Strange dynamics of domain walls and periodic stripes \\ along classical antiferromagnetic chains
Raphael Blumenfeld
This paper addressses the kinetics and dynamics of a family of domain wall solutions along classical antiferromagnetic Heisenberg spin chains at low energies. The equation of motion is derived and found to have long range position- and velocity-dependent two-body forces. A 'quiescent' regime is identified where the forces between walls are all repelling. Outside this regime some of the interactions are attractive, giving rise to wall collisions whereupon the colliding walls annihilate. The momentum of the system is found to be conserved in the quiescent regime and to suffer discontinuous jumps upon annihilation. The dynamics are illustrated by an exact solution for a double wall system and a numerical solution for a many-wall system. On circular chains the equations support stable periodic stripes that can rotate as a rigid body. It is found that the stripes are more stable the faster they rotate. The periodic structure can be destabilised by perturbing the walls' angular velocity in which case there is a transition to another periodic structure, possibly via a cascade of annihilation events.
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\begin{document}
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\title{Strange dynamics of domain walls and periodic stripes \\
along classical antiferromagnetic chains}
\author{Raphael Blumenfeld}


\address{Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK} 
\address{Email: rbb11@phy.cam.ac.uk} 
\maketitle 
\date{\today} 
\maketitle 

\begin{abstract} 
This paper addressses the kinetics and dynamics of a family of  domain wall 
solutions along classical antiferromagnetic Heisenberg spin chains at low energies. 
The equation of motion is derived and found to have long range position- and 
velocity-dependent two-body forces. 
A 'quiescent' regime is identified where the forces between walls are all repelling.
Outside this regime some of the interactions are attractive, giving rise to wall 
collisions whereupon the colliding walls annihilate.
The momentum of the system is found to be conserved in the quiescent regime and to 
suffer discontinuous jumps upon annihilation.
The dynamics are illustrated by an exact solution for a double wall system and a 
numerical solution for a many-wall system. 

On circular chains the equations support stable periodic stripes that can rotate as 
a rigid body. It is found that the stripes are more stable the faster they rotate.
The periodic structure can be destabilised by perturbing the walls' angular velocity 
in which case there is a transition to another periodic structure, possibly via 
a cascade of annihilation events.
 
\end{abstract}
\pacs{75.60.Ch}
\narrowtext


Spin chains with antiferromagnetic Heisenberg exchange coupling have attracted much 
attention in recent years due to the remarkable progress in synthesis methods, 
new experimental techniques, and a variety of possible applications. 
These chains exhibit a rich behaviour, much of which is not fully understood, either 
in the quantum or the classical regimes \cite{Botti}. 
Of particular interest is the low-energy regime which is amenable to some analytical 
treatment. It has been known for some time that this regime accommodates spin waves and 
instanton solutions \cite{SpinWavesInstantons} \cite{BPE}. 
Recently, however, it has been shown to also support a family of multitwist solutions 
\cite{BB} \cite{BBcurve} that extend a single twist found earlier \cite{BBD}. 
These solutions represent an arbitrarily number of domain walls that can travel along 
the chain. The multitwists are found by starting from the discrete Hamiltonian for the 
exchange coupling between nearest neighbour spins on the antiferromagnetic chain 

\begin{equation} 
H = \sum_i J \bS_i\bullet\bS_{i+1} 
\label{eq:Oi}
\end{equation}
and distinguishing between the even, $\bS_e$, and odd, $\bS_o$, sublattices. Defining the total, 
$\bzeta \propto \bS_e + \bS_o$, and staggered, $\boeta \propto \bS_e - \bS_o$ magnetisations  
(properly normalised) and taking the continuum limit, leads in the low energy regime to 

\begin{equation} 
{{\partial\boeta}\over{\partial t}} = \boeta \times {{\partial\boeta}\over{\partial x}}
\label{eq:Oii}
\end{equation} 
where $t$ is renormalised time and $x$ is the arclength parameter along the chain.
This equation translates into two coupled equations for the polar and azimuthal 
angles, $\theta$ and $\phi$, of the unit vector $\boeta$ which, in turn, are identical to 
the so called Belavin-Polyakov equations \cite{BPE} in the context of the nonlinear 
sigma model. The resemblance is not complete because in this context the equations are 
derived via the Bogomol'nyi derivation \cite{Bogo} and consequently come complete with 
uniform boundary conditions. Not do in the context of the antiferromagnetic chain where 
the boundary conditions can be arbitrary.

Introducing $\cos\theta = {\rm tanh}\psi$, whereupon 
$\boeta = (\cos\phi,\sin\phi,{\rm sinh}\psi)/{\rm cosh}\psi$ \cite{BB}, the coupled equations 
are transformed into 

\begin{equation}
{{\partial\psi}\over{\partial x}} = {{\partial\phi}\over{\partial t}} \ \ \ ; \ \ \
{{\partial\psi}\over{\partial t}} = - {{\partial\phi}\over{\partial x}}
\label{eq:Oiv}
\end{equation}
In the absence of sources, which give rise to logarithmic terms (corresponding to n-instanton 
solutions \cite{BPE} \cite{BBcurve}), the solutions are all the analytic functions in 
the $x-t$ plane. 
The smooth behaviour of these functions, once operated on with the hyperbolic function, yields 
multitwists in the angle $\theta$ along the chain where the nodes of $\psi$ are. 
These multitwists correspond to domain walls in the staggered magnetisation $\boeta$.

The aim of this paper is to analyse the dynamics of the domain walls corresponding to these twists.
Specifically, I will derive the equation of motion of the walls, showing that they interact via long 
range terms that are position and velocity dependent. It will be argued that there is a 'quiescent' 
regime where the system of walls conserves its global momentum. 
When wall velocities become sufficiently high the system exits this regime and domain walls 
develop mutual attarction, leading to potential annihilation events and consequently to rich 
dynamics. Upon annihilation, momentum conservation is broken for the entire system but remains 
intact for the remaining subsystem.
These results will be demonstrated on a double- and a many-wall systems.
Next, I will show that closing the chain to form a ring gives rise to a family of periodic stripe 
solutions in the quiescent regime. These lattice structures are stable to small perturbations but 
large perturbation effect annihilations, which lead to transitions to other 
periodic solutions. It will be also argued that a cascade of such transitions may occur resulting in 
intriguing dynamics.

Consider an open-ended infinite chain prepared with an initial set of domain walls 
that occupy a finite region. The general analytic solution for $\psi$ can 
be represented in the form

\begin{equation}
\psi = \sum_{n=1}^N P_{N-n}(t) x^n = \prod_{n=1}^N (x - x_n(t))   
\label{eq:Ai} 
\end{equation}
Where $N$ is the total number of domain walls determined by the initial data. The coefficients 
$P_m$ are polynomials of order $m$ in $t$ whose structure has been discussed in reference \cite{BB}.
Real roots, $x_n$ correspond to physical domain walls in the field $\boeta$ where the 
angle $\theta$ goes through an abrupt change of $\pi$. 
The aim in the following is to find and analyse the dynamics of these walls.

First note that the field $\psi$ must satisfy Laplace's equation and therefore, carrying out 
the explicit derivatives, we get 

\begin{equation}
\nabla^2\psi = \sum_{n=1}^N \sum_{m\neq n}^N 
{{\dot x_n \dot x_m + 1}\over{(x - x_n)(x - x_m)}} - 
\sum_{n=1}^N {{\ddot x_n}\over{x - x_n}} = 0
\label{eq:Aiii} 
\end{equation}
where $x_n$ is the location of the $n$th wall.  
The solution for $\psi$ is valid for all $x$ and in particular when $x$ approaches the location of 
the $n$th wall, $x_n$. It follows that the residues of the poles at $x_n$ must all vanish. 
A straightforward calculation yields the equation of motion for the $n$th wall 

\begin{equation}
\ddot x_n - \sum_{m\neq n} {{1 + \dot x_n \dot x_m} \over {x_n - x_m}} = 0
\label{eq:Aiv} 
\end{equation}
This innocent-looking set of equations turns out to describe very rich dynamics and the rest of the 
paper is devoted to an analysis of its physical consequences.

There are several features to be noted about the set of equations (\ref{eq:Aiv}):
(1) The force on a wall is the sum of {\it two-body} forces between it and all the other walls 
along the chain. These forces are long ranged and depend on both position and velocities.
(2) Denoting the force on wall $n$ due to wall $m$ by $f_{nm}$ then $f_{nm} = -f_{mn}$, which 
is reminiscent of Newton's third law of action and reaction in translational invariant systems 
that this one is not.
(3) As a consequence of the above property, the system of walls conserves momentum. 
To see this define the centre of mass, $\gamma = \sum_{k=1}^N x_k/N$ and summ over $n$. 
The terms in the sum over $m$ cancel in pairs leading to $\ddot\gamma = 0$ and hence to 
$\dot\gamma = p_0 =$ constant. 
(4) Eqs (\ref{eq:Aiv}) are symmetric under time reversal $tt\to -t$. Therefore, for each  
forward solution $\psi_f(t)$ there exists a bacward one $\psi_b(t) = \psi_f(-t)$.

To illustrate the kinetics consider a double wall system. The equations of motion read 

\begin{equation}  
\ddot x_1 - {{\dot x_1 \dot x_2 + 1}\over{x_1 - x_2}} = 0
\label{eq:Avi} 
\end{equation} 
\begin{equation}
\ddot x_2 - {{\dot x_1 \dot x_2 + 1}\over{x_2 - x_1}} = 0
\label{eq:Avii} 
\end{equation} 
Defining the centre of mass velocity $\dot\gamma = (\dot x_1+\dot x_2)/2=p_0$ and their separation 
$\delta = (x_2 - x_1)/2$, one arrives at an equation for $\delta$

\begin{equation}
2\delta\ddot\delta + \dot\delta^2 - c^2 = 0
\label{eq:Aviii} 
\end{equation}
where $c = \sqrt{(1 + p_0^2)}$. This equation is symmetric not only under time reversal, but also under 
$\delta\to -\delta$. A particular solution of this equation is 

\begin{equation}
\delta_1 = \pm c t + \delta_0 
\label{eq:Aix}
\end{equation} 
where the positive (negative) sign corresponds to a solution that propagates forward (backward) in time.
To find the general solution define $\delta = c\ \eta^{2/3}$ and substitute into eq (\ref{eq:Aviii}), 
which gives

\begin{equation}
\ddot\eta - {3\over 4}\eta^{-1/3} = 0 
\label{eq:Bi}
\end{equation}
Integrating this equation we have

\begin{equation}
\dot\eta^2 - {9\over 4}\eta^{2/3} = E
\label{eq:Bia}
\end{equation}
with $E$ the integration constant. This equation can now be solved by quadratures 

\begin{equation}
t = t_0 \pm {{\chi}\over c} \left[\sqrt{{{\delta}\over{\chi}}\left(
1 + {{\delta}\over{\chi}}\right)} - {\text arcsinh}\left({{\delta}\over{\chi}}\right)^{1/2} \right] 
\label{eq:Bii} 
\end{equation}
where $\chi = 4cE/9$. Again the positive (negative) sign corresponds to forward (backward) propagation, 
which increases (decreases) monotonically with time. The case $E=0$ yields the linear solution (\ref{eq:Aix}). 
There can be three scenarios for the motion of the two walls: (a) moving apart, (b) approaching each other 
and eventually colliding, or (c) first approaching and then moving away. Which scenario takes place  
depends sensitively on initial conditions. Solutions (\ref{eq:Aix}) and (\ref{eq:Bii}) are 
real for all $E > 0$ when $\delta > 0$ and represent either the two walls moving apart forever 
(forward propagation) or moving into an eventual collision (backward propagation). In the latter case 
the time to collision is $t_{coll} = \delta_0 / c$ for the linear case and 
${{\chi}\over c} \left[\sqrt{{{\delta_0}\over{\chi}}\left(
1 + {{\delta_0}\over{\chi}}\right)} - {\text arcsinh}\sqrt{{{\delta_0}\over{\chi}}} \right]$
in the general case.
A careful analysis of $E < 0$ gives that in this case the walls first approach each other until 
$delta = \mid\chi\mid$ whereupon the solution switches to another branch and they move back out.
This happens when the approach is at sufficiently low velocities such that their interaction is 
always repelling. Once the walls have collided in the second scenario they annihilate. Whether they 
will reappear at a later time can be figured out from eq (\ref{eq:Ai}) for $\psi$.
It is worth noticing that on forward propagation the separation between the walls converges 
asymptotically to a steady increase at a rate of $\dot\delta = c$ that is independent of all 
initial conditions except the total momentum of the system $p_0$.   

Having understood the double wall case let us turn to many-wall systems.  
Consider two walls in such a system, say $n$ and $m$, whose velocities are, respectively,
$\dot x_n$ and $\dot x_m$. 
The interaction between these walls is repulsive when $\dot x_n \dot x_m > -1$ and is 
attractive otherwise. The attraction comes about when the two walls either move away from, 
or towards, each other at sufficiently high speeds. In the former case the attraction slows down 
too fast a separation, as is demonstrated in the double wall system where $\dot\delta\to c$. 
In the latter case, attraction between walls that approach each other too fast is bound to  
lead to a collision unless deflected by other walls in the system. On slow approaches walls
do not collide but rather achieve a minimal separation before departing in opposite 
directions.
The regime where all pairs satisfy $\dot x_n \dot x_m > -1$ is 'quiescent' in the sense 
that no walls can collide and annihilate. The number of walls in this regime remains constant 
and therefore so does the total momentum. Once the system 
exits the quiescent regime the conservation laws do not hold. Upon each annihilation event 
the number of walls changes and the system's momentum undergoes a  
discontinuous jump. The momentum lost in such a jump is exactly that of the 
two annihilating walls and it is the momentum of the remaining subsystem which remains 
conserved from the moment of annihilation onward.

\begin{figure}
\centerline{\psfig{file=fig1.eps,width=9.cm,height=6.5cm}}
\caption{
The dynamics of ten walls along the antiferromagnetic spin chain in the quiescent regime.
The walls initial positions are normally distributed around an arbitrary point along of the chain 
and they are assigned low velocities chosen from a Gaussian distribution of zero mean.
Note how the initial closure of the gap between some walls is reversed due to walls repulsion.}
\end{figure}

This analysis is supported by numerical studies where the above kinematic scenarios can be clearly 
observed in the dynamics of the walls. Figure 1 shows a system of ten
domain walls in the quiescent regime. The walls initial positions are randomly chosen from a Gaussian 
distribution around an arbitrary location along the chain. 
Their initial velocities are chosen from a Gaussian distribution of zero mean and a standard deviation
of 0.5 (arbitrary units). 
As expected, the walls fan out with time, demonstrating the repulsive interactions.
It can be observed that some neighbouring walls initially start approaching each other but their 
momentum is too low to effect collision and eventually they pull apart, as discussed above. 
Figure 2 shows those walls starting from the same initial positions but at initial velocities that 
are chosen from a Gaussian distribution whose standard deviation is three times as large. 
In this case some walls (eg, the third and fourth from the right) start off approaching at velocities 
that are sufficiently high to give rise to collision, upon which they collide and annihilate.
Computing the momentum of the system indeed gives that these are constant before and after annihilation 
at respective values of -0.39 and -1.72 (arbitrary units) with the change occuring abruptly  
at the moment of annihilation. 
 
\begin{figure}
\centerline{\psfig{file=fig2.eps,width=9.cm,height=6.5cm}}
\caption{
The dynamics of ten walls along the antiferromagnetic spin chain outside the quiescent regime.
The walls are initially positioned as in figure 1, but their random initial velocities are chosen from 
a Gaussian distribution that is three times as wide. 
Note the collision and annihilation of two neighbouring walls. Before (after) annihilation the total 
momentum of the system is constant at -0.39 (-1.72) with the change occuring abruptly at the 
moment of annihilation.}
\end{figure}

Let us turn now to finite chains that are closed to form rings. 
Examination of the equations of motion shows that they support periodic solutions of $N$ 
walls around the ring that are located at intervals of $a=L/N$, where $L$ is the chain length. 
These are stripe solutions that are stable in the quiescent regime. 
This can be seen by applying a perturbation to one of the walls in this state, moving it from its 
lattice position by a distance smaller than $a$ while keeping its angular velocity low enough 
so as not to exit the quiescent regime. 
The repulsive interaction with the closer neighbour dominates and pushes the wall back to its  
lattice position, as can be readily verified by linear perturbation analysis.  
This means that in the quiescent regime the stripe lattice dynamics consists only of oscillations 
around the periodic structure. The spectrum of these oscillations would be interesting to analyse 
but this is outside the thrust of this paper.

A most intriguing feature of these solutions is that rotating the entire stripe lattice rigidly 
at an arbitrary angular velocity $\omega$ is also a solution. Moreover, 
the amplitude of the repulsive forces between the walls increases with the angular velocity as 
$ 1 + (\omega L/2\pi)^2$. 
The rotating solution is again stable under a perturbation of the walls from their lattice positions. 
However, the stability of the periodic structure depends more on perturbations to their angular 
velocities then positions. Suppose we give wall $n$ an excessive angular velocity $\delta\omega$. 
If $\delta\omega$ is in the same direction as $\omega$ this wall experiences a net restoring force 
from its two neighbours. The same applies if $\delta\omega$ is in the opposite direction to 
$\omega$ but its magnitude is smaller than $\omega\left[ 1 + \left( 2\pi / L\omega \right)^2 \right]$. 
Being stable, the dynamics of the lattice in the rotating frame due to low kinetic energy excitations 
will again consist of oscillations of the walls around the lattice positions. 
Once $\delta\omega$ exceeds the critical value of $\omega\left[ 1 + \left( 2\pi / L\omega \right)^2 \right]$ 
in the $ - \omega$ direction, the net force on the $n$th wall becomes attractive and pulls it 
further in that direction. The $n$th wall then runs into the $(n-1)$th wall and they annihilate. 
Thus, too high excessive angular velocities to individual walls destabilises the lattice. 

The dynamics ensuing in the wake of such an instability are fascinating:
Upon annihilation a gap appears in the periodic structure and the remaining $N-2$ walls must shift 
to accomodate the new conditions. If this readjustment gives rise only to low excessive angular velocities 
then the system settles, without further ado, into a new structure of period $a = L/(N-2)$. 
If, however, the readjustment gives rise to high excessive velocities in the $-\omega$ direction 
then another annihilation may occur, leading to further damage of the lattice. 
Thus, a cascade of annihilation events may follow the initial instability leading to strange and 
rich dynamics. 
This author believes that such behaviour can be observed by monitoring directly the spectrum of 
the stripe lattice to vibrational excitations.
At low kinetic energies \cite{energy} one expects plasmon-like excitations due to the long range Coulomb 
interaction. But, at higher kinetic energies the spectrum should become muddled by the above dynamics.

To conclude, the dynamics of domain walls on classical antiferromagnetic Heisenberg chain have been found to 
be surprisingly rich. The equation of motion of the walls along the chain has been derived from first 
principles, showing that the walls experience only two-body forces. 
These forces consist of both a Coulomb-like term and a long range velocity dependent term. 
The interactions can be either repulsive, in a quiescent regime, or attractive at high velocities.
The system of walls has been found to conserve momentum as long as the number of walls is conserved, which 
is always the case in the quiescent regime. Outside this regime walls may collide and annihilate, upon 
which the total momentum of the system changes discontinuously to a new level that remain constant until 
another annihilation event occurs.
An explicit solution of a double wall system has been presented, giving insight into the dynamics of 
many-wall systems.
A numerical solution of the latter has been discussed both inside and outside the quiescent regime and 
annihilation events have been demonstrated with increasing wall velocities.  
This solution also demonstrated the above principle of momentum conservation between annihilations.

On circular chains the equations of motion give rise to periodic stripe solutions that are stable for 
small perturbations. These periodic structures can roitate as a rigid body with the structures 
becoming {\it more stable} with increasing angular velocity. This observation may have direct application
in rotating magnetic memory devices. 
The dynamics of the stripe lattice has been found to be very rich. In the quiescent regime it  
oscillates and at low kinetic energies the system should resemble a confined electron gas due to the 
long range Coulomb-like repulsion. Increasing the kinetc energy brings into play the nonlinearity of 
the equation of motion distorting the oscillation spectrum into potentially new and interesting 
configurations. Nevertheless, the dynamics should still be confined to oscilatory vibrational 
excitations. 
As the kinetc energies increase further the system exits the quiescent regime, interactions turn 
attractive and walls can collide and annihilate. 
Annihilations on a ring lead to rearrangements of the walls, which in turn, can either settle into a new 
periodic structure or result in a cascade of further annihilation events, the dynamics of which 
should be very interesting to study.

\begin{thebibliography}{99}
\bibitem{Botti} S. Botti, A. Rosso, R. Santachiara and F. Tedoldi, Phys. Rev. B {\bf 63}, 012409 (2001)
\bibitem{SpinWavesInstantons} See e.g. H. J. Mikeska and M. Steiner, Adv. Phys. {\bf 40}, 191 (1991)
\bibitem{BPE} A. A. Belavin and A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. {\bf 22}, 503 (1975) 
[JETP Lett. {\bf 22} (1975)]
\bibitem{BB} R. Blumenfeld and R. Balakrishnan, J. Phys. A: Math. Gen. {\bf 33}, 2459 (2000) 
\bibitem{BBcurve} R. Balakrishnan and R. Blumenfeld, J. Math. Phys. 38, 5878 (1997)
\bibitem{BBD} R. Balakrishnan, A. R. Bishop and R. Dandoloff , Phys. Rev. {\bf B 47}, 3108 (1993)
\bibitem{Bogo} E. B. Bogomol'nyi, Sov. J. Nucl. Phys. {\bf 24}, 449 (1976) 
\bibitem{energy} Although energy is not conserved in the system, the kinetic energy is nevertheless a 
useful parameter to characterise the quiescent regime

\end{thebibliography}


\end{document}

------------------------------------------------------------------------

Two dimensional and quasi-two dimensional ferromagnetic systems have a 
long history as a model to study magnetic systems. Recent years saw renewed 
interest in these systems due to advances in processing and manufacturing technology 
of magnetic thin films. 

Here I report recent progress in the understanding of magnetic domains formation 
in magnetic layers. The main thrust of this paper is the suggestion that 
the macroscopic configuration of the domain wall structure should be 
hierarchical. The theoretical results discussed here bear relation to 
several other problems where similar mathematical formalism applies. For 
example, in the 
dynamics of spin fluctuations along antiferromagnetic spin chains, in 
the study of geometric phases and in the dynamics of line 
curves in three dimensions, to name a few. The common 
denominator of all these problems is that the governing equations have 
the form of the Belavin-Polyakov equations, with the exception of line 
curves moving in three dimensions where a more generalized form of these 
equations is involved \ref\babl{R. Balakrishnan and R. Blumenfeld, 
"Transformation of general curve evolution to a modified Belavin-Polyakov 
equation", J. Math. Phys. {\bf 38}, 5878 (1997)}.
From the practical point of view, these results have applications in magnetic 
thin films such as 
Giant Magnetoresistive (GMR) materials for microactuators and 
microsensors. There also seems to be relevance to the nanotubes technology 
when these systems are augmented with 
magnetic properties. This latter application has been suspected 
following recent results that the same theoretical analysis can be made to 
apply to non-planar geometries \ref\bs{R. Blumenfeld and A. Saxena, 
unpublished}.\ref\babla{R. Blumenfeld and R. Balakrishnan, 
"Exact multi-twist solutions to the Belavin-Polyakov equation and applications 
to magnetic systems", J. Phys. A: Math. Gen. {\bf 33}, 2459 (2000)}

------------------------------------------------------------------------

\bigskip
\newsec{The system}
\bigskip

We start from a discrete spin system whose Hamiltonian is 
\eqn\Ii{H = -J\sum_{<mn>} \bS_m\cdot\bS_n \ ,}
where $J$ is a uniform scalar couplings constant, $\bS_n$ is a Heisenberg 
spin of index $n$, where the index runs over all possible spins, and the 
angular brackets denote summation over $m$ and $n$ that are nearest neighbors 
in real space. For positive values of $J$ 
the system is ferromagnetic since the lowest energy is when $\bS_m$ and 
$\bS_n$ are parallel. The ground state is when all the spins 
point in the same direction and the entire system is uniformly 
magnetized.  When $J$ is negative the system is antiferromagnetic and 
spins favor local antiparallel alignment, in which case it depends 
of the details of the lattice structure whether the state is ordered or 
frustrated. Here we focus on two dimensional ferromagnetic systems. 
Although the spins are embedded in a two dimensional plane, the 
spin vector, $\bS$, is three dimensional as it is not constrained to 
point only in the plane.

At low energies above the ground state, the spins are almost parallel 
locally and the divergence from parallelism is slow over large distances  
and involve many spins. In this regime one can approximate the discrete 
system with a continuous model whose equivalent Hamiltanaion is
\eqn\Iii{H = -J\int \mid \nabla\bS(x,y)\mid^2 dxdy \ .}

In 1975, Belavin and Polyakov \ref\bp{A.Belavin and A.M.Polyakov, 
Pis'ma Zh. Eksp. Teor. Fiz. {\bf 22} 503 (1975). [JETP Lett. {\bf 22}, 245
(1975)]}\ derived the equation of motion for this Hamiltonian, which reads 
\eqn\Iiii{\bS_t = \bS\times\nabla^2\bS \ ,}
where the subscript $t$ stands for partial differentiation with respect to 
time. Here I would like 
to concentrate on the stationary solutions to $\Iiii$ where the time 
derivative on the left hand side vanishes
\eqn\Iiv{\bS\times\nabla^2\bS = 0 \ .}
In two dimensions it can be readily shown that one can rewrite Eq. $\Iiv$ 
in the form
\eqn\Iv{\bS_y = -\bS\times\bS_x \ .}

This equation .....

\bigskip
\newsec{The structure of magnetic domains}
\bigskip

To analyze Eq. $\Iv$ we recall that the local spin vector is normalized 
and we set the normalization constant to
$\mid\bS\mid = 1$. Writing $\bS$ in term of its two angular variables
\eqn\Ivi{\bS = \left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta 
\right) }
and substituting the form $\Ivi$ into the vectorial equation $\Iv$ one 
obtains two coupled equations for $\theta$ and $\phi$:
\eqn\Ivii{\theta_y = -\sin\theta\phi_x \ \ \ ;
\ \ \ \theta_x = \sin\theta\phi_y \ .}
The coupled Eqs. $\Ivii$ are very reminiscent of the so-called 
Belavin-Polyakov equations (BPE), also known as the duality relations, 
e.g., in the context of antiferromagnetic chains and strings. 
There is, however, a marked difference: In those systems the BPE are 
derived only under strictly uniform boundary conditions on $\theta$ and 
$\phi$. This restricts the possible solutions to the well known n-instantons.
In our case the boundary conditions are not important for obtaining 
this set of equations and therefore remain arbitrary at this stage. 
As will be seen below, the lifting of this constraint allows for a new type 
of kink (i.e., domain wall) solutions that are not available in the 
aforementioned traditional systems.

To make progress let us first transform the angle variable 
$\theta$ through
\eqn\Iviii{\tanh\psi = \cos\theta \ .}
In the new variables Eqs. $\Ivii$ become:
\eqn\Iix{\psi_y = -\phi_x  \ \ \ ;
\ \ \ \psi_x = \phi_y \ , }
which one recognizes as the Cauchy-Riemman relations (CRR). The boundary 
conditions on $\theta$ transform to boundary conditions on $\psi$ as 
follows:
\eqn\Ix{\theta = \left(\uparrow, \leftarrow, \downarrow, 
\rightarrow\right) = (\pi/2, \pi, -\pi/2, 0) \Rightarrow 
\psi = \left(\infty, 0^+, -\infty, 0^-\right) \ .}
It is convenient to represent the coordinates in complex form z=x+iy. 
Defining then the complex function
\eqn\Ixi{\Phi = \psi + i\phi \ ,}
the CRR become
\eqn\Ixii{\psi_z = \-i\phi_z }
and the complex conjugate of this equation.

The solutions to the CRR are well known to be all the harmonic functions 
and can contain at most simple poles in the complex plane. The poles can 
be regarded as charges (of both signs). The harmonic functions 
can be most conveniently presented by expansions in complete sets of 
functions, e.g. the Fourier expansion, Legendre polynomials or Bessel 
functions
depending on the 
boundary conditions or the symmetry of the 
system. Assuming, for example, a rectangular system it is convenient to 
use the Fourier transform
$$\psi = \sum_{k=1}^Q a_k \cosh(kx)\cos(ky) + \dots $$
\eqn\Ixiii{\phi = \sum_{k=1}^Q a_k \sinh(kx)\sin(ky) + \dots \ ,}
 where the dots stand for all combinations of hyperbolic and 
trigonometric functions (e.g., $b_k\cosh(kx)\sin(ky)$, etc.).  
$Q$ is the highest wave number and 
therefore determines the maximal number of the nodes that the 
solution can have. This parameter is usually governed by the boundary 
conditions and the distribution of the charges.

While these solutions are well known and therefore look uninterestingly 
innocent for the variables, $\psi$ and $\phi$, we have to remember that 
the physically relevant fields are $\theta$ and $\phi$, 
the two spin angles. 
Consider then an arbitrary solution for $\psi$ which consists 
of a set of peaks and troughs in the plane. The function 
$\tanh \psi$ takes this form and sharpens the transition from every 
trough to every peak at the position where a node, $\psi=0$, occurs. 
Away from these locations $\tanh\psi\approx\pm 1$ almost uniformly. 
Thus $\theta\approx\pm\pi$ on most of the plane and the transition from 
$+\pi$ to $-\pi$ occurs sharply at $\psi=0$. It follows that the solution 
describes a configuration of domain walls, located at the contours where 
$\psi=0$. The number and the locations of the domain walls is determined by the 
boundary conditions and by the locations of charges in the system. A 
positive (negative) point charge corresponds to a point where the spin 
is fixed at an up (down) position, and this corresponds to 
$\psi=\infty(-\infty)$. In this context the 'up' direction is determined 
arbitrarily once and for all. A spin can also be fixed locally at an an 
arbitrary orientation resulting in a finite value of $\psi$. 
The number $Q$ is determined by the solution and 
gives, in turn, the largest number of domain walls that the system may 
develop.

\bigskip
\newsec{Hierarchical structure of domain boundaries}
\bigskip

Having discussed the domain wall solutions,
let us rewrite the Hamiltonian in the variables $\psi$ and 
$\phi$:
\eqn\Ixiv{H = -J\int {{\mid\nabla\psi(x,y)\mid^2 + \mid\nabla\phi(x,y)\mid^2}
\over{\cosh^2\psi(x,y)}} dxdy \ .}
This Hamiltonian resembles that of a free particle but with a field dependent 
mass $1/\cosh^2\psi$. The denominator obtains its minimum on the lines where 
$\psi=0$ ($\cosh^2\psi=1$) and it increases exponentially fast as $\psi$ 
becomes finite on either side of these lines, in which case the integrand in 
$\Ixiv$ is exponentially small. Since $\psi=0$ only along domain walls then the 
energy practically vanishes away from the lines that the domain boundaries 
outline in the plane. As expected, the energy 
is concentrated along the domain walls and the total energy in the 
system is proportional to the cumulative length of the domain boundaries, $l$. 

If the system's linear size is $L$ then the system area 
is $A = O(L^2)$ and the  
average energy density is 
$\epsilon = l/A$. 
If the spatial distribution of domain walls in the system is uniform in 
the sense that the density of domain walls is constant across the system 
on some length scale that is well below $L$ then $l$ 
also scales as $0(L^2)$ and the average energy density is a constant 
{\it independent} of 
the system size. If, however, that the density of walls is hierarchical 
or fractal. Namely, 
\eqn\Ixvi{l \sim L^D \ .}
The number $D$ is the fractal dimension of the domain boundaries and must take 
a value between 1 and 2, $1\le D \le 2$. The average 
energy density of the system is then 
\eqn\Ixvii{\epsilon \sim  L^{D-2} }
and it {\it decreases} with system size to become very small 
for macroscopic sizes. It cannot vanish altogether 
because of  boundary conditions and local 'charges' that 
prevent the energy from dropping to the ground state altogether.
Consider now what happens in realistic situations, where the system size is 
fixed at a given value. The question is then what pattern will the domains assume 
upon formation via a dynamic process. From the preceding discussion we expect 
that the domains will organize in a structure that {\it minimizes} the 
energy density. Under the assumption that any domain forming process 
acts on energy minimization principle, the above argument suggests that the 
domains will arrange in a hierarchical structure since this 
gives the lowest energy. Of course, there is always some uniform distribution 
of {\it large} domains that can give the same value for $\epsilon$. I argue 
that the formation of such a distribution is unlikely. The reason is that for 
such a distribution to form one needs an information correlation length that 
is of order of the system size. In other words, local regions in the system 
need to `know' that in far away regions the domain size is the right one. 
During the dynamic processes the system is far from equilibrium and 
magnetically disordered. 
It follows that the flow of information (e.g., in the form of spin density 
waves or any other dynamics that are the solution of Eq. $\Iiii$) is hindered 
by localization and damping. This, in turn, restricts the flow of information 
in the system and the information correlation length is much 
smaller than the system size. Thus it would be awfully difficult to achieve a 
global uniform distribution. Therefore, the system has to settle in one of 
the many (most probably even metastable) hierarchical structures. 
The exact pattern that the domain structure will assume cannot be predicted 
from the analysis presented here. Rather, it will be determined by the specific 
process that governs the formation dynamics.


\bigskip
\newsec{Conclusion}
\bigskip


To summarize, I have addressed formation and spontaneous arrangement of magnetic 
domains in two dimensional ferromagnetic Heisenberg systems. 
The functional form of these solutions was found exactly and 
their nature has been discussed. I have shown that the magnetic energy in 
the system is concentrated along the domain boundaries and is exponentially 
small away from them. I argued that this suggests that minimization of the 
energy in the system is equivalent to minimization of the domain walls 
length and went on to predict that the pattern that the domain walls will 
assume is hierarchical or fractal due to energetic and dynamic considerations. 
Such structural arrangement can be made visible by magnetic force 
microscopy, as indeed seems to have been recently observed by Wuttig and 
coworkers \ref\wuttig{M. Wuttig, These proceedings}.
There are a couple of points that need to be mentioned: 

\noindent i. There emerges some evidence \bs\ that the solutions discussed here 
persist even when there is coupling between the magnetic and elastic properties 
in the layer (as in magneto-elastic thin films). Depending on the coupling 
strengths, the occurrence of magnetic domain walls in such systems may lead 
to deformations, resulting in structural kinks that may be visible through 
traditional atomic force microscopy.

\noindent ii. It has been shown recently \bs\ that the sharp domain wall 
solutions survive even if the system is made nonplanar, namely, when the plane 
is wrapped around a cylinder, or distorted into any other topology (torus, sphere). 
This suggests that many real-world magnetic systems can be used to test these 
predictions. For example, it would be very interesting to carry out an experiment
on nanotubes. We 
conjecture that if nanotubes can be made magnetic then it is possible that such a 
domain structure, whether hierarchical or not, may be observed. 
Furthermore, if the nanotubes can be made magneto-elastic it may be possible 
to observe structural kinks on the cylinders. This, in turn, can have an enormous 
impact on magneto-mechanical control of these exotic systems, an issue that 
is currently being explored.

Finally, it should be emphasized that the solutions discussed here are different 
from the regular instanton solutions that usually occur when the 
Belavin-Polyakov Equations are involved. The reason is that in our system the 
boundary conditions are arbitrary, while in the traditional systems that follow 
the BPE the boundary conditions have to be uniform, thus restricting the possible 
solutions and eliminating the ones discussed here. 

\vfill
\eject

\noindent{\bf Acknowledgement}

This work was supported in its early stages by the Theoretical Division at 
Los Alamos National Laboratory. Many of the results obtained here were 
obtained in collaborations with R. Balakrishnan and A.B. saxena. 
Discussions with D. Sherrington and Jorge Sobehart are gratefully acknowledged.

\listrefs
\vfill
\bye

(\ref{eq:Biv})

magnetically ordered materials, 75.30.E
Antiferromagnetic materials, 75.50.Ee
Domain walls and domain structure 75.60.Ch

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