Order–disorder transition in a linear system of anti-parallel magnetic dipoles with long-range interactions

In the present study, we investigate the magnetic ordering of a system with a large number of nanomagnetic dipoles, approximated as point dipoles, arranged on a regular ring, and having anti-parallel (anti-ferromagnetic) ordering at zero external magnetic fields. We can impose different restrictions on the degrees of freedom of the dipoles: dipoles along the ring and dipoles perpendicular to the ring. When the dipoles are along the ring, the model is called theϕ-model (the dipoles are along the ϕ-coordinate of the cylindrical coordinate system) or head-to-tail model. When the dipoles are perpendicular to the plane of the ring the model is called the z-model (the dipoles are along the z-axis of the cylindrical coordinate system), and when the dipoles are along the radials of the ring, the model is called the r-model (the radial model). The z-and the r- models are similar to the one-dimensional Ising chain model (spins ‘up’ or ‘down’): the only difference is that in the z-and r-models the interactions are dipolar long-range ones. By a combination of numerical and analytical methods, we have found the exact ground states of the models, which are degenerated. A perfectly ordered system is possible only at zero temperature: by increasing the temperature the perfect order is destroyed. Employing the Monte Carlo simulation, we show that in a system of magnetic dipoles forming a linear chain, similar to the one-dimensional Ising chain model, there are no phase transitions in the thermodynamics limit, when N→∞, except at zero temperature. The system continuously transforms from a perfectly-ordered state at zero temperature to a perfectly-disordered state at very high temperatures. However, the disordered states of the models consist of perfectly ordered domains, where the ordering corresponds to one of the degenerated ground states. With the temperature, the average size of the domains decreases to zero (at very high temperatures), and numerically this size is equal to the correlation length. For finite-sized models the perfectly ordered state is kept to some critical temperature depending on the number of dipoles: T1≈4.8/lnN/2 for the head-to-tail model, Tc≈1.8/lnN/2 for the z- and radial models, Tc is the temperature when the first defect appears in the perfectly-ordered ground state,measured in J=μ0m24πr03 units, where m is the dipole moment of the dipoles, r0 the nearest distance between them. Introducing quasi-dipoles, in the present study we investigate the basic properties of the z-models, such as the internal energy, the heat capacity, the order parameter, the correlation length, etc. Employing Boltzmann’s statistics, we have obtained the exact solution of the z- and 1d Ising models. The zero-temperature state of the z-model is a perfectly ordered configuration of anti-parallel pairs of dipoles, and it is degenerated because inverting the directions of the dipoles does not change the energy of the configuration. When the model consists of N odd number of dipoles, in the ground state there is a natural ‘defect’ – a non-paired dipole, which degenerates additionally N-times the ground state and increases the energy per dipole of the ground state. For any large number of dipoles N∼100 this effect becomes negligible. We have obtained exact analytical expressions for the energy and the magnetic field of the z-model at zero temperature.