, the DMI induced boundary condition leads to a rigid twist of the magnetization profile around for cross sections close to the edge. As we substitute this modified ansatz for the edges into the micromagnetic Hamiltonian from Eq. (II), we obtain that the energy potential due to edge effects does, as expected, not depend on the position . The main contributions are functions of and defined only on a small region around the edges. Within the above formalism, the Zeeman interaction due to an external magnetic field can also be incorporated. In this case the boundary condition will depend also on the relative positions of the string ends. Boulle et al. (2013); Muratov et al. (2017) In the following we will restrict our analysis to periodic boundary conditions.

For 2d magnetic textures it is possible to define a topological charge of the form Nagaosa and Tokura (2013b)

 Q=14π∫d2x^m⋅(∂x^m×∂y^m). (5)

The conservation of topological charge in the continuous approximation, is guaranteed by the boundary conditions. In a magnetic lattice the conservation of topological charge holds only for textures that are bigger than the discretization scale. By substituting the ansatz of the DW string into Eq. (5) we obtain,

 Q=12π∫ds(k−Φ′). (6)

For open DWs in thin films without periodic boundary conditions, the topological charge neither needs to be an integer number nor a conserved quantity. This means that it is possible to induce topological charge into these textures through boundary dynamics. Müller et al. (2016a); Du et al. (2015) In contrast, for DWs with periodic boundary conditions, this charge must be conserved in the continuum approximation as the integrals over and are quantized. Connecting the ends of a DW, see Fig. 2, leads to a skyrmion. For such closed DW strings without knots, the contribution from the integral over is , where we use Gauss-Bonnet theorem. Thus, by interpreting skyrmions as closed DW strings we, of course, also obtain their quantized topological charge of depending on their center magnetization. Eqs. (II) and (6) reveal that a possible mechanism to produce skyrmions out of DW strings is to manipulate the azimuthal angle. Milde et al. (2013)

In the following we will consider explicitly the dynamics of extended DWs in Sec. III and the physics of closed DWs in Sec. IV. For the case of open DWs the solution with minimum energy is given by a straight DW, where is a constant and ; while for a closed DW the state with minimum energy is given by a circular skyrmion with radius and a constant azimuthal angle along the curve. Note that plugging in a circular skyrmion ansatz into Eq. (II) leads to the effective energy known from the literature for skyrmions. Rohart and Thiaville (2013) These two minimal solutions do not have local degrees of freedom for the soft mode fields. Thus, the long range dynamics of a straight DW is globally defined by the soft mode pair Rodrigues et al. (2017) In the case of skyrmions, also two globally defined soft modes are enough to capture their dynamics, , the radius of the skyrmion and the azimuthal angle of the in-plane magnetization along the radius. Rodrigues et al. (2018); Makhfudz et al. (2012)

## Iii Effective dynamics of extended DWs

The model that best describes the magnetization dynamics in a ferromagnet is given by the Landau-Lifshitz-Gilbert (LLG) equation Gilbert (2004)

 ˙^m=γMs^m×δH[^m]δ^m+α^m×˙^m, (7)

where is the gyromagnetic constant, is the Hamiltonian of the system, is the dimensionless Gilbert damping parameter, and we define for any function . The first term on the right-hand side corresponds to the precession of the magnetization due to an effective magnetic field and it conserves energy. The second term correspond to a damping term which promotes the alignment of the magnetization with the effective magnetic field. The energy conserving part of the LLG Eq. (7) may be obtained from varying the action Papanicolaou and Tomaras (1991)

 A=∫dt(dMsγ∫d2x(1−cosθ)˙ϕ−H), (8)

where the first part is the spin Berry phase expressed in a standard spherical polar representation of the magnetization. Braun and Loss (1996) Plugging in Eq. (7) the ansatz of a curved DW in terms of the fields of soft modes, we obtain

 SB=−dcγ∫dt∫ds|X′|(˙X⋅^en)Φ, (9)

where , see App. B. From the spin Berry phase action, Eq. (9), and the effective Hamiltonian, Eq. (II), it is possible to obtain the equations of motion for the soft mode fields to study the dynamics of DWs in thin films, such as the propagation of extended DWs in different geometries, the influence of curvatures and the formation of cusps. A full general description is however rather complicated as the equations of motion are heavily influenced by the boundary conditions.

In the case of , i.e. a constant azimuthal angle along the DW, the Berry phase from Eq. (9) becomes

 SB=−dcγ∫dtΦ˙A, (10)

where by we define the area of the ferromagnetic domain with . This reveals that and are conjugated soft modes, Rodrigues et al. (2017) with a Poisson bracket given by . An important remark is that, since an external out-of-plane magnetic field couples directly to , it produces, as expected, a precession of the angle .

## Iv Effective dynamics of closed DWs

In this section we study the dynamics of closed DWs that can be parameterized within the polar coordinate representation

 X(s,t)=r(s,t)(cos(s),sin(s),0), (11)

where , and is a smooth function with . Such an ansatz includes circular and distorted magnetic skyrmions as shown in the right panel of Fig. 2. For a curve given by Eq. (11), the effective spin Berry phase is

 SB=−dcγ∫dt∫2π0dsΦr˙r. (12)

### iv.1 Equations of motion for closed DWs

The equations of motion for the local radial distance and the azimuthal angle of the closed DW are

 cγr˙r= (2Jcκ|X′|(Φ′−k))′+DsinΦ(π|X′|−cdk), (13a) −dcγ˙Φ= 1rδHeffδr, (13b)

where and . Here, we did not include damping dynamics. The addition of the Gilbert-damping term corresponds to adding a term proportional to the damping constant that mixes the evolution of and , see App.C.

Eq. (13a) has two interesting properties that we would like to point out: i) we find that the only contribution to the time evolution of the area enclosed by the DW is due to the DMI. In the case of we obtain after integrating the equation of motion along the curve

 ˙A=πDcγsinΦ(2cd+∫2π0ds|X′|). (14)

ii) In the absence of DMI, Eq. (13a) has the form of a continuity equation, i.e. . Here, analogous to the charge density, it is proportional to the area per unit length and corresponds to the density of topological charge per local length change. This shows that the existence of Bloch points, which corresponds to a concentration of topological charge, can generate strong deformations of the DW.

Eq. (13b) is rather complicated as it contains higher order derivatives. Integrating the equation of motion along the entire curve gives

 cγ∫ds˙Φ =∫ds(2(EcE+DπcosΦ)|X′|−Jcκ|X′|3(Φ′−k)2 −(r′r|X′|2)(2Jcκ|X′|(Φ′−k)+DcdcosΦ)′), (15)

where some of the terms with higher order derivatives vanished due to the periodic boundary conditions.

These equations provide a good intuition to the dynamics of extended DWs as well as deformed skyrmions, and convey the general dynamics of a broad range of magnetic textures. In particular, this effective theory allows for the study fo continuous deformations of skyrmions without requiring the knowledge of magnonic modes. In the following we will explicitly explore three examples, the propagation of deformations along a skyrmion, the eigenmodes of closed DWs and breathing dynamics.

### iv.2 Wave propagation along the closed DW

Here we consider circular skyrmions with radius with a small and smooth deformation of size , i.e.  with . We assume that the angle and the radius are fixed to the values minimizing the energy,Rohart and Thiaville (2013) i.e.  and where is the DW width. In this case we obtain from Eq. (13a) a drift equation

 (R30cγ2cκJ˙d+d′)≈0. (16)

This describes the propagation of the deformation along the DW. Since the curve is closed, this motion is periodic with frequency .

### iv.3 Rotational eigenmodes of a closed DW

In this example we consider solutions of Eq. (13a) that have a rigid shape and obey the following conditions: i) the curve has a maximum and a minimum radius, which are of the order of the equilibrium radius of a circular skyrmion; ii) the solution depends on the combination , where is the parameter along the curve, is time, and is a constant frequency; iii) the solution is periodic; and iv) . With these conditions, we find that the curve describing the DW string, is given by

 r(s)≈rmin+Δrsin2(ns2), (17)

where , and is an integer. The first four solutions of the above equation are shown in Fig. 3. The frequency for the n-th mode is given by

 ωn≈−2Jcκcγr3min(n2+1). (18)

The solutions from Eq. (17), which we obtained within our effective theory, correspond to the skyrmion excitation modes reported in the literature. Lin et al. (2014); Makhfudz et al. (2012)

### iv.4 Breathing modes of a closed DW

Here we consider the smooth breathing dynamics of a closed DW with along the curve. This means that its shape is fixed and the dynamics is given by a scaling factor , i.e. , where is the equilibrium fixed shape. From Eqs. (14) and (15), we obtain the following equations in terms of the scalling factor ,

 ˙ϵ =πD2A0cγsinΦ(∫2π0ds|X′0|+2cd(1−ϵ)), (19a) 2πcγ˙Φ =(∫ds2(EcE+DcosΦ)|X′0|−∫dsJcκk20|X′0|3) −ϵ(∫ds2(EcE+DπcosΦ)|X′0|−3∫dsJcκk20|X′0|3), (19b)

where , and are only functions of the initial curve of the DW given by . Note that in the case of smooth breathing dynamics the last term in Eq. (15) is small and therefore is neglected. From the first equation and first term in the second equation we obtain the condition for the equilibrium configuration. They fix and .

We define , where is dimensionless and at the average radius . The equilibrium condition fixes for any deformed skyrmion to

 ~R0=Δ√~cκ~cE−~cπ(D/E), (20)

where is the DW width and we have rescaled the constants , and . This equilibrium radius is analogous to the case of a circular skyrmion with rescaled parameters. Rohart and Thiaville (2013)

If we consider small perturbations of around the equilibrium configuration of the skyrmion, , we obtain the following equations of motion

 ˙ϵ =−φπD2A0cγ(∫2π0ds|X′0|+2cd), (21a) ˙φ =ϵ∫dsJcκk20πcγ|X′0|3, (21b)

describing the breathing dynamics of deformed skyrmions with frequency

 ω2=DJ~cκ2A0c2γ(∫2π0ds|X′0|+2cd). (22)

In case of a circular skyrmion, this breathing mode corresponds to the zero order excitation mode. Makhfudz et al. (2012)

## V Discussion and Conclusion

Motivated by the experimentally observed richer configurations of topological magnetic textures in thin films, in this paper we extended the soft mode formalism of DWs beyond the effective one dimensional theory. We provided an ansatz for the effective description of a DW in terms of soft mode fields. From the micromagnetic Hamiltonian for ferromagnetic thin films we obtained an effective energy for these topological textures revealing analogies of the DW string to elastic rubber bands. An important remark is that our results are invariant under reparametrization. We have shown that circular skyrmions are (meta-)stable configurations of the Hamiltonian and analyzed their stability. We calculated the effective spin Berry phase from which, associated with the effective energy, and obtain the equations of motion describing the dynamics of these structures. Finally we have applied our formalism to three examples of skyrmion dynamics.

We would like to emphasize that our developed formalism takes into account several aspects of topological textures dynamics, such as curvatures and variations of the in-plane magnetization. We have derived the topological charge for the extended DWs. Our theory allows the study of more complex dynamics like the propagation of extended DWs in different geometries with inhomogeneous perturbations, the local interaction of skyrmions and DWs, as well as the the annihilation and creation of skyrmions mediated by DWs. Furthermore, it is possible to analyze the stability of Bloch points Slonczewski (1975); Jantz et al. (1981); Da Col et al. (2014) in DWs and the creation of skyrmion lattices by excitation of worm domains. Jiang et al. (2015); Koshibae and Nagaosa (2014)

For a more general case, however, a full treatment of the boundary contributions is required.Rohart and Thiaville (2013) These boundary conditions can be treated effectivelyMuratov et al. (2017) and lead to additional terms in Eq. (II). Micromagnetic simulations show that the edges may be sources of both curvature and Bloch points,Müller et al. (2016b) and with the presented formalism we can analyze their propagation along the DW.

We conclude by mentioning that our formalism is extendible with regard to different types of interactions or more generalized types of DMI,Hoffmann et al. (2017); Hals and Everschor-Sitte (2017) which will cover the dynamics of more complex topological textures like antiskyrmions.

## Vi Acknowledgments

During the preparation of the manuscript we have learned that Oleg Tchernyshyov and Shu Zhang have independently developed a similar formalism. Tchernyshyov and Zhan (2017) We thank the authors for the discussions. We also thank Ben McKeever for helping revising the text. D.R.R. is a recipient of a DFG fellowship/DFG-funded position through the Excellence Initiative by the Graduate School Materials Science in Mainz (GSC 266). J. S. acknowledges funding from the Alexander von Humboldt Foundation, the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X, and Grant Agency of the Czech Republic grant no. 14-37427G. K.E.-S. acknowledges funding from the German Research Foundation (DFG) under the Project No. EV 196/2-1.

## Appendix A Derivation of the effective energy for the extended DW

In this paper we describe an extended DW in terms of a curve and a local unitary basis along the DW given by the longitudinal, , and the normal, , vectors to the curve. We make the ansatz that the DW can be described along the normal direction by a rigid one-dimensional DW. We can transform the Cartesian basis via a local rotation into the basis along the curve

 x,y→n,l, (23)

where are the coordinates along the vectors , respectively,

 n(s)=(x−X(s))⋅^en(s),l(s)=(x−X(s))⋅^el(s). (24)

Here denotes any position in space and the parameter takes values in between zero and the length of the extended DW. The transformation between the coordinate and any string parameter is given by

 dl=ds|X′|. (25)

The curvature of the DW is defined by do Carmo (1976)

 ∂l^el≡κ^en=k|X′|^en, (26)

where in the last equality we have used Eq. (25). According to our ansatz, we separate the micromagnetic Hamiltonian from Eq. (II) into three terms,

 H=Hn+Hl+HD (27)

where

 Hn =d∫dl∫dn(J2(∂n^m)2+K(1−m2z)), (28a) Hl =d∫dl∫dn(J2(∂l^m)2), (28b) HD =d∫dl∫dnD(mz(^en∂n+^el∂l).(mn^en+ml^el) −(mn∂n+ml∂l)mz). (28c)

are the normal, longitudinal and DMI contribution to the Hamiltonian respectively. We now show that our ansatz, where each cross section corresponds to a rigid one-dimensional DW, minimizes with the given DW boundary conditions. This can be directly seen by rewriting the exchange part in a spherical representation of the magnetization

 Hn =d√JK√2∫dl∫d~n((∂~nθ)2+sin2θ(∂~nΦ)2 +(1−m2z)), (29)

where is dimensionless. The state with minimum energy and DW boundary conditions, and , corresponds to a solution with and a specific dependence. This solution is invariant under a rigid rotation of . Furthermore, in the linear approximation does not depend explicitly on . The magnetization is then given by

 ^m =cosΦ(l)sinθ(n)^en(l)+sinΦ(l)sinθ(n)^el(l) +cosθ(n)^ez, (30)

where corresponds to a rigid DW centered at position . Plugging the magnetization configuration of Eq. (A) into Eq. (28) and perfoming the integration over gives

 Hn =dcE√JK∫dl (31a) Hl =dJcκ∫dl1|X′|2(Φ′−k)2, (31b) HD =dD∫dl(π−cdk|X′|)cosΦ, (31c)

where are dimensionless constants depending on the precise shape of the DW. Note that the DMI part of the Hamiltonian has the following boundary dependent azimuthal angle contribution . Upon integration along the length of the DW curve this term vanishes for periodic boundary conditions.

## Appendix B Spin Berry phase in the extended DW ansatz

The spin Berry phase in a standard spherical representation of the magnetization in thin film is Braun and Loss (1996)

 SB=dMsγ∫dt∫d2x(1−cosθ)˙Φ. (32)

Performing a partial integration we obtain

 SB=−dMsγ∫dt∫d2xΦsinθ˙θ. (33)

This representation has the advantage that the non-vanishing contributions come only from regions within the DW, as for the ferromagnetic domains. For the time variation of , we note that . Any variation of correspond to a translation along the local direction. Using Eq. (24), we get

 ˙θ=−(˙X⋅^en)∂nθ. (34)

Plugging this into Eq. (33) and changing to the system of coordinates gives

 SB=dMsγ∫dt∫dndl(Φ(˙X⋅^en)sinθ∂nθ). (35)

Since is fixed by the DW solution, we may integrate the dependence and we obtain

 SB=−dcγ∫dt∫dlΦ(˙X⋅^en), (36)

where and with the reparametrization from to we obtain Eq. (9) of the main text.

A simpler case is when meaning that is globally defined along the DW curve. Then the effective spin Berry phase (36) takes the form

 SB=−dcγ∫dtΦ∫dl(˙X⋅^en). (37)

We can further simplify this equation by the following considerations: The infinitesimal area spanned by and is given by A time variation of this area corresponds to and , such that

 δdA=(δn^en×dl^el)⋅^ez+(dn^en×δl^el)⋅^ez. (38)

We want to study the area change upon changing the DW string with a fixed parameterization, i.e. .

 ˙dA=((˙X⋅^en)^en×dl^el)⋅^ez=dl(˙X⋅^en) (39)

Note, that the second term vanishes as for a fixed parameterization and in the first term we have calculated the variation of the DW string along the normal direction.

Therefore, the second integral in Eq. (37) becomes

 ∫dl(˙X⋅^en)=∫˙(dA)=˙A, (40)

and we obtain Eq. (10) of the main text.

## Appendix C Damping terms and Rayleigh functional

The damping term in the Landau-Lifshitz-Gilbert equation, Eq. (7) of the main text, can be derived from a Rayleigh functional. Gilbert (2004) This means that we can identify

 α^m×˙^m↔δR[˙^m]δ˙^m, (41)

where the Rayleigh functional is given by

 R[˙^m]=αdMs2γ∫d2x(˙^m)2. (42)

If we plug in the ansatz of a DW curve, we obtain

 R[˙^m] =αdMs2γ∫d2x(sin2θ(˙Φ)2+(˙θ)2) =αdMs2γ∫dndl(sin2θ(˙Φ)2+(∂nθ)2(˙X⋅^en)2) =αdMs2γ∫dl(cΦ(˙Φ)2+cX(˙X⋅^en)2)). (43)

As the only non vanishing contribution comes from within the DW curve, we can change the basis to the local basis and then integrate the expression along the normal direction. The constants are positive and dimensionless. They depend only on the specific shape of the profile.

The effect of damping occurs as additional terms in the equations of motion of the closed DW, Eq. (13a) of the main text

 cγr˙r= (2Jcκ|X′|(Φ′−k))′ (44a) +DsinΦ(π|X′|−cdk)+γr/d, (44b) −dcγ˙Φ= 1rδHeffδr+γΦ. (44c)

Here and for the curve of Eq. (11) are given by

 γr =1rαdMs2γcΦ˙Φ, (45a) γΦ =−1rαdMs2γcX˙r. (45b)

This reveals that the effect of damping is to couple the dynamics of and . Note that in general for collective coordinates the damping parameters can be calculated via the expression

 γξi=∑j(∫d2x∂˙ξjR[˙m])/∫d2x(^m⋅∂ξj^m×∂ξi^m). (46)

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