Characterizing breathing dynamics of magnetic (anti-)skyrmions within the Hamiltonian formalism
We derive an effective Hamiltonian system describing the low energy dynamics of circular magnetic skyrmions and antiskyrmions. Using scaling and symmetry arguments we model (anti-)skyrmion dynamics through a finite set of coupled, canonically conjugated, collective coordinates. The resulting theoretical description is independent of both micromagnetic details as well as any specificity in the ansatz of the skyrmion profile. Based on the Hamiltonian structure we derive a general description for breathing dynamics of (anti-)skyrmions in the limit of radius much larger than the domain wall width. The effective energy landscape reveals two qualitatively different types of breathing behavior. For small energy perturbations we reproduce the well-known small breathing mode excitations, where the magnetic moments of the skyrmion oscillate around their equilibrium solution. At higher energies we find a breathing behavior where the in-plane angle of the skyrmion continuously precesses, transforming Néel to Bloch skyrmions and vice versa. For a damped system we observe the transition from the continuously rotating and breathing skyrmion into the oscillatory one. We analyze the characteristic frequencies of both breathing types, as well as their amplitudes and distinct energy dissipation rates. For rotational (oscillatory) breathing modes we predict on average a linear (exponential) decay in energy. We argue that this stark difference in dissipative behavior should be observable in the frequency spectrum of excited (anti-)skyrmions.
Topological magnetic textures have attracted substantial attention in spintronics Wolf et al. (2006); Chappert et al. (2007); Jonietz et al. (2010); Iwasaki et al. (2013) in light of prospects to harness their favorable properties for magnetic memory technologies, Parkin et al. (2008); Nagaosa and Tokura (2013); Fert et al. (2013) information processing Zhang et al. (2015a); Pinna et al. (2018); Luo et al. (2018); Zázvorka et al. (2018); Chauwin et al. (2018) and novel approaches to computation. Huang et al. (2017); He and Fan (2017a); Li et al. (2017); He and Fan (2017b); Prychynenko et al. (2018); Azam et al. (2018); Bourianoff et al. (2018) An important building block in this direction is understanding their dynamical excitations in order to test stability and to devise ways to efficiently manipulate them. Among the magnetic textures studied are domain walls Yamanouchi et al. (2004); Tatara and Kohno (2004); Hayashi et al. (2006); Parkin et al. (2008) and skyrmions. Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Mühlbauer et al. (2009); Schulz et al. (2012); Iwasaki et al. (2013); Sampaio et al. (2013); Nagaosa and Tokura (2013) With respect to possible applications in memory devices, skyrmions present several advantages over domain walls due to their smaller sizes, lower threshold for current driven mobility, Jonietz et al. (2010); Yu et al. (2012); Schulz et al. (2012); Fert et al. (2013) and their tendency to avoid obstacles and boundaries. Zhang et al. (2015b) Recently, a growing collection of other exotic relatives are also receiving attention. This includes magnetic solitons of higher topological order, Zhang et al. (2016) chiral bobbers, Zheng et al. (2018) non-topological counterparts such as the skyrmionium particle, Komineas and Papanicolaou (2015) and antiskyrmions Koshibae and Nagaosa (2016); Camosi et al. (2017); Everschor-Sitte et al. (2017); Hoffmann et al. (2017) where the winding number is opposite in sign compared to the skyrmion.
Following the pioneering work by Schryer and Walker, Schryer and Walker (1974) effective descriptions for current-driven and field-driven domain wall motion have been considered widely, Yamanouchi et al. (2004); Thiaville et al. (2005); Tretiakov et al. (2008) including recent models leveraging canonically conjugated variables derived from the spin Berry phase action. Berry (1984); Rodrigues et al. (2017); Ghosh et al. (2017) For a magnetic skyrmion, it has been shown Papanicolaou and Tomaras (1991); Zang et al. (2011); Komineas and Papanicolaou (2015) that its motion can be described with two conjugated variables describing its position. Effective descriptions for the internal dynamics of magnetic skyrmions, however, are few in number. Previous works have often focused on eigenmode analysis of magnons to obtain the small amplitude internal excitations of isolated magnetic skyrmions. Lin et al. (2014); Schütte and Garst (2014); Kravchuk et al. (2018); Garst et al. (2017) Skyrmion breathing modes, in which the core of the spin structure grows and shrinks periodically in time, were first described theoretically by numerical simulations of skyrmion lattices Mochizuki (2012) and later found experimentally Onose et al. (2012) in the insulator CuOSeO from microwave response experiments.
In this work, we derive a non-linear effective model for rotationally symmetric (anti-)skyrmion breathing modes in terms of two collective coordinates whose validity extends beyond approaches based on an eigenmode analysis, similar to an earlier effective breathing model found in the context of dynamically stabilized skyrmions. Zhou et al. (2015) The novelty of this work lies in the proposal of a Hamiltonian formalism that is independent of microscopic details, applicable to the circular internal modes of both skyrmions and antiskyrmions. We consider an experimentally relevant model for chiral thin films to study the equilibrium and breathing properties of (anti-) skyrmions in detail and we show that all material details can be collapsed into a single effective parameter. In modeling the non-linear excitations above equilibrium, we describe two dynamical regimes of coherent magnetization behavior: (i) oscillation around the local equilibrium magnetization direction, and (ii) rotational breathing mode dynamics where the local magnetization continuously rotates. By analogy, these regimes may be thought of as a pendulum which, depending on its energy, either (i) swings about its equilibrium position or (ii) fully rotates around its pivot point.
This paper is organized as follows. In Sec. II we describe how to pass from a micromagnetic dynamical description to a Hamiltonian mechanics in terms of collective coordinates. In Sec. III we derive the Hamiltonian mechanics describing the dynamics of soft modes for (anti-)skyrmions. We first review the Poisson bracket for the translational motion of rigid textures in two dimensions and then derive its analog for circular breathing modes. In Sec. IV we introduce a micromagnetic model for chiral thin films with perpendicular magnetic anisotropy, construct its effective energy as a one parameter physical model and study its equilibrium properties. Lastly, in Sec. V, as an application of the effective Hamiltonian formalism, we study the breathing modes of circularly shaped (anti-)skyrmions and make precise predictions regarding their dissipative behavior.
Ii Collective coordinates and Hamiltonian formalism
The magnetization dynamics below the critical temperature in ferromagnetic materials is well described in a continuum approximation by the Landau-Lifshitz-Gilbert (LLG) equation, Gilbert (2004)
where the magnetization configuration is represented by the unit vector . Here is the angular momentum density, is the gyromagnetic constant, is the constant saturation magnetization, the Hamiltonian of the system is , is the Gilbert damping parameter, and the overdots indicate total time derivatives .
The LLG is a non-linear differential equation with an infinite number of degrees of freedom. This poses an obstacle for the comprehensive study of the magnetization dynamics, prohibiting a full analytical solution and usually requiring extensive micromagnetic simulations. However, the low energy dynamics of magnetic textures depend only on the system’s soft modes which can be described by conjugated variables in a Hamiltonian formalism. Clarke et al. (2008); Tretiakov et al. (2008); Rodrigues et al. (2017); Ghosh et al. (2017) This allows access to generic features of the magnetization textures independently from the microscopic characteristics of the material.
The conservative, precessional, part of the LLG equation may be derived from the first variation of the action
with the constraint of a constant magnetization amplitude. Here is the Lagrangian and is the spin Berry phase action. Berry (1984); Kosevich et al. (1990); Papanicolaou and Tomaras (1991) The spin Berry phase couples the dynamical degrees of freedom of the local magnetization. In a pherical representation of the magnetization field, , using the “north-pole” parametrization, Braun and Loss (1996) it is possible to write the spin Berry phase as,
Two canonically conjugated fields, in this case given by and , are sufficient to describe the magnetization dynamics. In addition to the energy conserving part, the phenomenological damping term may be introduced directly into the Euler-Lagrange equations by use of a Rayleigh dissipative functional, Gilbert (2004) such that the full Eq. (1) is derived from
where the Rayleigh functional is
Since the magnetization is characterized by a field, it has infinitely many modes. It is possible to map the dynamics to an infinite number of time dependent functions , i.e. . The unique equation of motion with infinite degrees of freedom, Eq. (1), becomes an infinite set of equations of motion for these dynamical parameters in this approach. The utility of this mapping is that these different parameters may have different time scales. Therefore the low energy excitations can be described as a reduced set of collective coordinates, describing the soft modes which dominate the dynamics, Tretiakov et al. (2008) and whose relaxation time is much longer than the rest of the infinite set. Examples include the position and tilt angle of a domain wall when subjected to small driving currents, or the position of a rigid homogenous domain in steady translational motion as described originally by Thiele. Thiele (1973) Identifying collective coordinates therefore offers the possibility to work with a reduced number of degrees of freedom, i.e. a minimal number of equations of motion just for the corresponding soft modes, rather than the full LLG field equation, which is usually analytically intractable.
There are several methods to obtain the equations of motion for the collective coordinates. Thiele (1973); Tretiakov et al. (2008); Everschor et al. (2011); Rodrigues et al. (2017) In this paper we outline the Hamiltonian approach. Within this formalism, the equations of motion are obtained in an explicit and direct manner from an effective energy which is a function of collective coordinates for the soft modes. We provide the comparison between this Hamiltonian approach and the generalized Thiele approach in Appendix B. Recasting Eq. (1) in a Hamiltonian language yields Papanicolaou and Tomaras (1991)
containing an energy conserving part expressed using Poisson brackets for two canonically conjugated fields and , and a damping contribution described by . The conventions for Poisson brackets used throughout this paper and their properties are listed in Appendix A.
The effective equations of motion for the collective coordinates in the Hamiltonian language are
where the Poisson brackets are now defined for pairs of canonical variables () in terms of the independent collective coordinates and their time derivatives .
In general, the canonical coordinates may always be taken as collective coordinates themselves, ; meanwhile, the momenta are gauge-dependent functionals so that the corresponding Lagrangian is . Zakharov and Kuznetsov (1997) The canonical momenta arising from the vector potential correspond to a monopole field in the spin Berry phase action, i.e. , Clarke et al. (2008); Tchernyshyov (2015) and are related to the gyrotropic tensor by Tretiakov et al. (2008); Clarke et al. (2008) . For the commonly-used gauge choice giving Eq. (3) the functional is simply . In terms of the magnetization, the gyrotropic tensor is
In the following, we will only consider Hamiltonians that are not explicitly time dependent. In this case the dissipative part is derived by comparing the rate of energy dissipation that is given by the dissipative functional, , to its expansion , where
is the viscosity tensor. Solving for yields the result,
We point out that equations (7) and (10) taken together are not specific to magnetization dynamics; rather, they are generic for conservative mechanical systems to which frictional forces (force terms linear in velocities) are included in the equation of motion by the use of a Rayleigh function. Using the identity , we may write Eqs. (7) and (10) together for a reduced set of soft modes,
which is our first main theoretical result. As an explicit matrix equation the above result becomes
by use of the relation between the Poisson brackets and the gyrocoupling tensorTchernyshyov (2015) .
Iii Effective Hamiltonian descriptions for magnetic skyrmions
The LLG equation contains topologically non-trivial solutions. In 1D they include domain walls, and in 2D they include different types of solitons that are distinguished by their integer topological charge or winding number,
An important example of magnetic solitons in 2D are skyrmions. In the following we will apply the Hamiltonian description given by Eqs. (7) and (10) to the dynamics of skyrmions and antiskyrmions. First we review the steady translational motion for rigid topological structures using this approach, and second we apply the technique to study the breathing mode.
iii.1 Translational modes of rigid topological textures
The translational motion of a rigid structure can be described in terms of a position where and are collective coordinates describing a soft mode.Papanicolaou and Tomaras (1991); Moutafis et al. (2009); Tchernyshyov (2015) The following discussion does not require a specific definition of . To obtain the Poisson bracket structure from Eq. (3), we need to peturbatively expand the fields and in terms of small deviations in , and respectively. Considering a rigid texture ansatz for the magnetization, , implies Thiele (1973) and . Performing an expansion in the spin Berry Phase action (3) up to quadratic order and discarding terms that do not contribute to the dynamics leads to, (see Appendix B)
where is the thickness of the system and the spatial integral is proportional to the topological charge defined in Eq. (13). Hence the effective action is
Noting that the canonical momentum to is , we therefore read off the Poisson bracket for topologically non trivial textures, Papanicolaou and Tomaras (1991)
which are equivalent to the traditional Thiele equations for skyrmions, Thiele (1973); Everschor (2012) and in the case of circular states, like for a simple skyrmion, the off-diagonal elements vanish.
iii.2 Internal dynamics of skyrmions and antiskyrmions
For a study of the internal dynamics of magnetic solitons, such as skyrmion breathing modes, one must go beyond the rigid texture (or traveling-wave) approximation used in the last section. We will consider a thin film system that is translationally invariant along the direction and contains localized rotationally symmetric states,Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Nagaosa and Tokura (2013)
where and are the polar coordinates in the – plane. Since the magnetic angle only changes with distance from the skyrmion core, these field configurations are referred to as circular in this work. In view of describing the breathing modes of skyrmions as well as antiskyrmions we parameterize the azimuthal angle of the magnetization by,
where is the vorticity and is the relative azimuthal angle. For a simple skyrmion one has and the angle describes its helicity: for a Bloch-skyrmion and for a Néel-skyrmion where , see Fig. 1. This corresponds to with the boundary conditions and . In other words the ferromagnetic background points in the direction. For circular (anti-)skyrmions there are two characteristic length scales: the radius and the width for the twisted domain over which varies, see Fig. 1(c). We define the radius by the circle where . We assume in the following that during the breathing dynamics retains its smooth and monotonic variation, ensuring the definition of to be unique. This work will consider only large (anti-)skyrmions in the regime where where the skyrmion’s wall width can be considered constant even as its radius is allowed to vary.
The relevant soft mode for the breathing dynamics of a circular (anti-)skyrmion is described by the radius and the relative azimuthal angle . Zhou et al. (2015) Unlike in Sec. III.1, the magnetic texture is not rigid but soft in its overall shape. For skyrmions, the phase is the global in-plane angle of the local magnetization pointing away from the radial direction (such that correspond to Néel and to Bloch configurations respectively), while for antiskyrmions, changing corresponds to a rigid rotation of the entire magnetic texture.
in terms of the softmodes and . The length scale is given by , where is a dimensionless constant (with ) arising from the integral over . Defining the canonical momentum conjugate to as , we read off the Poisson bracket as
Via the identity: we obtain . Exploiting Eq. (11), the dynamical equations for and are readily derived
Above we used that by virtue of the rotationally symmetric ansatz Eqs. (18) and (19). Eq. (22) describes the effective internal dynamics of a rotationally symmetric magnetic texture subject to the ansatz Eqs. (18) and (19) for time-independent Hamiltonians .
Iv Effective energy for circular skyrmions
Previous works have assumed an explicit domain wall ansatz for the skyrmion’s radial profile. Sheka et al. (2001); Rohart and Thiaville (2013); Romming et al. (2015) By using scaling arguments, however, one does not need to assume a specific ansatz for the skyrmion. The energy can in fact be expanded in powers of the collective coordinate for skyrmions satisfying . We illustrate this procedure for a micromagnetic model including exchange, anisotropy and interfacial Dzyaloshinskii-Moriya interaction (DMI), whose magnetic free energy is given by
where the + (-) sign stands for the isotropic (anisotropic) DMI which stabilizes circular (anti-)skyrmions. Performing the expansion of Eq. (23) in for a large radius skyrmion we obtain the following dimensionless effective energy in units of ,
where is the dimensionless (anti-)skyrmion radius in units of the one-dimensional domain wall width . The single coupling constant is the reduced DMI strength selecting for either a ferromagnetic or helical mean-field ground state. Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Rohart and Thiaville (2013) All rescaling constants are summarized in Table 1. The dimensionless values and are uniquely determined by the material parameters through the coupling constant (see Appendix C), Kravchuk et al. (2018) since we have chosen to focus on the limit where the skyrmion wall width is time-independent. The effective energy is identical for both skyrmions and antiskyrmions as a consequence of Eq. (19) taken with the appropriate choice of for skyrmions (antiskyrmions). As such, we will reduce our discussion to skyrmions only from now on even though the results derived are valid for antiskyrmions as well. In the Appendix C, we discuss how to introduce other interactions, such as dipole-dipole and bulk DMI into this framework.
iv.1 Energy landscape analysis
The energy landscape described by the effective model (24) has two extrema. The first is a global energy minimum with corresponding equilibrium coordinates
The second represents an energy saddle point with coordinates:
where we note that and that their respective effective energies are .
Upon rescaling the radii by the equilibrium radius () and energy by the equilibrium energy () in (24), the effective energy reduces to the simplified form
where we have defined as the single parameter which encapsulates the entire contribution from the material properties on the physics of the system. In these units the saddle point radius is exactly the inverse of the corresponding saddle point energy ().
A cut along the plane partitions the energy landscape into three distinct regions. A schematic view of the energy landscape is shown in Fig. 2 where individual energy sectors have been color coded to guide the reader. The coordinates in the bowl and horn regions both correspond to high energy states () as opposed to the basin’s low energy states (). In the inset of Fig. 2 we show a cut through and to emphasize the structure of the extrema introduced above. The constant energy trajectories followed by skyrmions in their configuration space in the absence of damping (see Fig. 3(a)) are obtained by solving Eq. (27) for the rescaled radius
While the two solutions in (28) represent distinct horn and bowl orbits in the regime, they represent the two branches of the same basin orbit in the case. In all scenarios, constant energy orbits describe skyrmion breathing motions as the radius grows and shrinks as a function of . The qualitative nature of orbits in the basin and bowl/horn regions are however very different from each other as the dynamical range of is limited in the basin orbits while it takes all values from to in the horn/bowl orbits. This leads us to denominate the high and low energy breathing dynamics as rotating and oscillating modes respectively. The degeneracy of the rotating modes disappears as their energy is lowered through the saddle energy and into the basin. The horn rotations are unphysical however as they would predict uncollapsable skyrmions in the limit of very small skyrmion sizes. Furthermore, since our theory is only applicable for skyrmion much larger than the profile wall width , it cannot reliably describe their behavior at such small radii. Overall, the dynamical spectrum of this model is reminiscent of that of a simple pendulum which exhibits rotations and oscillations around its suspension point depending on whether the kinetic energy is greater or less than the potential energy of its “upside down” unstable equilibrium.
Lastly, the full range of the radial oscillations can be readily obtained from Eq. (28) by noting that all radial maxima/minima in the orbits appear on the line (as shown in Fig. 2). For the rotating modes have and whereas in the oscillating phase one has and . Since the lower energy branch on this line is independent of the coupling , the amplitude for oscillating breathing modes below the saddle energy is insensitive to material properties (see discussion below).
V Skyrmion breathing modes
v.1 Equations of Motion
In terms of the variables and , the Poisson bracket Eq. (21) becomes
where we assume a constant shape factor coming from the dimensionless integral in the spin Berry phase action. Moreover the dimensionless dissipation factors both scale linearly with the radius ,
where and are time-independent proportionality constants that may depend on the coupling strength . Using Eqs. (29) and (30) in the Hamiltonian formalism (12) gives the two mode dynamical system for skyrmion breathing
with as defined in Eq. (27) and is the rescaled time,
Numerical solutions of Eqs. (31) with and without damping are shown in Fig. (4a) where the initial energy is set above the saddle energy. These results show the characteristic transition from rotating to oscillating phase in the , and damped evolutions as compared to their undamped, constant energy analogs. All three variables relax towards the equilibrium state as expected. The rate of energy loss is qualitatively different in the two phases, transitioning from a linear-like to an exponential decay as further discussed below. Micromagnetic simulations confirm the qualitative predictions of the effective energy model as it pertains to the skyrmion breathing modes. In particular, we observe the rotating (Fig. 4b) and oscillating (Fig. 4c) breathing regimes and their distinct energy loss behavior.
In the following we present analytical results pertaining to the expected dynamical periods, breathing amplitudes and energy decay rates.
v.2.1 Dynamical Periods
where the limits of integration are and with if (or otherwise) being the Heaviside function. For the case of the positive and negative signs refer to bowl-like and horn-like rotations respectively. Direct numerical integrations of these formulas show that the period scales linearly in energy for both the rotating and oscillating breathing modes (see Fig. 5). Our theory allows to explore dynamics beyond the small amplitude limit in an ansatz independent manner. It includes, however the small radius perturbation calculated previously in the literature derived by using a non- energy-minimizing ansatz. For small oscillations around equilibrium the period can be computed up to as
Upon converting back to physical time using (32) and recalling from (25) how the physical equilibrium radius scales with the material parameters , one recovers that the period of small oscillations around equilibrium scales as in agreement with previous literature. Kravchuk et al. (2018); Rodrigues et al. (2018)
In Figure 6 we emphasize the predictive power of the effective model by comparing theory to micromagnetic modeling in the undamped dynamical limit. Radial and angular trajectories obtained by integrating (31) match very well with equivalent micromagnetic simulations in both rotating (a) and oscillating (b) regimes. While the oscillating breathing modes match almost perfectly, a small deviation is seen between physical and predicted radial dynamics in the rotating regime. This is due to a breakdown in the large skyrmion approximation underpinning the theory whenever the skyrmion contracts to sizes comparable to the profile’s wall width. To illustrate this, the inset of subfigure (a) shows the different profiles observed for maximum and minimum skyrmion radii throughout one rotational period.
v.2.2 Breathing Amplitudes
The maximum and minimum possible skyrmion radii derived from the model are,111From here on we select for definiteness.
which may be obtained directly from Eq. (28) upon setting . At energies below the saddle energy (), is the maximum radius of oscillations and is the minimum radius of oscillations; meanwhile, for energies above the saddle energy (), is the maximum radius of rotations on the bowl, is the minimum radius of rotations on the bowl, is the maximum radius of rotations on the horn, and is the minimum radius of rotations on the horn. We see from Eq. (35a) that the stationary points of the -oscillations for breathing modes below the saddle energy are independent of the material properties. This may be likened to a mass-on-a-spring system where the amplitude is fully determined by the initial extension from equilibrium even though the specific dynamics connecting the two extrema of motion do depend on the size of the spring constant and the mass. In this case this is not a trivial consequence of small harmonic oscillations around equilibrium however because it is true for the entire range.
v.2.3 Energy Decay Rates
which we will use to analytically explain the distinction between linear and exponential decay observed in the numerical calculations (see Fig. 4). Since Eq. (36) is globally negative, except at the energy minimum where it vanishes, it correctly describes a dissipative process that relaxes the skyrmion to its equilibrium state.
If the skyrmion’s rotational and oscillatory modes precess on timescales sufficiently small compared to those for energy dissipation, the two dynamics can be effectively decoupled by averaging (36) over constant energy orbits to obtain a single ordinary differential equation describing the energy lost by the system over time. One then has:
where the integrals are performed over one complete oscillatory/rotational orbits (28). In what follows we will leverage the fact that orbital periods (33) scale linearly with the orbit’s energy (see Fig. 5). The above integrals are not exactly solvable but a series of upper and lower bounds can still be constructed (see Appendix D). For the rotational breathing mode, one can show that:
where we absorb all constants into . By construction, the solution of this new bounding equation is guaranteed to decay faster than the true solution of Eq. (36). Since for rotational modes, this upper bound guarantees at most a linear decay to the skyrmion’s energy.
Following a similar reasoning for the oscillatory breathing modes by constructing a lower bound to the energy dissipation rate, one finds (see Appendix D),
where is the maximum range of domain wall tilt angle attained during a single constant energy oscillation. Since the solution of (39) is guaranteed to decay slower than that of (36), the true energy loss in the oscillating regime must decay at least exponentially. These arguments confirm the sharp transition observed in the dissipation rate when the skyrmion breathing dynamics cross the saddle separatrix when transitioning between and states, see Fig. (4).
In this work we have derived the Hamiltonian system for the low energy excitations of rotationally symmetric magnetic (anti-)skyrmions in an ansatz-independent manner. By means of scaling and symmetry arguments we modeled the breathing mode of (anti-)skyrmions in terms of collective coordinates, where the area of reversed spins in the skyrmion core and the skyrmion phase are conjugated variables in phase space. As seen from the form of the energy landscape, our model exhibits a rich behavior which is confirmed by micromagnetic simulations of (anti-)skyrmion structures in magnetic substrates with translational invariance along the anisotropy axis. The main results presented in this paper include the analytical and numerical investigation of the well-known oscillatory breathing mode where small amplitude oscillations in the radius and skyrmion phase around equilibrium proceed in tandem, as well as the description of rotational breathing behavior, characterized by large radius oscillations and a continuous non-uniform precession of the phase. Furthermore, we predict two distinct regimes of energy dissipation where the average power loss of large amplitude rotational breathing modes decays linearly as opposed to exponentially for the oscillating modes. We expect that these distinctive energy decays will allow to detect the different modes experimentally. It must be stated that the limit of our model lies in the implicit assumption of fixed skyrmion wall profiles. The next order approximation would be to incorporate the skew of the wall profile by introducing an additional pair of collective coordinates. Doing so would allow extension of this analysis to skyrmion radii much smaller than those allowed by this work. We would like to emphasize that the results described here hold for both skyrmions and antiskyrmions. Therefore a perfect test system will be one where both of them occur simultaneously. This is for example naturally the case when skyrmion and antiskyrmion pairs are created,Everschor-Sitte et al. (2017); Leonov and Mostovoy (2017); Stier et al. (2017) or in systems with certain symmetries. Hoffmann et al. (2017)
The groups at Mainz acknowledge funding from the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X, the Graduate School of Excellence Materials Science in Mainz (MAINZ, GSC 266), the German Research Foundation (DFG) under the Project No. EV 196/2-1 and the Alexander von Humboldt Foundation.
B. M. , D. R. and D. P. contributed equally to this work.
Appendix A Poisson brackets
In the main text we applied techniques from Hamiltonian mechanics. Poisson brackets entered at the level of the collective coordinates where we were concerned with just time dependent functions, and also at the level of the LLG field equation where the magnetization field depends on both space and time. Below we review equations for Poisson brackets relevant to this work.
Time dependent functions
The Poisson bracket convention for time dependent functions and is
corresponding to the simple action . For example, a basic result is the Poisson bracket between the canonical coordinates and momenta,
Using the Poisson bracket, the time derivative of any function can be calculated,
where Hamilton’s equations were used in the second line.
A further rule can be derived for the Poisson bracket between quantities that have a known dependence on functions of the canonical variables, e.g. and similar for :
corresponding to an action of the form . So the Poisson bracket between the canonical fields is
The time derivative for a field , similar to Eq. (42), is
by use of Hamilton’s equations and . Finally the useful identity in analogy to Eq. (43), for the Poisson bracket between quantities that have known dependence on a set of functions, say , is
Example: Hamiltonian formulation of the LLG
where . This may be verified explicitly. For example, using the spherical parameterization of the magnetization employed in the main text, we may identify the two canonical fields as and from the spin Berry phase action Eq. (3). Then the three non-zero Poisson brackets between the magnetization components , and in Eq. (48) are straightforwardly verified using Eq. (44) with these canonical fields. Moreover in this situation Eq. (47) reduces to a cross product structure
which is the anticipated precessional term for undamped motion.
Appendix B Comparison between generalized Thiele method and the Hamiltonian formalism
The generalized Thiele method is based on the idea of describing the dynamics of certain magnetic configurations just in terms of the time evolution of a finite number of collective coordinates describing the soft modes, Tretiakov et al. (2008); Clarke et al. (2008) for which we have that,
This decomposition has been successfully applied in the description of the low energy excitations of topological magnetic textures. For example, for the field-driven or current-driven motion of domain walls the physics is well described up to a certain magnitude of the applied driving field or current Tatara and Kohno (2004) by a soft mode described by the two collective coordinates: the domain wall position and the tilt angle of the magnetization inside the wall. Another example is the dynamics of the position of a rigid homogenous domain in steady translational motion as described by Thiele. Thiele (1973) Considering the expansion Eq. (51), performing the projection of the LLG Eq. (1) onto and integrating over volume gives a generalization of Thiele’s equations Thiele (1973); Tretiakov et al. (2008); Everschor et al. (2011)
where are the elements of the inverse matrix. Comparing this equation with the Hamiltonian Eq. (7), we obtain that the equivalence between the generalized Thiele approach and Hamiltonian approach is embedded in the following identities
The result for the dissipative term in terms of the Poisson brackets is also derived in a more general manner (see the main text) and therefore this structure is general for including viscous damping into Hamilton’s equations, regardless of the system of study.
To further illustrate the Hamiltonian approach, we present as an example the Poisson bracket for and , describing the position of a skyrmion, for the soft mode associated with translational motion.
Example: Derivation of the Poisson bracket for the translational modes from the spin Berry phase action
By virtue of Thiele’s traveling wave ansatz, , the unit magnetization has the properties