In-plane (IP) vortices

So far we have seen the isotropic case at zero field, which is the only case in which static solutions with non trivial IPand OP components have been found. Apart from making easier the introduction of topological concepts, this is not the case of our interest.

As soon as we allow for anisotropy of the easy-plane type, the simplest case being K >0 , δ= 0, in (3.13a), the equation for the OP structure get too complicate. The most important difference is that now the uniform state with minimum energy lies on the plane, and, therefore, the conditions (3.15) at spacial infinity are changed to

S^z(x, y)→0 , for |~x| → ∞ , (3.22) while the angleφ remains arbitrary. It is immediately seen that planar solutions (this is, ˙φ= 0 and ˙ψ =ψ = 0) do exist, with the non-trivial angular distribution (3.19). We refer to them as “IP vortices”. For them it is clear that Γ = 0, whileQ6= 0.

The condition (3.22) has a drastic consequence: the energy of a single IP vortex, which from (3.8) withψ = 0 is simply

E_IP = JS_c² 2

Z

dx dy(∇φ)~ ² , (3.23) scales with the logarithm of the size of the system, and becomes infinite in an infinite medium⁵. This prevents its existence as a solitary excitation, which would extend over the whole system, but not their appearance in pairs of opposite vorticity. In Fig. 3.1 we see how the 2-D fields of (3.19) look for Q = 1, called a “vortex” (left), Q = −1, called “antivortex” (center). It is enough to look at the diagonals to see the opposite arrows in opposite directions.

The pair formed by linear superposition of these two fields in the right panel shows, instead, that the field far from the vortex “dipole” tends to be uniform. Therefore, a vortex-antivortex pair has a finite energy (relative to the energy of the uniform state) and is in this sense “localized”, in contrast to single vortices, in an infinite medium.

It is interesting to note that astationary solution where the IP vortex structure (3.19) rotates as a rigid whole, can be obtained by changing to a reference frame (in the spin

5In addition, (3.23) must be regularized by some means, since the integrand is singular at the center of a vortex. This is not necessary for the OP vortices to be studied in Sec. 3.1.2.

-8 -6 -4 -2 0 2 4 6 8 linear superposition of the last two cases.

space) which rotates with a constant angular velocity, say ω, i.e.,

φ˜=φ−ω t (3.24)

While eq. (3.13a) in terms of ˜φ is modified only by the constant term ω on the l.h.s.

(recall that by now~b = 0), the eq. (3.13b) remains unchanged. Thus, adding a term ω t toφ in (3.19) results in a constant precession of the IP vortex, which still satisfies the Laplace equation for each instant.

Correspondingly, there exists an underlying O(2)-symmetry of the Hamiltonian upon rotations in the spin space which let S^z invariant. Note that the vortex (3.19) is a solution which has broken theO(2)-symmetry in the spin space, i.e., it isless symmetric than the Hamiltonian.

Rotations in the spacial plane XY, instead, let the Hamiltonian invariant only for an infinite system, or for a finite system with a border which is circular. And, of course, it is a symmetry operation only for the continuum system, not for the discrete one.

Precisely a finite, circular system is what we are interested in.

A border will break the translational invariance of the infinite system, and the as-sociated total momentum conservation⁶. The rotational invariance, instead, will be preserved, for a circular border, under proper boundary conditions.

The usual boundary conditions (BC) for the Laplace equation which are satisfied by (3.19), are the Dirichlet-BC and the Neumann-BC. The Dirichlet-BC fix the phase of the spin field at the border to some definite function, for instance, through the requirement

6A vortex in an infinite system experiences no force and cannot move [58].

where ϕ = arctan(y/x) is the polar angle in the plane, and L denotes the radius of the system. The Neumann-BC simply require a null derivative of the field in direction normal to the border,

The effect of these two types of borders can be described, in analogy with a 2-D electrostatic potential problem, by introducing an “image” vortex outside the system, whose field superpose to the vortex inside the system, to assure either of the above conditions. I would like to emphasize that an “image vortex” is a pictorial way to speak about the effect of a border on the only vortex which is present inside the system (perhaps it would not be a bad idea to call the outer one “imaginary” vortex).

The effect of the border is, precisely, to deform the field (3.19) such that the conditions (3.25) or (3.26) are fulfilled. This also produces an effective “force” and a consequent movement of the vortex. A minute of algebra shows that, for a vortex at X~ = (X, Y) with vorticity q, the necessary deformation has the form

Φ(x, y) =q arctan

is interpreted as the position of the image (outside the system), since (3.27) has the form of a superposition (the Laplace equation is linear!). Later on, after giving the form of the interaction between vortices, we will describe the “force” that the border produces over the vortex in the system, as the force exerted over it by the image vortex. We will see that the results of numerical simulations completely support this identification.

To conclude about IP vortices, let us show the result for the energy (3.23) in a circular system with a field distribution (3.27). There are a couple of tricks to calculate this integral as a line integral, which I do not reproduce here, but refer the reader to [46, 45].

In the Appendix A.4 we calculate the integral in the (mobil) frame which is fix to the coordinates of the vortex center. The energy of a single IP vortex, without the image term in (3.27), at a distance R from the center of a circular system of radius L, is (its self-energy, in units ofJS_c²)

while the total energy,with the image contribution, is (see App. A.4)

Here rv is a small cut-off parameter, to be identified in the next Section with the characteristic size of vortex excitations which have a non-trivial out-of-plane structure.

The excess energy of a single vortex

∆E =−π

as initial condition for simulations, over an initial condition including the image vortex, appears in the form of spin waves generated during the vortex movement, and the interaction between these waves and the vortex plays a role in the observed dynamics, as will be seen later. Finally, a simple inspection of (3.13a) shows that in order that an IP vortex can move, so that the position of its center translates, it must develop an OP structure [22], because otherwise ˙m = ˙φ = 0 and the IP structure would not change in time (letting aside a possible precession of all the spins with the same frequency, produced by a perpendicular field b_⊥).