Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films
Abstract
Microwave magnetodynamics in ferromagnets are often studied in the smallamplitude or weakly nonlinear regime corresponding to modulations of a welldefined magnetic state. However, strongly nonlinear regimes, where the aforementioned approximations are not applicable, have become experimentally accessible. By reinterpreting the governing LandauLifshitz equation of motion, we derive an exact set of equations of dispersive hydrodynamic form that are amenable to analytical study even when full nonlinearity and exchange dispersion are included. The resulting equations are shown to, in general, break Galilean invariance. A magnetic Mach number is obtained as a function of static and moving reference frames. The simplest class of solutions are termed uniform hydrodynamic states (UHSs), which exhibit fluidlike behavior including laminar flow at subsonic speeds and the formation of a Mach cone and wavefronts at supersonic speeds. A regime of modulational instability is also possible, where the UHS is violently unstable. The hydrodynamic interpretation opens up novel possibilities in magnetic research.
Magnetodynamics in thin film ferromagnets have been studied for many decades. Advances in nanofabrication and the advent of spin transfer Berger (1996); Slonczewski (1996) and spinorbit torques Hals and Brataas (2013) have opened a new frontier of experimentally accessible nonlinear physics Hoefer, Silva, and Keller (2010); Mohseni et al. (2013); Macià, Backes, and Kent (2014); Chung et al. (2016); Mistral et al. (2008). Largeamplitude excitations negate the use of typical linear or weakly nonlinear analyses Slavin and Tiberkevich (2009); L’vov (1994); Bertotti, Mayergoyz, and Serpico (2008), necessitating instead either micromagnetic simulations Vansteenkiste et al. (2014) or analytical approaches suited to strongly nonlinear dynamics. Therefore, an interpretation of the LandauLifshitz (LL) equation that includes full nonlinearity, yet is amenable to analytical study, would be insightful.
A hydrodynamic interpretation was proposed by Halperin and Hohenberg Halperin and Hohenberg (1969) to describe spin waves in anisotropic ferro and antiferromagnets. Recently, theoretical studies of thin film ferromagnets with planar anisotropy have identified a relationship to superfluidlike hydrodynamic equations König, Bønsager, and MacDonald (2001); Sonin (2010); Takei and Tserkovnyak (2014); Chen et al. (2014); Skarsvåg, Holmqvist, and Brataas (2015); Flebus et al. (2016) supporting largeamplitude modes beyond strongly nonlinear spin wave and macrospin modes L’vov (1994); Bertotti, Mayergoyz, and Serpico (2008). However, these studies are limited to the longwavelength, lowfrequency regime where linear and weaklynonlinear approaches apply. The relaxation of these approximations along with the identification of a deep connection between magnetodynamics and fluid dynamics brings new perspectives on magnetism and reveals novel physical regimes. Indeed, nonlinear, dispersive physics are required to describe superfluids and exotic structures such as solitons, quantized vortices, and dispersive shock waves (DSWs) Hoefer et al. (2006); Carusotto et al. (2006); Fetter (2009); El and Hoefer (2016), as exemplified by BoseEinstein condensates (BECs) Pethick and Smith (2002); Donley et al. (2001); Hoefer et al. (2006); El, Gammal, and Kamchatnov (2006); Carusotto et al. (2006); Cornish, Thompson, and Wieman (2006); Gladush et al. (2007); Kevrekidis, Frantzeskakis, and CarreteroGonzàlez (2008); Frantzeskakis (2010); Fetter (2009); Kevrekidis, Frantzeskakis, and CarreteroGonzález (2015); Demokritov et al. (2006); Qu, Pitaevskii, and Stringari (2016); Congy, Kamchatnov, and Pavloff (2016). To obtain an analytical description of largeamplitude magnetic textures, we introduce dispersive hydrodynamic (DH) equations for a thinfilm ferromagnet.
This letter shows that the LL equation exactly maps into a DH system of equations, without longwavelength and lowfrequency restrictions. The conservative equations map to the Euler equations of a compressible, isentropic fluid. The DH equations for a planar ferromagnet admit spincurrentcarrying, spatially periodic magnetization textures termed “uniform hydrodynamic states” (UHSs), providing a continuous and generalized description of spin superflows König, Bønsager, and MacDonald (2001); Sonin (2010); Takei and Tserkovnyak (2014) up to smallamplitude spin waves. Within the DH formulation, we prove that planar ferromagnets break Galilean invariance and elucidate their referenceframedependent dynamics by identifying the linear dispersion relation for spin waves propagating on top of a UHS background. Such symmetry breaking at the level of linear spin waves is striking and fundamentally different from the nontrivial referenceframedependent dynamics of topological textures due to their inherent nonlinearity, e.g., Walker breakdown for domain wall propagation Beach, Tsoi, and Erskine (2008) and core reversal in magnetic vortices Yamada et al. (2007). In this letter, we also show that static textures can break Galilean invariance for infinitesimal spin wave excitations that ride on a textured background. To emphasize this novel result, we suggest a Brillouin light scattering experimental test where broken Galilean invariance manifests itself as a spinwave dispersion shift in the presence of a UHS.
We consider the nondimensionalized LL equation (see Supplementary Material Sup ())
(1) 
Damping is parametrized by the Gilbert constant , is the magnetization vector normalized to the saturation magnetization, and is the normalized effective field including: ferromagnetic exchange, ; total anisotropy determined by , where is the perpendicular magnetic anisotropy field, such that () represents a material with easyplane (perpendicular magnetic) anisotropy; and a perpendicular applied field, . This nondimensionalization of a twodimensional (2D) thin film provides a parameterfree description of materials with planar or uniaxial anisotropy. We consider an idealized case where inplane magnetic anisotropy is negligible, i.e., its symmetrybreaking contribution only perturbs the leading order solution, similar to domain wall Brownian motion Kim, Takei, and Tserkovnyak (2015).
Fluidlike variables are introduced using the canonical Hamiltonian cylindrical transformation Papanicolaou and Tomaras (1991)
(2) 
where is the azimuthal phase angle. We identify () as the longitudinal spin density and as the fluid velocity. There are two important features of Eq. (2). First, the flow is irrotational because the velocity originates from a phase gradient, i.e., only quantized circulation, such as a magnetic vortex Sonin (2010), is possible. Second, is undefined when , corresponding to fluid vacuum.
Utilizing the transformation (2) and standard vector calculus identities, the LL equation (1) can be exactly recast as two DH equations Sup ()
(3a)  
Equation (3a) is reminiscent of spin density continuity Zhang and Yang (2005) from which we identify the spin density flux as the spin current
(4) 
Vacuum carries zero spin current. However, maximal spin current is reached when , identified as the saturation density. This implies that ferromagnetic textures ( are better spin current conductors than smallamplitude spin waves Hoffmann (2013). The hydrodynamic equivalents for the fluid velocity Eq. (3) are displayed. When , Eq. (3) becomes , a diffusion equation for the velocity, hence acts similar to a viscosity. Previous works Halperin and Hohenberg (1969); König, Bønsager, and MacDonald (2001); Sonin (2010); Takei and Tserkovnyak (2014) have neglected exchange dispersion and nonlinearity in Eqs. (3) by assuming the longwavelength, near saturation density, lowvelocity limit, i.e., , , and . As we show below, the full nonlinearity and exchange dispersion included in Eqs. (3a) and (3) are required to describe the existence and stability regions of magnetic hydrodynamic states and broken Galilean invariance.
Insight can be gained from the homogeneous field , conservative limit, where Eqs. (3) become conservation laws for and . Notably, the nonnegative deviation from vacuum , or fluid density, is not conserved. A conservation law for the momentum can also be obtained
(5) 
where the magnetic pressure is defined as
(6) 
Equations (3a) with , and (Breaking of Galilean invariance in the hydrodynamic formulation of ferromagnetic thin films) are analogous to the timereversed Euler equations expressing conservation of mass and momentum for a 2D, compressible, isentropic fluid with a density and velocitydependent pressure .
Additionally, the onedimensional conservative limit of Eqs. (3a) and (3) are exactly the equations describing polarization waves in twocomponent spinor Bose gases Qu, Pitaevskii, and Stringari (2016); Congy, Kamchatnov, and Pavloff (2016) and, in the near vacuum (), longwavelength, and lowfrequency regime, approximate the mean field dynamics of a BEC Pethick and Smith (2002); Satija and Balakrishnan (2011). This suggests that thin film ferromagnets are ripe for the exploration of nonlinear structures observed in BECs, e.g., “Bosenovas” Donley et al. (2001); Cornish, Thompson, and Wieman (2006) in attractive (); and dark solitons Frantzeskakis (2010), vortices Fetter (2009), and DSWs Hoefer et al. (2006) in repulsive () BECs. Some of these structures have been observed in uniaxial (dissipative droplets Mohseni et al. (2013); Chung et al. (2016); Macià, Backes, and Kent (2014)) and planar (vortices Mistral et al. (2008)) thin film ferromagnets. As we demonstrate, hydrodynamic states are also supported.
Consider an ideal planar thin film ferromagnet () and a homogeneous field. Equations (3) admit a static () solution with nonzero fluid velocity, , , , and . These are ground states known as spindensity waves (SDWs) Grüner (1994) or soliton lattices Sonin (2010) that minimize both exchange and anisotropy energies, i.e., any configuration with is stable when lies purely inplane. SDWs exhibit a periodic structure that affords them topological stability whereby the phase rotation can be unwound only by crossing a magnetic pole () Sonin (2010); Sup (). For a nonzero field, , SDWs are also supported for any when due to the longitudinal spin density relaxation effected by Eq. (3a). Such a relaxation maintains and thus the topology of and finite spin current carried by a SDW. This property is identical to that of equilibrium transverse spin currents in other magnetic textures including domain walls and vortices [Ref. Sonin, 2010, Eq. (4) in Ref. Tserkovnyak, Hankiewicz, and Vignale, 2009].
For no damping, Eqs. (3) admit dynamic solutions parametrized by the constants , where , , called uniform hydrodynamic state (UHS). The fluid velocity is the wavenumber of the UHS whose frequency is
(7) 
obtained from the magnetic equivalent of Bernoulli’s equation Sup (). Here, positive implies clockwise spatial increase of the azimuthal phase whereas negative implies clockwise temporal precession about the axis defining forward and backward wave conditions, schematically shown in Fig. 1. This is in contrast to magnetostatic forward and backward volume waves established by the relative direction between their wave vector and the external applied field.
The magnetization in a UHS can exhibit large angle deviations from the axis, making it a strongly nonlinear texture. Near saturation density, , a UHS limits to a spin superflow König, Bønsager, and MacDonald (2001); Sonin (2010); Takei and Tserkovnyak (2014) whereas near vacuum, , the UHS frequency Eq. (7) becomes the exchange spinwave dispersion . Thus, a UHS is the generalization of periodic magnetic textures from large (spin superflow) to small (spinwave) amplitudes. It is important to recognize that the ground state for the UHS is a SDW, i.e., the ground state of planar ferromagnets is not defined by a single orientation except for the vacuum state. In this sense, the UHS is significantly different from the conventional theory of spin waves based on the HolsteinPrimakoff transformation Mattis (2006) and their strongly nonlinear dynamics. L’vov (1994); Bertotti, Mayergoyz, and Serpico (2008).
Smallamplitude perturbations of a UHS can be regarded as spin waves with a generalized dispersion relation obtained from the linearization of Eqs. (3a) and (3)
(8) 
where is the wave vector, and the velocity reflects a Doppler shift, i.e., the velocity of an external observer with respect to the UHS. The dispersion relation shows that magnetic systems lack Galilean invariance. In other words, an observer velocity does not generally result in a reference frame where the relative fluid velocity is zero. Galilean invariance is recovered near vacuum with dispersion , i.e. exchangemediated spin waves and the BEC limit Pethick and Smith (2002); Satija and Balakrishnan (2011); and for spin superflow, . More importantly, the fluid velocity confers a spectral shift in Eq. (8) due to the UHS’s intrinsically chiral topology, similar to interfacial DzyaloshinskiiMoriya interaction Nembach et al. (2015).
The long wavelength limit of Eq. (8) leads to coincident spinwave phase and group velocities, i.e., magnetic sound velocities
(9) 
Here, we assume collinear and opposite to (). Subsonic flow occurs when spin waves can propagate both forward and backward: . However, when , the flow is supersonic and some spin waves are convected away. These conditions can be quantified in terms of the Mach numbers , when , , respectively
(10) 
For both, the flow is subsonic or laminar when . In the supersonic regime, , it is energetically favorable to generate spin waves, thus leading to the Landau breakdown of superfluidlike flow Landau (1941). The resulting phase diagrams are shown in Fig. 2. Interestingly, Eq. (10) predicts that is independent of , implying that only the UHS longitudinal spin density and its nontrivial topology, , set the supersonic transition, not the frequency . It must be noted that broken Galilean invariance causes the Landau criterion concept Pethick and Smith (2002) to give an incorrect sonic curve.
A qualitatively distinct flow regime occurs when and the sound velocities Eq. (9) are complex. This corresponds to a change in the mathematical type of the long wavelength Eqs. (3) from hyperbolic (wavelike) to elliptic (potentiallike). Consequently, the UHS is unstable in the sense that small fluctuations lead to drastic changes in its temporal evolution or modulational instability (MI) Whitham (1974); Zakharov and Ostrovsky (2009). Note that , does not result in MI.
The aforementioned regimes were validated by performing micromagnetic simulations with damping Vansteenkiste et al. (2014). We simulate dynamics for an ideal Permalloy nanowire ( T) of nondimensional width with transverse free spin boundary conditions and horizontal periodic boundary conditions (PBCs). We initialize with a SDW, include only local dipolar fields (zero thickness limit), and set . A homogeneous field stabilizes the SDW at a specific and a quantized that satisfies the PBC. This enables us to numerically probe along a horizontal line in the phase diagram of Fig. 2(a) by implementing a slowly increasing . By inserting a point defect (a magnetic void), the SDW spontaneously generates spin waves when is large enough to cross the supersonic transition, leading to a breakdown in the spatial homogeneity of the SDW Sup (). Due to the SDW’s topology and the PBC, the change in symmetry is accommodated by annihilating a single phase rotation and reducing in a quantized fashion. Topologically, this is possible in planar ferromagnets by crossing a magnetic pole, e.g., nucleating a vortex, as shown in the Supplementary Video 1. This was also observed in wires of width . The sonic curve estimated this way is shown in Fig. 2(a) by black circles, demonstrating good agreement with . We attribute any discrepancy to boundary and finite size effects Frisch, Pomeau, and Rica (1992), as further explored below.
We use the same numerical method described above with the addition of thermal fluctuations and the symmetrybreaking nonlocal dipolar fields to study the stability of a SDW in a nanowire of nondimensional thickness . In this case, the SDW topological structure completely collapses at the boundary shown in Fig. 2(a) by squares. In contrast to a recent report where stable spin superflow was predicted only for nanowires shorter than the material exchange length Skarsvåg, Holmqvist, and Brataas (2015), we observe stable SDWs over a wide range of parameters in phase space (Supplementary Video 2).
The supersonic transition in the moving frame is estimated by use of a numerical method described elsewhere Hoefer and Sommacal (2012). A moving, perpendicular, localized, weak magnetic field spot with velocity is used to perturb a homogeneous state in the bias field . The obtained sonic curve is in good agreement with , shown in Fig. 2(b) by red circles.
We now explore the effect of finitesized obstacles on a UHS. As observed in BECs, obstacles can generate vortices, wavefronts, and DSWs in a fluid flow Carusotto et al. (2006); Fetter (2009); Hoefer et al. (2006). Note that wavefronts, i.e., “spinCerenkov” radiation, were previously observed via micromagnetic simulations in homogeneous (), thick ferromagnets in the moving reference frame () Yan et al. (2013). The wavefronts studied here are different, resulting from both moving (, ) and static (, ) reference frames; yet another manifestation of broken Galilean invariance. We illustrate these features with simulations where and local dipolar fields are included, shown in Fig. 3 as a grayscale map and vector field for and , respectively [see Supplement for the corresponding inplane magnetization map]. First, we consider the subsonic conditions x of Fig. 2(a) for a SDW in the static reference frame () with a magnetic defect within a circular area of in diameter. The static configuration in Fig. 3(a) is analogous to Bernoulli’s principle for laminar flow.
A different situation occurs at supersonic conditions x (), Fig. 3(b). Here, the density develops a distinct Mach cone (dashed), delimiting static wavefronts and the nucleation of propagating vortexantivortex pairs, shown far from the defect in Fig. 3(c). In the moving reference frame, a homogeneous state is perturbed by a moving, weak, localized field. Utilizing the subsonic conditions x (), the flow is laminar, Fig. 3(d) [c.f. supersonic x in Fig. 2(a)]. Wavefront radiation outside the Mach cone is observed for the supersonic condition x () in Fig. 3(e). However, the field spot amplitude is too weak to excite vortexantivortex pairs. Animations are in Supplementary Videos 3 to 6.
The MI regime for UHSs with exhibits a violent instability (see Supplementary Video 7). Notably, for a uniaxial ferromagnet with , MI is always operative. This is consistent with the focusing of spin waves and the formation of dissipative droplets in spin torque devices utilizing materials with perpendicular magnetic anisotropy Hoefer, Silva, and Keller (2010); Mohseni et al. (2013); Macià, Backes, and Kent (2014); Chung et al. (2016).
We now discuss an experimental test for the hydrodynamic predictions. As mentioned above, the dispersion relation Eq. (8) features a spectral shift with nonzero fluid velocity. This shift can be observed e.g., by means of Brillouin light scattering (BLS), as already shown for DzyaloshinskiiMoriya interactions Nembach et al. (2015). For a given fluid velocity, the magnitude of the shift, , can be tuned by an externally applied field. Use of such tuning, in combination with BLS, will allow a direct test of the predicted breaking of Galilean invariance, insofar as the nonlinear properties of the dispersion relation Eq. (8) can be quantitatively investigated. In particular, if one plots spinwave frequency vs. wavenumber squared for both the Stokes and antiStokes BLS peaks in the shortwavelength limit, , the modulus of the zerowavenumber intercepts from linear regression will not be equal in the case of broken Galilean invariance, in contrast to the case of Galilean invariance, where the intercepts should be equal.
In summary, the dispersive hydrodynamic (DH) formulation permits us to quantify the manner in which thin film ferromagnets lack Galilean invariance in the context of linear spin wave propagation on a dynamic UHS or static SDW background. The breaking of Galilean invariance is often associated with relativistic phenomena wherein the Lorentz transformation conjoins spacetime into a single coordinate system, replacing the Galilean transformation. Instead, the present case ultimately reflects the counterintuitive ability of exchangecoupled, topological spin textures to support spin currents, even in the static case. The predictions are robust to damping, nonlocal dipolar fields, and finite temperatures for a large portion of phase space. The exact representation of the LL equation in DH form along with associated mathematical tools Whitham (1974); Kevrekidis, Frantzeskakis, and CarreteroGonzález (2015); El and Hoefer (2016) enables new magnetodynamic predictions and a frontier of magnetic research, for example the observation of a Mach cone, wavefronts, and vortex nucleation, suggesting the existence of coherent structures such as oblique solitons and DSWs.
Acknowledgements.
The authors thank Leo Radzihovsky for beneficial discussions. E.I. acknowledges support from the Swedish Research Council, Reg. No. 63720146863. M.A.H. partially supported by NSF CAREER DMS1255422.References
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Supplementary material
Appendix A Nondimensionalization of the LandauLifshitz equation
The LandauLifshitz (LL) equation can be written as
(11) 
where the damping is in Gilbert form, is the magnetization vector, is the saturation magnetization, is the gyromagnetic ratio, is the vacuum permeability, and the effective field is
(12) 
where , is the exchange length, is the exchange constant with units Jm, , and is the perpendicular magnetic anisotropy energy density with units Jm. In order to generalize our results for materials favoring either planar or uniaxial anisotropy, we introduce the quantity , where is the perpendicular magnetic anisotropy field. It is then possible to rescale time , space , and field . As a result, we obtain the dimensionless LandauLifshitz equation
(13) 
[Eq. (1) in the main text]. Note that the special case of an isotropic ferromagnet leads to a divergence from our nondimensionalization. However, we consider that an isotropic thin film ferromagnet is experimentally unlikely, in which case the divergence does not affect the generality of the analytical results.
Appendix B Derivation of hydrodynamic equations
The dispersive hydrodynamic representation can be derived by inserting the canonical Hamiltonian transformation
(14a)  
(14b) 
into Eq. (13). Solving for we obtain the exact transformation
(15) 
Solving for and we obtain
(16a)  
(16b)  
Interpreting as the longitudinal spin density, we observe that Eq. 16a is reminiscent of the continuity equation Zhang and Yang (2005) in the absence of damping
(17) 
where is the spin current, is spin torque, and is a relaxation process. Therefore, we identify the first term of the righthandside of Eq. (16a) as the spin current, acknowledging the sign difference arising from our definition of .
The term appearing in Eq. (16a) is nonnegative, quantifying the fluid density deviation from vacuum, . However, one can verify that this is not a conserved quantity. In fact, neglecting damping,
(18) 
whose righthandside cannot be written as a divergence. This agrees with the notion that spin current itself is not a conserved quantity even when damping is zero, whereas the longitudinal spin density is always conserved Zhang and Yang (2005) in the absence of specific processes. Indeed, this is a restatement of the fact that spin currents in ferromagnets have both a coherent and diffusive character, where the coherent form is typified by spin wave propagation, whereas the diffusive character is associated with thermal magnon kinetics. As such, the lack of conservation for the fluid density in ferromagnets is analogous to the fact that the energyconserving coherence lifetime in the Bloch formulation of spin dynamics can be substantially shorter than the thermodynamic relaxation time Sparks (1965).
In the longwavelength, lowfrequency, near saturation density limit, , , and , we obtain
(19a)  
(19b) 
These equations are consistent with the equations describing spin superfluidity Sonin (2010); Takei and Tserkovnyak (2014).
Considering a UHS with a uniform precessional frequency , no damping, and neglecting spatial variation in , the magnetic analogue of Bernoulli’s equation can be obtained by multiplying Eq. (15) by
(20) 
where is pressure, the second term is the kinetic energy, the third term is a potential (analogous to gravity).
Appendix C Micromagnetic simulations
We perform micromagnetic simulations to validate our analytical results. For the magnetic defect in the static frame, , we use the GPUaccelerated code Mumax3 Vansteenkiste et al. (2014). In order to maintain a SDW in the absence of external forcing, e.g., spin injection, we set a SDW with an integer number of periods as an initial magnetization state and implement periodic boundary conditions (PBC) along the nanowire’s length and free spin boundary conditions transversely. The integer period requirement sets the length of the simulation area. We use idealized physical parameters consistent with infinitely permeable Permalloy, kA/m, pJ/m, and . In this case, space is normalized by nm according to the nondimensionalization introduced above. Three types of simulations were performed:

To estimate the sonic curve, we set a nanowire of in width, in length, and in thickness. A SDW at a particular is defined as an initial condition. Taking advantage of the field relaxation of the longitudinal spin density, we set and sweep the field at a normalized rate of (on the order of T/s). This rate is slow enough to sweep as a function of field. A pointdefect is introduced as an absence of magnetization. The supersonic condition is obtained at the point when the linear increase in changes slope, corresponding to a quantized drop of the SDW topological charge.

The supersonic transition is also estimated by including nonlocal dipolar fields. In contrast to case 1 above, we do not consider any defect and instead add room temperature ( K) thermal fluctuations. The supersonic transition is identified as a dramatic drop in the longitudinal spin density , corresponding to the generation of spin wave as the UHS decays. To take into account the noise introduced by thermal fluctuations, a linear fit is applied to the averaged . The error is obtained from the standard deviation throughout the simulation time.

The effect of finitesized obstacles is obtained from simulations on a domain spanning in width, in length, and in thickness. We set the damping to and utilize an external field to maintain the desired longitudinal spin density. This size guarantees that the generated features damp before reaching the simulation physical boundary.
In the case of the moving obstacle, a localized applied field spot, we use a pseudospectral method to solve the twodimensional LandauLifshitz equation in the moving frame Hoefer and Sommacal (2012). The simulation is initialized with an homogeneous state, . The localized field spot magnitude is set to in normalized units. Two types of simulations are performed:

To estimate the sonic curve, we set a simulation area of in width and in length. A pointfield is implemented as a hyperGaussian of order 4 and width . Transient structures are damped during the simulation time. Thus, the supersonic transition is estimated by the existence of a welldefined wave front, i.e., wavefront radiation.

The effect of finitesized obstacles is obtained from a simulation following the geometry described in point 3 above.
Appendix D Fluidlike behavior for the inplane magnetization
In the main text, Fig. 3 shows the fluidlike behavior of a thin film planar ferromagnet, both in the static and moving reference frames. A hydrodynamic visualization is chosen, where the longitudinal spin density and fluid velocity are represented by a grayscale map and vector field, respectively. Here, we show the corresponding magnetization component () in Fig. 4 as a grayscale map. Note that the SDW, Fig. 4(ac), is visualized as a periodic variation of . The features arising in the subsonic and supersonic conditions distort this periodic configuration. Notably, the vortexantivortex, Fig. 4 breaks the periodicity of the SDW. This can be understood by topological arguments, as outlined in the next section. In contrast, the homogeneous state in the moving reference frame exhibits similar features for and .
Appendix E Hydrodynamic topology
The notion of topology in our hydrodynamic framework is tightly linked to the definition of the magnetic ground state. The concept of topology is typically used to mathematically classify states or textures into groups separated by an infinite barrier (see Ref. Braun, 2012 for details). Thus, a texture belonging to a certain topological group cannot change its topological group i.e., it is topologically protected. Within a topological group, textures are allowed to be continuously deformed, i.e., such a deformation does not cross a barrier.
Physically, topological barriers correspond to a large, but finite, energy. In a magnetic system, such a barrier can be interpreted as the magnetization hard axis. For example, a domain wall separating two magnetic states crosses a hard axis, and is therefore topologically protected. In fact, domain walls are usually annihilated by hitting a pinning site or a physical boundary, where the energy barrier can be easily surpassed.
In the hydrodynamic context, we are interested in solutions such as a UHS, exhibiting phase rotations in the  plane. Therefore, in the magnetization unit sphere, a UHS traces a circle Sonin (2010). At this point, it is important to distinguish between uniaxial and planar ferromagnets. For uniaxial ferromagnets, the UHS circle can be continuously deformed along the unit sphere latitude until it reaches its ground state, the axis. This is schematically shown in Fig. 5(a) for a UHS circle (red) in the northern hemisphere. Consequently, UHSs in uniaxial ferromagnets do not cross the hard axis to return to the homogeneous ground state, , which is typically referred to as a state with trivial topology. It is important to point out that other topological structures are allowed in uniaxial ferromagnets, such as Bloch domain walls, where the ground states are separated by the hard plane.
In the case of planar ferromagnets, the ground state is the isotropic easy plane. Thus, a UHS can continuously deform to a circle along the equator, i.e., a SDW when , as shown in Fig. 5(b). Since this ground state is not homogenous, it possesses a nontrivial topology. For this reason, UHSs in planar ferromagnets exhibit topological protection. The only mechanism to annihilate a UHS is to cross the hard axis i.e., the poles. As shown in the Supplementary Video 1, this is achieved by creating a magnetic vortex whose core points exactly along the hard axis and it is therefore an avenue to unwind a phase rotation. From this topological argument, it is also clear that the state with does not possess topology and it is therefore the hydrodynamic vacuum state.