Maria Gloria Pini - ISC-CNR Firenze #
FINITE SIZE EFFECTS ON THE SYMMETRY OF METASTABLE CONFIGURATIONS
IN THE CLASSICAL ONE-DIMENSIONAL PLANAR SPIN MODEL
WITH COMPETING EXCHANGE INTERACTIONS #
The classical one-dimensional (1D) planar spin model with competing nearest neighbor 
(nn) and next nearest neighbor (nnn) exchange interactions (\(Jnn>0\) and \(Jnnn<0\), 
respectively) was introduced decades ago [1] to account for the observation of a 
modulated phase (a spiral or helicoid) in a class of magnetic crystals and alloys, 
including rare-earth elements and manganese compounds. In the thermodynamic limit, the 
modulated phase was proved to exist provided that \(G=Jnn/(4|Jnnn|)<1\) and the 
relative 
angle between neighboring spins is given by \(+arccos(G)\) or \(-arccos(G)\). Opposite 
signs 
correspond to equivalent helicoids with opposite sense of rotation (or chirality).

In the present work, we investigate the effect of finite size on the equilibrium 
states of such a model. We are driven by the interest for artificially created 
nanoscale magnetic structures: for example, an ultrathin film of Ho [2], made of N 
parallel ferromagnetic planes, where the vector magnetization of each atomic layer is 
confined to the film plane and is exchange coupled to the magnetization of neighboring 
layers, with opposite signs of the exchange constant depending on the layers' position 
(positive between nn layers and negative between nnn ones).

Finding the magnetization profile across the film thickness, while accounting for the 
discrete location of atomic layers, is a difficult task even in a mean field 
approximation, where the problem is reduced to a 1D one, since it requires the 
necessity to solve a system of \((N-1)\) equations, for the \((N-1)\) relative 
orientation 
angles, obtained after the minimization of the thermodynamic potential. Except for 
very small values of \(N\), finding the exact solution is quite demanding; thus, to 
obtain 
an estimate of the equilibrium configurations, most authors resorted either to 
time-consuming iterative procedures [2] or to a continuous approximation which allowed 
to obtain analytical results [3].

In this work we make use of a theoretical method [4], recently developed to find the 
noncollinear canted magnetic states of ultrathin ferromagnetic films with competing 
surface and bulk anisotropies [5], to calculate the magnetization profile in the case 
of our model with competing nn and nnn exchange interactions.  The essence of the 
method is to reduce the difficult problem of finding minima of the thermodynamic 
potential in the \((N-1)\)-dimensional space of the \((N-1)\) relative orientation 
angles, to 
the much simpler problem of finding the \((N-1)\) roots of a function in the 
one-dimensional space of the first relative orientation angle. Subsequently, the roots 
are analyzed in order to determine which of them correspond to stable, metastable or 
unstable states.

In this way, we were able to determine, in a very quick and quite accurate way, the 
equilibrium states of the model up to \(N=15\). In addition to the ground state, which 
is 
symmetric with respect to the center of the chain (or, equivalently, to the center of 
the film), we found metastable states of two kinds: either antisymmetric or without a 
definite symmetry ("ugly" states). In the ground state, the modulated configuration is 
non uniform along the finite size of the chain, but the chirality of the helicoid does 
not change. In contrast, the metastable states are characterized either by a change of 
chirality in the middle of the chain (antisymmetric state) or a change of chirality 
located away from the middle of the chain ("ugly" state). The above interpretation was 
confirmed performing a further analysis of the various modulated configurations in the 
framework of a discrete nonlinear mapping approach developed years ago [6].

The most interesting result, coming from our exact calculations, is that the 
antisymmetric states are metastable for even values of \(N\) and unstable for odd 
values 
of \(N\), while the "ugly" states are always metastable. This fact, being a 
consequence of 
discretization and finite size, can by no means be evidenced using a continuum model 
[3]. Clearly, as \(N\) grows, any difference between even and odd number of \(N\) is 
found 
to 
decrease, and for \(N\) tending to infinity it is expected to vanish.
<br/>
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[1] T. A. Kaplan, Phys. Rev. 116, 888 (1959); A. Yoshimori, J. Phys. Soc. Jpn. 14, 807 
(1959); J. Villain, J. Phys. Chem. Solids 11, 303 (1959). <br/>
[2] E. Weschke et al., Phys. Rev. Lett. 93, 157204 (2004). <br/>
[3] P. I. Melnichuk, A. N. Bogdanov, U. K. Roessler, and K.-H. Mueller, J. Magn. Magn. 
Mater. 248, 142 (2002). <br/>
[4] A. P. Popov, A. V. Anisimov, O Eriksson, and N. V. Skorodumova, Phys. Rev. B 81, 
054440 (2010). <br/>
[5] A. P. Popov, J. Magn. Magn. Mater. 324, 2736 (2012). <br/>
[6] L. Trallori, P. Politi, A. Rettori, M. G. Pini, and J. Villain, Phys. Rev. Lett. 
72, 920 (1994). <br/>
<br/><br/>
In collaboration with A.P. Popov and A. Rettori