Novel chiral metastable states in the discrete finite-size classical one-dimensional planar spin model with competing exchange interactions

The classical one-dimensional 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 (spiral or helix) in a class of magnetic crystals and alloys, including rare-earth elements and manganese compounds. In the thermodynamic limit, the helical ground state exists for ?=Jnn/(4|Jnnn|)<1 and the relative angle between neighboring spins is ±arccos(?); opposite signs correspond to helices with opposite chirality (i.e., clockwise/counterclockwise sense of rotation). In the present work, driven by the interest for artificially created nanoscale magnetic structures displaying a helical or spiral magnetic order (such as ultrathin films of rare-earth elements [2] or Fe chains deposited on the 5x1 reconstructed surface of Ir [3]), we investigate the effect of finite size and open boundary conditions on the equilibrium states of the above model. To calculate the non-collinear magnetization profile of a discrete, finite, open chain of N spins, we make use of a theoretical method recently developed [4]. 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 systematic and very accurate way, the equilibrium states of the model up to N=16. In addition to the ground state, which is symmetric with respect to the center of the chain, 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 helix 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 most interesting result, coming from our exact calculations [5], 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. As N grows the difference between even-N and odd-N configurations is found to decrease, and for N tending to infinity it is expected to vanish.[1] T. A. Kaplan, Phys. Rev. 116 (1959) 888; A. Yoshimori, J. Phys. Soc. Jpn. 14 (1959) 807; J. Villain, J. Phys. Chem. Solids 11 (1959) 303[2] E. Weschke et al., Phys. Rev. Lett. 93 (2004) 157204[3] M. Menzel et al., Phys. Rev. Lett. 108 (2012) 197204[4] A. P. Popov, A. V. Anisimov, O Eriksson, and N. V. Skorodumova, Phys. Rev. B 81 (2010) 054440[5] A. P. Popov, A. Rettori, and M. G. Pini, Phys. Rev. B 90 (2014) 134418