Fig. 4
From: Order–disorder phase transitions of phosphorene and their application to adiabatic quantum computing
Mean length of domain boundaries (mean Domain wall Length (DL)) as a function of temperature for the 2D Ising model with nearest next-nearest neighbor interactions at different values of the parameter \(\alpha = J/J^{\prime }\), where \(J\) and \(J^{\prime }\) are the interaction strengths of nearest neighbors and next-nearest neighbors. The DL parameter is described by Eq. (s22). The results of figures (a) to (f) are obtained from a Monte Carlo simulation with the annealing procedure. For \(\alpha < 2\), the mean wall length tends to DL = 1 at low temperatures and for \(\alpha > 2\) it tends to DL = 0. It is because of different orders at these two regions (checker-stripy for \(\alpha < 2\), and checkerboard for \(\alpha > 2\)). In the annealed system, there is no evidence of a glasslike state, except for \(\alpha = 2\), where the crossover between different types of ordered ground states occurs. It is because that as we lower the temperature, the system restores an ordered state, in which it remains. This behavior arises from the fundamental difference between annealed and quenched Monte Carlo processes. In annealed Monte Carlo, as the temperature is gradually lowered, each simulation begins with the final spin configuration obtained at the previous (higher) temperature. As a result, once the system reaches a temperature where the previous state is already ordered, it effectively starts with an ordered configuration. At these low temperatures, the system lacks sufficient thermal energy to escape from this state, making it unlikely to explore other configurations. In \(\alpha = 2\), where the crossover between ordered phases occurs, DL is between 0 and 1 at low temperatures. The results of figures (g) to (l) are obtained from a Monte Carlo simulation without the annealing procedure (quenched system). The steep deviation from 0 at \(\alpha > 2\) is a signature of the onset of the glasslike state. It is because that at low temperatures, the system is in a configuration with a high energy barrier and there is not enough energy to escape this situation and restore the order. simulations are done on a \(32 \times 32\) lattice.
