Competition between magnetic frustration and spin dimensionality in the ground state of the classical antiferromagnetic $n$-vector model with arbitrary $n$

Abstract

A new method to characterize the strength of magnetic frustration is proposed by calculating the dimensionality of the absolute ground state of the classical nearest-neighbor antiferromagnetic n-vector model with arbitrary n, describing interactions between spins that reside on the vertices of frustrated molecules. Spins having more than three components may minimize the energy more effectively against frustration by allowing bigger angles between nearest-neighbors, leading to ground states in higher than three-dimensional spin space. Platonic solids in three and four dimensions, which have equivalent sites and consist of only one type of polygon, have lowest-energy configurations in a number of spin dimensions equal to their real-space dimensionality. The same is true for the Archimedean solids, which have equivalent sites but consist of more than one type of polygon. Fullerene molecules and geodesic icosahedra, whose sites are not equivalent, can produce ground states in as many as five spin dimensions. It is also found that the introduction of interactions in the next higher dimension does not necessarily lower the energy immediately, but it typically takes a finite coupling for this to happen. Frustration is also characterized by the maximum value of the ground-state energy when the exchange interactions are allowed to vary, which reveals symmetry patterns in the magnetic behavior and that symmetry and connectivity can play a more important role than molecular size in alleviating frustration.