week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017 Breaking of Galilean Invariance in the Hydrodynamic Formulation of Ferromagnetic Thin Films Ezio Iacocca,1,2,* T. J. Silva,3 and Mark A. Hoefer1 1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA 2Department of Physics, Division for Theoretical Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden 3National Institute of Standards and Technology, Boulder, Colorado 80305-3328, USA (Received 30 June 2016; published 5 January 2017) Microwave magnetodynamics in ferromagnets are often studied in the small-amplitude or weakly nonlinear regime corresponding to modulations of a well-defined magnetic state. However, strongly nonlinear regimes, where the aforementioned approximations are not applicable, have become exper- imentally accessible. By reinterpreting the governing Landau-Lifshitz 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 wave fronts 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. DOI: 10.1103/PhysRevLett.118.017203 Magnetodynamics in thin film ferromagnets has been This Letter shows that the LL equation exactly maps into studied for many decades. Advances in nanofabrication and a DH system of equations, without long-wavelength and the advent of spin transfer [1,2] and spin-orbit torques [3] low-frequency restrictions. The conservative equations are have opened a new frontier of experimentally accessible analogous to the Euler equations of a compressible, isen- nonlinear physics [4–8]. Large-amplitude excitations neg- tropic fluid. The DH equations for a planar ferromagnet ate the use of typical linear or weakly nonlinear analyses admit spin-current-carrying, spatially periodic magnetiza- [9–11], necessitating instead either micromagnetic simu- tion textures termed “uniform hydrodynamic states” lations [12] or analytical approaches suited to strongly (UHSs), providing a continuous interpolation between large nonlinear dynamics. Therefore, an interpretation of the amplitude spin superflows [14–16] and small-amplitude Landau-Lifshitz (LL) equation that includes full nonlinear- spin waves. Within the DH formulation, we prove that ity, yet is amenable to analytical study, would be insightful. planar ferromagnets break Galilean invariance and elucidate A hydrodynamic interpretation was proposed by Halperin their reference-frame-dependent dynamics by identifying and Hohenberg [13] to describe spin waves in anisotropic the linear dispersion relation for spin waves propagating on ferro- and antiferromagnets. Recently, theoretical studies of top of a UHS background. Such symmetry breaking at the thin film ferromagnets with planar anisotropy have identified level of linear spin waves is striking and fundamentally a relationship to superfluidlike hydrodynamic equations different from the nontrivial speed-dependent dynamics of [14–19] supporting large-amplitude modes beyond weakly topological textures due to their inherent nonlinearity, e.g., nonlinear spin wave and macrospin modes [10,11]. However, Walker breakdown for domain wall propagation [35] and these studies are limited to the long-wavelength, low- core reversal in magnetic vortices [36]. In this Letter, we also frequency regime where linear and weakly nonlinear show that static textures can break Galilean invariance for approaches apply. The relaxation of these approximations infinitesimal spin wave excitations that ride on a textured along with the identification of a deep connection between background. To emphasize this novel result, we suggest a magnetodynamics and fluid dynamics brings new perspec- Brillouin light scattering experimental test where broken tives on magnetism and reveals novel physical regimes. Galilean invariance manifests itself as a spin-wave Indeed, nonlinear, dispersive physics are required to describe dispersion shift in the presence of a UHS. superfluids and exotic structures such as solitons, quantized We consider the nondimensionalized LL equation (see vortices, and dispersive shock waves (DSWs) [20–23], Supplemental Material [37]) as exemplified by Bose-Einstein condensates (BECs) ∂m ¼ −m × h − αm × m × h : ð1Þ [20–22,24–34]. To obtain an analytical description of ∂t eff eff large-amplitude magnetic textures, we introduce dispersive Damping is parametrized by the Gilbert constant α, hydrodynamic (DH) equations for a thin-film ferromagnet. m¼M=Ms ¼ðmx;my;mzÞ is the magnetization vector 0031-9007=17=118(1)=017203(6) 017203-1 © 2017 American Physical Society week ending PRL 118, 017203 (2017) PHYSICAL REVIEW LETTERS 6 JANUARY 2017 h ¼ Δm− 2 normalized to the saturation magnetization, and eff limit, i.e., j∇nj ≪ 1, jnj ≪ 1, and juj ≪ 1. As we show σmzzˆ þ h0zˆ is the normalized effective field including below, the full nonlinearity and exchange dispersion ferromagnetic exchange, Δm; total anisotropy determined included in Eqs. (3a) and (3b) are required to describe by σ ¼ sgnðMs − HkÞ, where Hk is the perpendicular the existence and stability regions of magnetic hydro- magnetic anisotropy field, such that σ ¼þ1 (σ ¼ −1) dynamic states and broken Galilean invariance. represents a material with easy-plane (perpendicular mag- Insight can be gained from the homogeneous field netic) anisotropy; and a perpendicular applied field, h0zˆ. ∇h0 → 0, conservative α → 0 limit, where Eqs. (3) become This nondimensionalization of a two-dimensional (2D) thin conservation laws for n and u. Notably, the non-negative film provides a parameter-free description of materials with deviation from vacuum ð1 − n2Þ, or fluid density, is not planar or uniaxial anisotropy. We consider an idealized case conserved. A conservation law for the momentum p ¼ nu where in-plane magnetic anisotropy is negligible; i.e., its can also be obtained, symmetry-breaking contribution only perturbs the leading ∂p ∇n∇nT order solution, similar to domain wall Brownian motion [40]. ¼ ∇ ½ð1 − n2ÞuuTþ∇Pðn; jujÞ þ ∇ ∂t · · 1 − n2 Fluidlike variables are introduced using the canonical Hamiltonian cylindrical transformation [41] nΔn þ 1 j∇nj2 n2j∇nj2 − ∇ 2 þ ; ð Þ 1 − n2 ð1 − n2Þ2 5 n ¼ mz; u ¼ −∇Φ ¼ −∇½arctan ðmy=mxÞ; ð2Þ where the magnetic pressure is defined as where Φ is the azimuthal phase angle. We identify n jnj ≤ 1 u 1 2 2 ( ) as the longitudinal spin density and as the fluid Pðn; jujÞ ¼ ð1 þ n Þðσ − juj Þ − σ: ð6Þ velocity. There are two important features of Eq. (2). First, 2 the flow is irrotational because the velocity originates from Equations (3a) with α ¼ 0, and (5) are analogous to the a phase gradient, i.e., only quantized circulation, such as a time-reversed Euler equations expressing conservation of magnetic vortex [15], is possible. Second, Φ is undefined mass and momentum for a 2D, compressible, isentropic when n ¼1, corresponding to fluid vacuum. fluid with a density- and velocity-dependent pressure P. Utilizing the transformation (2) and standard vector Additionally, the one-dimensional conservative limit of calculus identities, the LL equation (1) can be exactly Eqs. (3a) and (3b) are exactly the equations describing recast as two DH equations [37] polarization waves in two-component spinor Bose gases jnj ∼ 1 ∂n ∂Φ [33,34] and, in the near vacuum ( ), long-wavelength, ¼ ∇ ½ð1 − n2Þu þ αð1 − n2Þ ; ð Þ and low-frequency regime, approximate the mean field ∂t · |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ∂t 3a |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} dynamics of a BEC [24,44]. This suggests that thin film spin current spin relaxation ferromagnets are ripe for the exploration of nonlinear ∂u Δn nj∇nj2 structures observed in BECs, e.g., “Bosenovas” [25,27] in ¼ ∇½ðσ − juj2Þn − ∇ þ ∂t |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 1 − n2 ð1 − n2Þ2 attractive (σ ¼ −1), and dark solitons [30], vortices [22], and |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} velocity flux DSWs [20] in repulsive (σ ¼þ1) BECs. Some of these dispersion structures have been observed in uniaxial (dissipative drop- 1 – −∇h þ α∇ ∇ ½ð1 − n2Þu : ð Þ lets [5 7]) and planar (vortices [8]) thin film ferromagnets. |{z}0 1 − n2 · 3b As we demonstrate, hydrodynamic states are also supported. |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} potential force Consider an ideal planar thin film ferromagnet (σ ¼þ1) viscous loss and a homogeneous field. Equations (3a) admit a static Equation (3a) is reminiscent of spin density continuity (∂Φ=∂t ¼ 0) solution with nonzero fluid velocity, u ¼ uxˆ, [42] from which we identify the spin density flux as the juj < 1, n ¼ 0, and h0 ¼ 0. These are ground states known spin current as spin-density waves (SDWs) [45] or soliton lattices [15] that minimize both exchange and anisotropy energies; i.e., J ¼ −ð1 − n2Þu: ð Þ s 4 any configuration with juj < 1 is stable when m lies purely Vacuum carries zero spin current. However, maximal spin in plane. SDWs exhibit a periodic structure that affords current is reached when n ¼ 0, identified as the saturation them topological stability