Phase transitions and critical phenomena in the two-dimensional Ising model with dipole interactions: A short-time dynamics study

Abstract

The ferromagnetic Ising model with antiferromagnetic dipole interactions is investigated by means of Monte Carlo simulations, focusing on the characterization of the phase transitions between the tetragonal liquid and stripe of width h phases. The dynamic evolution of the physical observables is analyzed within the short-time regime for 0.5≤δ≤1.3, where δ is the ratio between the short-range exchange and the long-range dipole interaction constants. The obtained results for the interval 0.5≤δ≤1.2 indicate that the phase transition line between the h=1 stripe and tetragonal liquid phases is continuous. This finding contributes to clarifying the controversy about the order of this transition. This controversy arises from the difficulties introduced in the simulations due to the presence of long-range dipole interactions, such as an important increase in the simulation times that limits the system size used, strong finite size effects, as well as to the existence of multiple metastable states at low temperatures. The study of the short-time dynamics of the model allows us to avoid these hindrances. Moreover, due to the fact that the finite-size effects do not significantly affect the power-law behavior exhibited in the observables within the short-time regime, the results could be attributed to those corresponding to the thermodynamic limit. As a consequence of this, a careful characterization of the critical behavior for the whole transition line is performed by giving the complete set of critical exponents.

Evidence of Kosterlitz-Thouless phase transitions in the Ising model with dipolar interactions

“…This controversy is a consequence of the limitations introduced by the long-range dipolar interactions on the Monte Carlo simulation results, as will be discussed in the following section. Recently, by using the short-time dynamics approach (STD), Horowitz et al [33] showed that the h 1 -TL transition line presents a continuous character, in agreement with the results obtained by Fonseca et al [32] by studying the complex partition function zeros from multicanonical algorithms. Later, Bab et al studied the h 2 -TL transition line and demonstrated that it is of first order for the 1.2585 δ 1.36 interval and continuous for the 1.37 δ 1.9 interval, with a tricritical point at δ 1.37.…”

“…With an aim to avoid these hindrances, the short-time dynamics approach has already been successfully applied to this model [33,34]. In this method, it is not necessary to have a system in equilibrium, and the simulations proceed within a short-time interval (t mic , t max ).…”

“…In this way, the interactions in the Hamiltonian of Eq. ( 1) are replaced by effective interactions that take into account not only the interactions of the original system but also the interactions with its replicas [33].…”

“…It was found that for systems with L > 100, m = 5000 was determined as enough in order to obtain results that are independent of L within the error bars. From these results, the range of validity of the STD method was obtained by searching the longest time interval where the susceptibility exhibited the best power-law behavior defined by means of the local exponent [33]. In this way, t mic 400 and t max > 5000 MCS were estimated for the studied δ and temperature values.…”

“…The energy is measured in units of g. Since the ferromagnetic order is destroyed by the frustration originated by the dipolar interactions, an interesting but complex phase diagram comes up. The T -δ phase diagram, where T is the temperature of the thermal bath, has been widely studied analytically [15,18,25,26], and by Monte Carlo simulations [27][28][29][30][31][32][33][34]. The obtained results showed that both the ground-state (T = 0) [26] and the lowtemperature regimes from δ > 0.4403 present domain structures characterized by stripes of an integer width n measured in lattice units (h n ), which run along one axis of the lattice where the model is set.…”