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, $\gamma$, between nnn and nn exchange couplings, all the equilibrium configurations of the model were calculated with great accuracy for $N\le 16$, 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 $\gamma$, 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 $\chi \left(N\right)>0$ increases monotonically with increasing $N$, approaching the value $\left(\chi =1\right)$ pertinent to the ground state in the limit $N\to \infty$. 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 $\chi \left(N\right)=0$], 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 $\chi \left(N\right)\ne 0$; 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.

Figure 1

Schematic view of the zeros of the function $F\left({\alpha }_1\right)$, defined in Eq. (13), for an open chain of $N$ spins with competing nn and nnn exchange coupling. The value of $\gamma =J_2/J_1=-0.34$ was kept fixed while $N$ was varied. The various types of solutions are labeled on the basis of their symmetry with respect to the middle of the chain (“symmetric”, “antisymmetric”, or “ugly”) and of their stability/instability (blue/red color of the label). Note that “ugly” solutions always come in pairs $\left(U_{\pm }\right)$.

Figure 2

All (stable and metastable, in blue; unstable, in red) configurations for a chain with $\gamma =-0.34$ and $N=12$ spins. Left column: relative angles ${\alpha }_i \left(i=1,...,N-1\right)$. Right column: absolute angles ${\theta }_i \left(i=1,...,N\right)$. The lines are guides to the eye.

Figure 3

All (stable and metastable, in blue; unstable, in red) configurations for a chain with $\gamma =-0.34$ and $N=13$ spins. Left column: relative angles ${\alpha }_i \left(i=1,...,N-1\right)$. Right column: absolute angles ${\theta }_i \left(i=1,...,N\right)$. The lines are guides to the eye.

Figure 4

Chiral order parameter $\chi$ of the ground-state configuration of a chain with $\gamma =-0.34$ versus the number, $N$, of spins, as calculated from Eq. (7). One has $\chi \left(N\right)>0$ for a configuration with ${\alpha }_i>0 \left(i=1,...,N-1\right)$. The line is a guide to the eye.

Figure 5

Unstable “antisymmetric” configurations, $AS$-1, of a chain with $\gamma =-0.34$ and number of spins, $N$, ranging between 7 and 11. Left column: relative angles ${\alpha }_i \left(i=1,...,N-1\right)$ of the $AS$-1 configuration (red squares) with $\chi \left(N\right)=0$ and of the two degenerate ground state configurations (gray triangles), with equal and opposite values of $\chi \left(N\right)\ne 0$. Right column: absolute angles ${\theta }_i \left(i=1,...,N\right)$ of the $AS$-1 configuration. The lines are guides to the eye.

Figure 6

The same as in Fig. 5, but now the number of spins, $N$, ranges between 12 and 16. For these values of $N$, the “antisymmetric” configuration $AS$-1 turns out to be metastable for even $N$, and unstable for odd $N$.

Figure 7

“Ugly” configurations $U_{+}$, with positive chiral order parameter $\chi >0$, of a chain with $\gamma =-0.34$ and number of spins, $N$, ranging between 12 and 16. Whenever the number of spins is increased by 1, an additional “ugly” state $U_{+}$ is found, which is unstable for even $N$ and metastable for odd $N$.

Figure 8

Phase portrait, obtained from the discrete nonlinear map Eqs. (14), for the classical planar spin chain in the case $\gamma =-0.34$. The red points $P_0=(0,0)$ and $P_{\pm }=\pm (\overline{\alpha },\overline{s})$, with $cos\overline{\alpha }=-1/\left(4\gamma \right)$ and $\overline{s}=sin2\overline{\alpha }$, denote the elliptic and the hyperbolic fixed points of the map, respectively. In the limit $N\to \infty ,P_0$ and $P_{\pm }$ correspond to the unstable collinear ferromagnetic state and to the stable helical states with constant pitch $\pm \overline{\alpha }$, respectively. Red arrows denote the direction of the map trajectories, with slopes given by Eq. (18), in the neighborhood of $P_{\pm }$.

Figure 9

Nonlinear map representation of stable “symmetric” $S$-1 (ground state, GS), metastable “antisymmetric” $AS$-1, and metastable “ugly” $U_{\pm 1}^B$ configurations of a finite chain with $\gamma =J_2/J_1=-0.34$ and $N=14$ spins. The physical relative angles are ${\alpha }_i$ with $i=1,...,13$, while ${\alpha }_{14}$ is a fictitious angle introduced to take into account open boundary conditions, Eq. (19), in the map formalism, Eq. (14).