Discovery of metastable states in a finite-size classical one-dimensional planar spin chain with competing nearest- and next-nearest-neighbor exchange couplings

A theoretical method recently developed is used to find all possible equilibrium magnetic states of a finite-size classical one-dimensional planar spin chain with competing nearest-neighbor (nn) and next-nearest-neighbor (nnn) exchange interactions. The energy of a classical planar model with N spins is a function of N absolute orientational angles or equivalently, due to the absence of in-plane anisotropy, of (N-1) relative orientational angles. The lowest energy stable state (ground state) corresponds to a global minimum of the energy in the (N-1)-dimensional space, while metastable states correspond to local minima. For a given value of the ratio, ?, between nnn and nn exchange couplings, all the equilibrium configurations of the model were calculated with great accuracy for N<=16, and a stability analysis was subsequently performed. For any value of N, the ground state was found to be "symmetric" with respect to the middle of the chain in the relative angles representation. For the chosen value of ?, the ground state consists of a helix whose chirality is constant in sign along the chain (i.e., all the spins turn clockwise, or all anticlockwise), but whose pitch varies owing to finite-size effects; e.g., for positive chirality we found that the chiral order parameter ?(N)>0 increases monotonically with increasing N, approaching the value (?=1) pertinent to the ground state in the limit N->?. For finite but not too small values of N, we found metastable states characterized by one reversal of chirality, either localized just in the middle of the chain ["antisymmetric" state, with chiral order parameter ?(N)=0], or shifted away from the middle of the chain, to the right or to the left [pairs of "ugly" states, with equal and opposite values of ?(N)?0; the attribute "ugly" refers to the absence of a definite symmetry in the relative angles representation]. Concerning the stability of these states with one reversal of chirality, two main results were found. First, the "antisymmetric" state is metastable for even N and unstable for odd N. Second, an additional pair of "ugly" states is found whenever the number of spins in the chain is increased by 1; the states in each additional pair are unstable for even N and metastable for odd N. Analysis of stable and metastable configurations in the framework of a discrete nonlinear mapping approach provides further support for the above results.