Magnetic frustration and the n-vector model

Magnetic interactions are strongly linked with strongly correlated electronic behavior and have been extensively studied during the last thirty years [1,2]. More specifically, the Ising, the XY, and the Heisenberg model describe interactions between non-mobile spins. The antiferromagnetic Heisenberg model (AHM) involves interactions in three-dimensional spin space. When considered for spins mounted on the vertices of frustrated lattices or molecules, the combination of frustration, quantum fluctuations, and low dimensionality can lead to new phases differing from conventional order and possessing a non-trivial low-energy spectrum. Frustration has also been found to lead to non-trivial magnetic response of the AHM. There are magnetization and susceptibility discontinuities in an an external field, even though there is no magnetic anisotropy [3]. Frustration is directly related to the type of polygons that form a lattice or molecule, with the simplest frustrated unit the triangle. The lowest energy per bond for a triangle decreases as the spin-space dimensionality is continuously increased from one and reaches its absolute minimum exactly at two spin-space dimensions, or in other words when continuously going from the antiferromagnetic Ising to the antiferromagnetic XY model by increasing the exchange constant in the second spin dimension. This shows that the absolute magnetic lowest energy state of the AHM on a frustrated structure can be seen as being compressed by dimensionality, and that an increase in the latter relieves frustration toward an absolute energy minimum. This effect manifests itself in the absolute lowest-energy state, but also in the magnetization response in an external field. In this proposal we aim to characterize magnetic frustration in a new way, by the lowest-energy state of the classical n-vector model [4], which includes antiferromagnetic bilinear exchange interactions between spins in arbitrary spin-space dimension n. By varying n we will look for the highest dimension nmax above which the zero-field ground state energy and the magnetization response cease to change. The strength of frustration will then be directly related to nmax. Another goal is to characterize frustration by means of correlations of nmax with characteristics of the structures, such as the polygons they are made of, their spatial symmetry, and the equivalence or not of their vertices. The study will focus on frustrated molecules such as the fullerenes and their duals. These have already been shown to possess non-trivial magnetic properties, for example magnetization and susceptibility discontinuities in an external field [3,5-10]. It is a natural question to ask if these discontinuities become less in number or even completely disappear as the spin-space dimensionality is increased, allowing the spins more space to exist and optimize the correlations with their neighbors, while simultaneously optimally lowering their Zeeman energy. The method of calculation of the classical magnetization response was used extensively in the past [3,5-6,8-16]. The spins are unit vectors, and for an n-dimensional spin space, n-1 angles are needed to define each spin. Each angle is moved opposite its gradient direction until the energy minimum is reached. The program that does the calculation is ready to use.