Estudo dinâmico da criatividade do modelo XY em três dimensões

Abstract

In this work, we study the critical properties of the XY model defined in a simple cubic lattice through short-time critical dynamic relaxation. The system consists of planar rotors, located in the lattice sites. The rotors are endowed with magnetic moment and rotational inertia. There is interaction only among the first neighbors. Since the XY model does not have an intrinsic dynamic, we add to the interaction Hamiltonian a rotational kinetic energy term to allow Hamiltonian dynamics. Thus, the dynamic evolution of the system was obtained through the numerical integration of Hamilton’s equations of motion. We employ a symplectic algorithm, based on the Trotter-Suzuki decomposition of the exponential operator, for the integration of the system of coupled equations. At the initial time of the simulation, t = 0, the system is prepared in a completely ordered state and its total energy is fixed at a value in the critical region. Then it is released to evolve and the thermodynamic quantities are calculated over time. We monitor the magnetization, its second moment, and the Fourier transform of the two-point correlation function. This calculation was repeated for several values of the energy density at the critical regime. In this deterministic approach, the energy is conserved and the system configurations are distributed according to a microcanonical ensemble. On the other hand, traditionally the computational simulations of model systems are realized in the canonical ensemble, by the Monte Carlo method. Here, we compare our results obtained by deterministic relaxation with those obtained from stochastic simulations. We verified that the critical relaxation of the order parameter of the system satisfies the same scale law observed in stochastic methods. The calculation of static and dynamic critical exponents shows that the deterministic dynamics and the stochastic dynamics belong to the same universality class.