Magnetization of a classical XY spin trimer in a coplanar magnetic field

Advances in molecular magnetism have led to the realization of nanomagnets consisting of clusters of a small number of spins. Many times single quantum spins combine into an entity with a relatively large spin which, in turn, is coupled to similar peer entities. For this condition, one can see the spin vector operator as a classical Heisenberg spin vector of unit length. In systems where interactions favor alignment in the plane, such as in ultracold atoms in optical lattices or planar magnetic materials with anisotropic exchange coupling, clusters of spins with large-spin values can be viewed as classical Heisenberg XY spins. In this work we calculate exactly the magnetic properties of a trimer of $N=3$ classical Heisenberg XY spins in a coplanar magnetic field. Presence of a magnetic field makes this problem impossible to solve by standard transfer matrix methods used for decades. However, a different mathematical approach whose idea is the introduction of auxiliary spin variables into the starting expression of the partition function leads to interesting compact exact results. We illustrate the application of the method by calculating the total magnetic moment (spin) for the case of a classical XY spin trimer system in a coplanar magnetic field at arbitrary values of the magnetic field and arbitrary temperatures.