A semi-analytical solution of a system of dipoles placed along a ring tending to an exact solution at low temperatures

Monodomain nanomagnetic particles are widely used in the manufacturing of substances with new properties. In this study, we consider an ensemble identical nanomagnetic particles placed regularly along a ring in a zero external magnetic field, called the ring dipole model. Each magnetic moment is considered as a point spin with continuous degrees of freedom of its magnetic components; the spin–spin interactions are long-range. Based on the concept of point defects and the two-energy level model, we develop an exact analytical low-temperature solution and a phenomenological solution for higher temperature states of the system. The basic parameters of the model—the energy level of reference and the energy of a defect are obtained from numerical experiments conducted using the Monte Carlo method. The ground state of the system is destroyed with the appearance of the cylindrical radial and axial components of the spins, while the polar components of the moments are ordered to some extent. The order with respect to the polar components of the spins is kept up to some temperature which depends on the number of dipoles; above this temperature quadrupole point defects are formed which separate the system into domains of spins with the same sign of their polar components. On increasing the temperature, the system undergoes a continuous topological order–disorder transition with respect to the polar component of the magnetic moments, in a similar way to the Kosterlits–Thoulless topological phase transition.