Formation of couples of topological defects in one-dimensional magnetic dipole systems

In this study, emphasis is placed on the ordering of three-dimensional point dipoles, with continuous degrees of freedom, arranged on a regular ring at low temperatures and a zero external magnetic field, where interactions are long-range. The perfectly ordered state of the system at low temperatures is destroyed with the appearance of north–north (‘NN’-positive) or south-south (‘SS’-negative) point defects. These defects appear above a certain critical temperature T_c= ε_f/ ln (N/ 2 - 1) , where ε_f is the energy of defect formation at low temperatures, and N is the number of dipoles in the model. On increasing the temperature, the number of defects increases and as a result the system undergoes a continuous topological order–disorder transition, similar to the Kosterlits-Thoulless topological phase transitions found in 2-D systems: in one-dimensional systems, instead of vortex defects, we observe coupled, bounded NN-SS point defects. Based on the Boltzmann statistics, an exact law-temperature solution for the model is obtained which may also be extrapolated to high temperatures.