Universal lowest energy configurations in a classical Heisenberg model describing frustrated systems with wheel geometry

Abstract

We minimize the energy function of the classical Heisenberg model describing the frustrated wheel shape systems consisting of an odd number of spin vectors, where a single vector is located at the center and the remaining vectors occupy $n$ sites on a ring. Using the Lagrange manifold method developed recently, we find the exact geometrical configurations of the spin vectors corresponding to the global energy minima. We reveal two subsets of collinear $n/2$ spin vectors which are tilted by an angle $\ensuremath{\psi}$ in the opposite direction with respect to that fixed by the central vector. We prove that $\ensuremath{\psi}$ does not depend on $n$ and all the spin vectors are collinear, if the system is nonfrustrated or it is subject to weak frustration, otherwise the sublattices start to rotate. In this frustration region, the configurations are double degenerate and differ by chirality. Our findings confirm that the classification of spin frustration holds in the classical limit and allows us to discriminate different regions by the proper configurations. We demonstrate the correspondence between the total spin in the ground state of the quantum model and a component of the net total spin vector which can be exploited in analysis of the quantum models and physical complexes.