# Hydrodynamic theory of coupled current and magnetization dynamics in spin-textured ferromagnets

###### Abstract

We develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics in metallic ferromagnets. The collective spin density couples to the spin current through a U(1) Berry-phase gauge field determined by the local texture and dynamics of the magnetization. We determine phenomenologically the dissipative corrections to the equation of motion for the electronic current, which consist of a dissipative spin-motive force generated by magnetization dynamics and a magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque on the magnetic texture follows from the Onsager principle. We investigate the effects of thermal fluctuations and find that electronic dynamics contribute to a nonlocal Gilbert damping tensor in the Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including magnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.

###### pacs:

72.15.Gd,72.25.-b,75.75.+a## I Introduction

The interaction of electrical currents with magnetic spin texture in conducting ferromagnets is presently a subject of active research. Topics of interest include current-driven magnetic dynamics of solitons such as domain walls and magnetic vortices,Berger (1984); Zhang and Li (2004); Tserkovnyak et al. (2006); Tatara et al. (2008) as well as the reciprocal process of voltage generation by magnetic dynamics.Berger (1986); Volovik (1987); Stern (1992); Barnes and Maekawa (2007); Saslow (2007); Tserkovnyak and Mecklenburg (2008); Tserkovnyak and Wong (2009); Yang et al. (2009) This line of research has been fueled in part by its potential for practical applications to magnetic memory and data storage devices.Wolf et al. (2001) Fundamental theoretical interest in the subject dates back at least two decades.Berger (1986); Haldane (1986); Volovik (1987) It was recognized early onVolovik (1987) that in the adiabatic limit for spin dynamics, the conduction electrons interact with the magnetic spin texture via an effective spin-dependent U(1) gauge field that is a local function of the magnetic configuration. This gauge field, on the one hand, gives rise to a Lorentz force due to “fictitious” electric and magnetic fields and, on the other hand, mediates the so-called spin-transfer torque exerted by the conduction electrons on the collective magnetization. An alternative and equivalent view is to consider this force as the result of the Berry phaseBerry (1984) accumulated by an electron as it propagates through the ferromagnet with its spin aligned with the ferromagnetic exchange field.Barnes and Maekawa (2007); Tserkovnyak and Mecklenburg (2008); nBerry () In the standard phenomenological formalism based on the Landau-Lifshitz-Gilbert (LLG) equation, the low-energy, long-wavelength magnetization dynamics are described by collective spin precession in the effective magnetic field, which is coupled to electrical currents via the spin-transfer torques. In the following, we develop a closed set of nonlinear classical equations governing current-magnetization dynamics, much like classical electrodynamics, with the LLG equation for the spin-texture “field” in lieu of the Maxwell equations for the electromagnetic field.

This electrodynamic analogy readily explains various interesting magnetoelectric phenomena observed recently in ferromagnetic metals. Adiabatic charge pumping by magnetic dynamicsMoriyama et al. (2008) can be understood as the generation of electrical currents due to the fictitious electric field.Tserkovnyak et al. (2008b) In addition, magnetic textures with nontrivial topology exhibit the so-called topological Hall effect,Bruno et al. (2004); Binz and Vishwanath (2006) in which the fictitious magnetic field causes a classical Hall effect. In contrast to the classical magnetoresistance, the flux of the fictitious magnetic field is a topological invariant of the magnetic texture.Volovik (1987)

Dissipative processes in current-magnetization dynamics are relatively poorly understood and are of central interest in our theory. Electrical resistivity due to quasi-one-dimensional (1D) domain walls and spin spirals have been calculated microscopically.Viret et al. (1996); Marrows (2005); C. Wickles and W. Belzig. () More recently, a viscous coupling between current and magnetic dynamics which determines the strength of a dissipative spin torque in the LLG equation as well the reciprocal dissipative spin electromotive force generated by magnetic dynamics, called the “ coefficient,”Zhang and Li (2004) was also calculated in microscopic approaches.Tserkovnyak et al. (2006); Kohno et al. (2006); Duine et al. (2007) Generally, such first-principles calculations are technically difficult and restricted to simple models. On the other hand, the number of different forms of the dissipative interactions in the hydrodynamic limit are in general constrained by symmetries and the fundamental principles of thermodynamics, and may readily be determined phenomenologically in a gradient expansion. Furthermore, classical thermal fluctuations may be easily incorporated in the theoretical framework of quasistationary nonequilibrium thermodynamics.

The principal goal of this paper is to develop a (semi-phenomenological) hydrodynamic description of the dissipative processes in electric flows coupled to magnetic spin texture and dynamics. In Ref. Tserkovnyak and Wong, 2009, we drew the analogy between the interaction of electric flows with quasistationary magnetization dynamics with the classical theory of magnetohydrodynamics. In our “spin magnetohydrodynamics,” the spin of the itinerant electrons, whose flows are described hydrodynamically, couples to the local magnetization direction, which constitutes the collective spin-coherent degree of freedom of the electronic fluid. In particular, the dissipative coupling between the collective spin dynamics and the itinerant electrons is loosely akin to the Landau damping, capturing certain kinematic equilibration of the relative motion between spin-texture dynamics and electronic flows. In our previous paper,Tserkovnyak and Wong (2009) we considered a special case of incompressible flows in a 1D ring to demonstrate the essential physics. In this paper, we establish a general coarse-grained hydrodynamic description of the interaction between the electric flows and textured magnetization in three dimensions, treating the itinerant electron’s degrees of freedom in a two-component fluid model (corresponding to the two spin projections of spin- electrons along the local collective magnetic order). Our phenomenology encompasses all the aforementioned magnetoelectric phenomena.

The paper is organized as follows. In Sec. II, we use a Lagrangian approach to derive the semiclassical equation of motion for itinerant electrons in the adiabatic approximation for spin dynamics. In Sec. III, we derive the basic conservation laws, including the Landau-Lifshitz equation for the magnetization, by coarse-graining the single-particle equation of motion and the Hamiltonian. In Sec. IV, we phenomenologically construct dissipative couplings, making use of the Onsager reciprocity principle, and calculate the net dissipation power. In particular, we develop an analog of the Navier-Stokes equation for the electronic fluid, focusing on texture-dependent effects, by making a systematic expansion in nonequilibrium current and magnetization consistent with symmetry requirements. In Sec. V, we include the effects of classical thermal fluctuations by adding Langevin sources to the hydrodynamic equations, and arrive at the central result of this paper: A set of coupled stochastic differential equations for the electronic density, current, and magnetization, and the associated white-noise correlators of thermal noise. In Sec. VI, we apply our results to special examples of rotating and spinning magnetic textures, calculating magnetic texture resistivity and magnetic dynamics-generated currents for a magnetic spiral and a vortex. The paper is summarized in Sec. VII and some additional technical details, including a microscopic foundation for our semiclassical theory, are presented in the appendices.

## Ii Quasiparticle action

In a ferromagnet, the magnetization is a symmetry-breaking collective dynamical variable that couples to the itinerant electrons through the exchange interaction. Before developing a general phenomenological framework, we start with a simple microscopic model with Stoner instability, which will guide us to explicitly construct some of the key magnetohydrodynamic ingredients. Within a low-temperature mean-field description of short-ranged electron-electron interactions, the electronic action is given by (see appendix A for details):

(1) |

Here, is the ferromagnetic exchange splitting, is the direction of the dynamical order parameter defined by , is the local spin density, and is the spinor electron field operator. For the short-range repulsion discussed in appendix A, and , where is the local particle number density. For electrons, the magnetization is in the opposite direction of the spin density: , where is the gyromagnetic ratio. Close to a local equilibrium, the magnetic order parameter describes a ground state consisting of two spin bands filled up to the spin-dependent Fermi surfaces, with the spin orientation defined by . We will focus on soft magnetic modes well below the Curie temperature, where only the direction of the magnetization and spin density are varied, while the fluctuations of the magnitudes are not significant. The spin density is given by and particle density by , where are the local spin-up/down particle densities along . can be essentially constant in the limit of low spin susceptibility.

Starting with a nonrelativistic many-body Hamiltonian, the action (1) is obtained in a spin-rotationally invariant form. However, this symmetry is broken by spin-orbit interactions, whose role we will take into account phenomenologically in the following. When the length scale on which varies is much greater than the ferromagnetic coherence length , where is the Fermi velocity, the relevant physics is captured by the adiabatic approximation. In this limit, we start by neglecting transitions between the spin bands, treating the electron’s spin projection on the magnetization as a good quantum number. (This approximation will be relaxed later, in the presence of microscopic spin-orbit or magnetic disorder.) We then have two effectively distinct species of particles described by a spinor wave function , which is defined by . Here, is an SU(2) matrix corresponding to the local spatial rotation that brings the -axis to point along the magnetization direction: , so that . The projected action then becomes:

(2) |

where

(3) |

is the spin-texture exchange energy (implicitly summing over the repeated spatial index ), which comes from the terms quadratic in the gauge fields that survive the projection. In the mean-field Stoner model, the ferromagnetic exchange stiffness is . To broaden our scope, we will treat it as a phenomenological constant, which, for simplicity, is determined by the mean electron density.nExchange () The spin-projected “fictitious” gauge fields are given by

(4) |

Choosing the rotation matrices to depend only on the local magnetic configuration, it follows from their definition that spin- gauge potentials have the form:

(5) |

where . We show in Appendix B the well known result (see, e.g., Ref. Bazaliy et al., 1998) that is the vector potential (in an arbitrary gauge) of a magnetic monopole in the parameter space defined by :

(6) |

where is the monopole charge (which is appropriately quantized).

By noting that the action (2) is formally identical to charged particles in electromagnetic field, we can immediately write down the following classical single-particle Lagrangian for the interaction between the spin- electrons and the collective spin texture:

(7) |

where is the spin- electron (wave-packet) velocity. To simplify our discussion, we are omitting here the spin-dependent forces due to the self-consistent fields and , which will be easily reinserted at a later stage. See Eq. (29).

The Euler-Lagrange equation of motion for derived from the single-particle Lagrangian (7), , gives

(8) |

The fictitious electromagnetic fields that determine the Lorentz force are

(9) |

They are conveniently expressed in terms of the tensor field strength

(10) |

by and . is the antisymmetric Levi-Civita tensor and we used four-vector notation, defining and . Here and henceforth the convention is to use Latin indices to denote spatial coordinates and Greek for space-time coordinates. Repeated Latin indices are, furthermore, always implicitly summed over.

## Iii Symmetries and conservation laws

### iii.1 Gauge invariance

The Lagrangian describing coupled electron transport and collective spin-texture dynamics (disregarding for simplicity the ordinary electromagnetic fields) is

(11) |

, , and here is the spin of individual particles labelled by . The resulting equations of motion satisfy certain basic conservation laws, due to spin-dependent gauge freedom, space-time homogeneity, and spin isotropicity.

First, let us establish gauge invariance due to an ambiguity in the choice of the spinor rotations . Our formulation should be invariant under arbitrary diagonal transformations and on the rotated fermionic field , corresponding to gauge transformations of the spin-projected theory:

(12) |

respectively. The change in the Lagrangian density is given by

(13) |

respectively, where and are the corresponding charge and spin gauge currents. The action is gauge invariant, up to surface terms that do not affect the equations of motion, provided that the four-divergence of the currents vanish, which is the conservation of particle number and spin density:

(14) |

(The second of these conservation laws will be relaxed later.) Here, the number and spin densities along with the associated flux densities are

(15) |

and

(16) |

where and for spins up and down. In the hydrodynamic limit, the above equations determine the average particle velocity and spin velocity , which allows us to define four-vectors and . Microscopically, the local spin-dependent currents are defined, in the presence of electromagnetic vector potential and fictitious vector potential , by

(17) |

where is the electron charge.

### iii.2 Angular and linear momenta

Our Lagrangian (11) contains the dynamics of that is coupled to the current. In this regard, we note that the time component of the fictitious gauge potential (96), , is a Wess-Zumino action that governs the spin-texture dynamics.Volovik (1987); Braun and Loss (1996); Tatara et al. (2008) The variational equation gives:

(18) |

To derive this equation, we used the spin-density continuity equation (14) and a gauge-independent identity satisfied by the fictitious potentials: their variations with respect to are given by

(19) |

where

(20) |

One recognizes that Eq. (18) is the Landau-Lifshitz (LL) equation, in which the spin density precesses about the effective field given explicitly by

(21) |

Equation (18) also includes the well-known reactive spin torque: ,Tserkovnyak et al. (2006) which is evidently the change in the local spin-density vector due to the spin angular momentum carried by the itinerant electrons. One can formally absorb this spin torque by defining an advective time derivative , with respect to the average spin drift velocity .

Equation (18) may be written in a form that explicitly expresses the conservation of angular momentum:Bazaliy et al. (1998); Lifshitz and Pitaevskii (1980)

(22) |

where the angular-momentum stress tensor is defined by

(23) |

Notice that this includes both quasiparticle and collective contributions, which stem respectively from the transport and equilibrium spin currents.

The Lorentz force equation for the electrons, Eq. (8), in turn, leads to a continuity equation for the kinetic momentum density.Volovik (1987) To see this, let us start with the microscopic perspective:

(24) |

Using the Lorentz force equation for the second term, we have:

(25) |

utilizing Eq. (18) to obtain the last line. Coarse-graining the first term of Eq. (24), in turn, we find:

(26) |

Putting Eqs. (25) and (26) together, we can finally write Eq. (24) in the form:

(27) |

where

(28) |

is the magnetization stress tensor.Volovik (1987)

A spin-dependent chemical potential governed by local density and short-ranged interactions can be trivially incorporated by redefining the stress tensor as

(29) |

In our notation, , and is a symmetric compressibility matrix in spin space, which includes the degeneracy pressure as well as self-consistent exchange and Hartree interactions. In general, Eq. (29) is valid only for sufficiently small deviations from the equilibrium density.

Using the continuity equations (14), we can combine the last term of Eq. (27) with the momentum density rate of change:

(30) |

which casts the momentum density continuity equation in the Euler equation form:

(31) |

We do not expect such advective corrections to to play an important role in electronic systems, however. This is in contrast to the advective-like time derivative in Eq. (18), which is first order in velocity field and is crucial for capturing spin-torque physics.

### iii.3 Hydrodynamic free energy

We will now turn to the Hamiltonian formulation and construct the free energy for our magnetohydrodynamic variables. This will subsequently allow us to develop a nonequilibrium thermodynamic description. The canonical momenta following from the Lagrangian (11) are

(32) |

Notice that for our translationally-invariant system, the total linear momentum

(33) |

where we have used Eq. (5) to obtain the second equality, coincides with the kinetic momentum (mass current) of the electrons. The latter, in turn, is equivalent to the linear momentum of the original problem of interacting nonrelativistic electrons, in the absence of any real or fictitious gauge fields. See appendix A. While is conserved (as discussed in the previous section and also follows now from the general principles), the canonical momenta of the electrons and the spin-texture field, Eqs. (32), are not conserved separately. As was pointed out by Volovik in Ref. Volovik, 1987, this explains anomalous properties of the linear momentum associated with the Wess-Zumino action of the spin-texture field: This momentum has neither spin-rotational nor gauge invariance. The reason is that the spin-texture dynamics define only one piece of the total momentum, which is associated with the coherent degrees of freedom. Including also the contribution associated with the incoherent (quasiparticle) background restores the proper gauge-invariant momentum, , which corresponds to the generator of the global translation in the microscopic many-body description.

Performing a Legendre transformation to Hamiltonian as a function of momenta, we find

(34) |

where is the kinetic energy of electrons and is the exchange energy of the magnetic order. As could be expected, is the familiar single-particle Hamiltonian coupled to an external vector potential. According to a Hamilton’s equation, the velocity is conjugate to the canonical momentum: . We note that explicit dependence on the spin-texture dynamics dropped out because of the special property of the gauge fields: . Furthermore, according to Eq. (19), we have , so the LL Eq. (18) can be written in terms of the Hamiltonian (34) asTserkovnyak and Wong (2009)

(35) |

So far, we have included in the spin-texture equation only the piece coupled to the itinerant electron degrees of freedom. The purely magnetic part is tedious to derive directly and we will include it in the usual LL phenomenology.Lifshitz and Pitaevskii (1980) To this end, we redefine

(36) |

by adding an additional magnetic free energy , which accounts for magnetostatic interactions, crystalline anisotropies, coupling to external fields, as well as energy associated with localized or orbitals.nEnergy () Then the total free energy (Hamiltonian) is , and we in general define the effective magnetic field as the thermodynamic conjugate of : . The LL equation then becomes

(37) |

where is the total effective spin density. To enlarge the scope of our phenomenology, we allow the possibility that . For example, in the model, an extra spin density comes from the localized -orbital electrons. Microscopically, term in the equation of motion stems from the Wess-Zumino action generically associated with the total spin density.

In the following, it may sometimes be useful to separate out the current-dependent part of the effective field, and write the purely magnetic part as , so that

(38) |

and Eq. (37) becomes:

(39) |

For completeness, let is also write the equation of motion for the spin- acceleration:

(40) |

retaining for the moment the advective correction to the time derivative on the left-hand side and reinserting the force due to the spin-dependent chemical potential, . These equations constitute the coupled reactive equations for our magneto-electric system. The Hamiltonian (free energy) in terms of the collective variables is (including the elastic compression piece)

(41) |

where is the spin-dependent momentum that is locally averaged over individual particles.

### iii.4 Conservation of energy

So far, our hydrodynamic equations are reactive, so that the energy (41) must be conserved: . The time derivative of the electronic energy is

(42) |

The change in the spin-texture energy is given, according to Eq. (39), by

(43) |

The total energy is thus evidently conserved, . When we calculate dissipation in the rest of the paper, we will omit these terms which cancel each other. The total energy flux density is evidently given by

(44) |

## Iv Dissipation

Having derived from first principles the reactive couplings in our magneto-electric system, summed up in Eqs. (39)-(41), we will proceed to include the dissipative effects phenomenologically. Let us focus on the linearized limit of small deviations from equilibrium (which may be spin textured), so that the advective correction to the time derivative in the Euler Eq. (40), which is quadratic in the velocity field, can be omitted. To eliminate the quasiparticle spin degree of freedom, let us, furthermore, treat halfmetallic ferromagnets, so that and , where is the electron’s spin.nTwo () From Eq. (40), the equation of motion for the local (averaged) canonical momentum is:nCC ()

(45) |

in a gauge where , so that .nEM () . The Lorentz force due to the applied (real) electromagnetic fields can be added in the obvious way to the right-hand side of Eq. (45). Note that since we are now interested in linearized equations close to equilibrium, in Eq. (45) can be approximated by its (homogeneous) equilibrium value.

Introducing relaxation through a phenomenological damping constant (Drude resistivity)

(46) |

where is the collision time, expressing the fictitious magnetic field in terms of the spin texture, Eq. (45) becomes:

(47) |

Adding the phenomenological Gilbert dampingGilbert (2004) to the magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert equation:

(48) |

where is the damping constant. Eqs. (47) and (48), along with the continuity equation, , are the near-equilibrium thermodynamic equations for and their respective thermodynamic conjugates . This system of equations of motion may be written formally as

(49) |

The matrix depends on the equilibrium spin texture . By the Onsager reciprocity principle, , where if the th variable is even (odd) under time reversal.

In the quasistationary description of a nonequilibrium thermodynamic system, the entropy is formally regarded as a functional of the instantaneous thermodynamic variables, and the probability of a given configuration is proportional to . If the heat conductance is high and the temperature is uniform and constant, the instantaneous rate of dissipation is given by the rate of change in the free energy, :

(50) |

where we used Eq. (47) and expressed the effective field as a function of by taking of Eq. (48):

(51) |

Notice that the fictitious magnetic field does not contribute to dissipation because it does not do work.

So far, there is no dissipative coupling between the current and the spin-texture dynamics, and the macroscopic equations obey the global time-reversal symmetry. However, we know that dissipative couplings exists due to the misalignment of the electron’s spin with the collective spin texture and spin-texture resistivity.Marrows (2005); Tserkovnyak et al. (2006) Following Ref. Tserkovnyak and Wong, 2009, we add these well-known effects phenomenologically by making an expansion in the equations of motion to linear order in the nonequilibrium quantities and . To limit the number of terms one can write down, we will only add terms that are spin-rotationally invariant and isotropic in real space (which disregards, in particular, such effects as the angular magnetoresistance and the anomalous Hall effect). To second order in the spatial gradients of , there are only three dissipative phenomenological terms with couplings , , and consistent with the above requirements, which could be added to the right-hand side of Eq. (47).nViscosity () The momentum equation becomes:

(52) |

It is known that the “ term” comes from a misalignment of the electron spin with the collective spin texture, and the associated dephasing. It is natural to expect thus that the dimensionless parameter , where is a characteristic spin-dephasing time.Tserkovnyak et al. (2006) The “ terms” evidently describe texture-dependent resistivity, which is anisotropic with respect to the gradients in the spin texture along the local current density. Such term are also naturally expected, in view of the well-known giant-magnetoresistance effect,Baibich et al. (1988) in which noncollinear magnetization results in electrical resistance. The microscopic origin of this term is due to spin-texture misalignment, which modifies electron scattering.

The total spin-texture-dependent resistivity can be put into a tensor form:

(53) |

The last term due to fictitious magnetic field gives a Hall resistivity. Note that , consistent with the Onsager theorem. We can finally write Eq. (47) as:

(54) |

As was shown in Ref. Tserkovnyak and Wong, 2009, since the Onsager relations require that within the current/spin-texture fields sector, there must be a counterpart to the term above in the magnetic equation, which is the well-known dissipative “ spin torque:”

(55) |

The total dissipation is now given by

(56) |

The second law of thermodynamics requires the total dissipation to be positive, which puts some constraints on the allowed values of the phenomenological parameters. We can easily notice, however, that the dissipation (56) is guaranteed to be positive-definite if

(57) |

which may serve as an estimate for the spin-texture resistivity due to spin dephasing. This is consistent with the microscopic findings of Ref. C. Wickles and W. Belzig., .

## V Thermal Noise

At finite temperature, thermal agitation causes fluctuations of the current and spin texture, which are correlated due to their coupling. A complete description requires that we supplement the stochastic equations of motion with the correlators for these fluctuations. It is convenient to regard these fluctuations as being due to the stochastic Langevin “forces” on the right-hand side of Eq. (49). The complete set of finite-temperature hydrodynamic equations thus becomes:

(58) |

where . The simplest (while possibly not most realistic) case corresponds to a highly compressible fluid, such that . In this limit, and the last two equations completely decouple from the first, continuity equation. In the remainder of this section, we will focus on this special case. The correlations of the stochastic fields are given by the symmetric part of the inverse matrix ,Landau and Lifshitz (1980) which is found by inverting Eq. (58) (reduced now to a system of two equations):

(59) |

Writing formally these equations as (after substituting from the first into the second equation)

(60) |

we immediately read out for the matrix elements :

(61) |

According to the fluctuation-dissipation theorem, we symmetrize to obtain the classical Langevin correlators:Landau and Lifshitz (1980)

(62) |

where and

(63) |

are, respectively, the symmetric and antisymmetric parts of the conductivity matrix . The short-ranged, -function character of the noise correlations in space stems from the assumption of high electronic compressibility. Contrast this to the results of Ref. Tserkovnyak and Wong, 2009 for incompressible hydrodynamics. A presence of long-ranged Coulombic interactions and plasma modes would also give rise to nonlocal correlations. These are absent in our treatment, which disregards ordinary electromagnetic phenomena.

Focusing on the microwave frequencies characteristic of ferromagnetic dynamics, it is most interesting to consider the regime where . This means that we can employ the drift approximation for the first of Eqs. (59):

(64) |

Substituting this in Eq. (59), we can easily find a closed stochastic equation for the spin-texture field:

(65) |

where we have defined the “spin-torque tensor”

(66) |

The antisymmetric piece of this tensor modifies the effective gyromagnetic ratio, while the more interesting symmetric piece determines the additional nonlocal Gilbert damping:

(67) |

where

(68) |

In obtaining Eq. (65) from Eqs. (59), we have separated the reactive spin torque out of the effective field: . (The remaining piece thus reflects the purely magnetic contribution to the effective field.) The total stochastic magnetic field entering Eq. (65),

(69) |

captures both the usual magnetic Brown noiseBrown (1963) and the Johnson noise spin-torque contributionForos et al. (2008) that arises due to the substitution in the reactive spin torque . Using correlators (62), it is easy to show that the total effective field fluctuations are consistent with the nonlocal effective Gilbert damping tensor (68), in accordance with the fluctuation-dissipation theorem applied directly to the purely magnetic Eq. (65).

To the leading, quadratic order in spin texture, we can replace and in Eq. (68). This additional texture-dependent nonlocal damping (along with the associated magnetic noise) is a second-order effect, physically corresponding to the backaction of the magnetization dynamics-driven current on the spin texture.Tserkovnyak and Wong (2009) It should be noted that in writing the modified LLG equation (55), we did not systematically expand it to include the most general phenomenological terms up to the second order in spin texture. We have only included extra spin-torque terms, which are required by the Onsager symmetry with Eq. (52). The second-order Gilbert damping (68) was then obtained by solving Eqs. (52) and (55) simultaneously. (Cf. Refs. Tserkovnyak and Wong, 2009; Zhang et al., 2009.) This means in particular, that this procedure does not capture second-order Gilbert damping effects whose physical origin is unrelated to the longitudinal spin-transfer torque physics studied here. One example of that is the transverse spin-pumping induced damping discussed in Refs. Hankiewicz et al., 2008.

## Vi Examples

### vi.1 Rigidly spinning texture

To illustrate the resistivity terms in the electron’s equation of motion (52), we first consider 1D textures. Take, for example, the case of a 1D spin helix along the axis, whose spatial gradient profile is given by , where is the wave vector of the spatial rotation and . See Fig. 1. It gives anisotropic resistivity in the plane, , and along the direction, :

(70) |