2 where the effective field is ⃗ h ′ ef f = h 0 ˆ x ′ + f (cos θ )ˆ z ′ + 2 A µ 0 M s ∂ $^{2}$⃗ m ′ ∂z ′ 2 that does not depend on time (In general, the field due to the magnetic anisotropy depends on time in a rotation frame, but it is time-independent for the uniaxial wire). texture nature if the motion of its constitute local spins are synchronized. To demonstrate that this can indeed happen, let us study the motion of a HH DW in a uniax- ial wire under a CPMF ⃗ h ( t ) = h 0 (cos ωt ˆ x + sin ωt ˆ y ) with frequency ω and amplitude h 0 . The free energy density U ( M$_{z}$ ) is a functional function of M$_{z}$ , and the effective field takes a form of ⃗ h$_{ef f}$ = ⃗ h ( t ) + f ( M$_{z}$ )ˆ z + 2 A µ 0 M 2 s ∂ 2 ⃗ M ∂z 2 , where f is the anisotropy field and the last term is the ex- change field with the exchange coefficient A . The physics is that DW synchronize its motion with a CPMF so that DW plane rotates around wire axis at angular velocity ω . According to the LLG equation (Eq. (1)), the pre- cession motion around z − axis gives rise to a damping torque T$_{d}$ = α ( ⃗ m × ∂ ⃗ m ∂t ) = − αω sin θ⃗ e$_{θ}$ ( ⃗ e$_{θ}$ is unit vec- tor of θ direction) shown in Fig. 1. Also, because of the lag of the DW motion with the field ( δ is the angle between the DW plane and CPMF plane defined by ⃗ h 0 and z − axis), the field exerts a torque with θ -component T$_{p}$ = γ ( ⃗ m × ⃗ h$_{ef f}$ ) · ⃗ e$_{θ}$ = γh 0 sin δ shown in Fig. 1. Later analysis (Eq. (5) derived later) shows that T$_{p}$ al- ways overwhelms T$_{d}$ so that the HH DW propagates to the left along the wire (a collective motion of spins along θ -direction corresponds to the DW propagation along the wire). Eq. (3) does not have any explicit time-dependent term, and the original problem becomes a DW subjected to a transverse field h 0 ˆ x $^{′}$, a Slonczewski-like torque (b- term) α ⃗ m ′ × ( ⃗ m ′ × ⃗ ω ), and a field-like torque (c-term) ⃗ m ′ × ⃗ ω. Thus a DW under a CPMF behaves like the DW under a current-induced STT if one views ⃗ ω as a spin polarized current. Unlike the STT from a real spin- polarized current, c-term is much larger than b-term. Thus a CPMF is very efficient in driving DW propaga- tion along the wire. It should be noticed that the equiva- lent spin-polarized current is uniformly applied to a DW, instead of spatially dependent spin-polarized current re- lated to ∂ ⃗ m ∂z inside a DW[6, 7]. Interestingly, Eq. (3) is exactly the same as that in a recently studied system[12] of current-driven DW motion in a sandwiched long and narrow spin valve, in which the magnetization of refer- ence magnetic layer plays the role of field polarity ⃗ ω . Thus a DW of uniaxial wire under a CPMF is equivalent to the DW in composite spin valves under a current[12]. Early study[12] on Eq. (3) showed that DW propagates like a rigid body under small torques, corresponding to a perfect synchronized motion with the CPMF in the current case. It is noted that ⃗ ϝ = α γ ⃗ m ′ × ⃗ ω is a non- conservative field since ∇ ⃗ m ′ × ⃗ ϝ = − 2 α γ ⃗ ω = 0 . In order to find out how the DW propagation velocity depends on the amplitude and the frequency of the CPMF, we adopt the generalized analysis of Schryer and Walker[1]. It is required to first find the static DW solutions of Eq. (3) at ⃗ ω = 0 (the case of a constant transverse field ⃗ h 0 ). For the conventional uniaxial anisotropy U (cos θ ) = − 1 $_{2}$K cos 2 θ, a static DW centered at z ′ = Q exists with following DW profile θ = θ ( z ′ − Q ) , η = 0 , where[18] For the motion of a HH DW under the CPMF, it is convenient to use the rotation frame of d T T M y (z) x p 0 h z FIG. 1: Dashed arrow denotes the damping torque T d on the local spins (long arrows) when the DW plane moves around (indicated by the curved arrow) the wire axis during synchro- nization. Short arrow denotes torque T p arising from the lag of DW motion and the field. sin θ ( z $^{′}$) = sin θ 0 + cos 2 θ 0 cosh [ z ′ Δ cos θ 0 ] + sin θ 0 , (4)   ˆ x ′ ˆ y ′ ˆ z ′   =   cos ωt sin ωt 0 − sin ωt cos ωt 0 0 0 1     ˆ x ˆ y ˆ z   , (2) ˙ η = − ω − αγh 0 1 + α 2 sin η, (5) ˙ Q = − ω + ˙ η α (1 sin θ 0 ) Δ . (6) ∂ ⃗ m ′ ∂t = − γ ⃗ m ′ × ( ⃗ h ′ ef f + α γ ⃗ m ′ × ⃗ ω − ⃗ ω γ ) + α ⃗ m ′ × ∂ ⃗ m ′ ∂t , (3) here θ 0 = sin − $^{1}$( µ 0 M$_{s}$h 0 /K ) is the tilted polar angle of two domains due to transverse magnetic field ⃗ h 0 , and Δ = √ 2 A/K is the static DW width without external magnetic field. Assume the profile of the moving DW is the same as that of the static one, then the moving DW is given by Eq. (4) with collective coordinates[18– 20] ( Q, η ) being functions of time. Substituting Eq. (4) into Eq. (3), the equations for Q and η are where ˆ x $^{′}$, ˆ y $^{′}$, ˆ z ′ are the unit vectors of Cartesian co- ordinates in the rotation frame. ⃗ m becomes ⃗ m ′ = ( sin θ cos η, sin θ sin η, cos θ ) in the rotation frame, where η = − δ is the azimuthal angle of ⃗ m in the ro- tation frame which is related to azimuthal angle φ in the laboratory frame by η = φ − ωt . Polar angle θ is the same in the two frames. The time derivatives of ⃗ m in the two reference frames are connected to each other by ∂ ⃗ m ∂t = ∂ ⃗ m ′ ∂t − ⃗ m $^{′}$× ⃗ ω , here angular velocity vector ⃗ ω = ω ˆ z is assumed (Fig. 1). Eq. (1) in this rotation frame becomes