We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy , and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape. 64.60.Cn, 05.50.+q, 02.70.Lq Computer simulation has become an essential tool in condensed matter physics [1], particularly for the study of phase transitions and critical phenomena. The workhorse for the past half-century has been the Metropolis importance sampling algorithm, but more recently new, efficient algorithms have begun to play a role in allowing simulation to achieve the resolution which is needed to accurately locate and characterize phase transitions. For example, cluster flip algorithms, beginning with the seminal work of Swendsen and Wang [2], have been used to reduce critical slowing down near 2nd order transitions. Similarly, the multicanonical ensemble method [3] was introduced to overcome the tunneling barrier between coexisting phases at 1st order transitions, and this approach also has utility for systems with a rough energy landscape [4][5][6]. In both situations, histogram reweighting techniques [7] can be applied in the analysis to increase the amount of information that can be gleaned from simulational data, but the applicability of reweighting is severely limited in large systems by the statistical quality of the "wings" of the histogram. This latter effect is quite important in systems with competing interactions for which short range order effects might occur over very broad temperature ranges or even give rise to frustration that produces a very complicated energy landscape and limit the efficiency of other methods.In this paper, we introduce a new, general, efficient Monte Carlo algorithm that offers substantial advantages over existing approaches. Unlike conventional Monte Carlo methods that directly generate a canonical distribution at a given temperature g(E)e −E/KBT , our approach is to estimate the density of states g(E) accurately via a random walk which produces a flat histogram in energy space. The method can be further enhanced by performing multiple random walks, each for a different range of energy, either serially or in parallel fashion. The resultant pieces of the density of states can be joined together and used to produce canonical averages for the calculation of thermodynamic quantities at essentially any temperature. We will apply our algorithm to the 2-dim ten state Potts model and Ising model which have 1st-and 2nd-order phase transitions, respectively, to demonstrate the efficiency and accuracy of the method.Our algorithm is based on the observation that if we perform a random w...

We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at essentially any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but can also estimate the Gibbs free energy and entropy, quantities which are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both 1st and 2nd order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 × 200 and Ising models on lattices up to 256 × 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a 3D ±J spin glass model we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space. 05.50.+q, 64.60.Cn, 02.70.Lq

Using the Wang-Landau sampling method with a two-dimensional random walk we determine the density of states for an asymmetric Ising model with two- and three-body interactions on a triangular lattice, in the presence of an external field. With an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite-size analyses at, and away from, the critical endpoint. We observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint, and the exponents for both divergences agree well with previous theoretical predictions.

We use a two-dimensional Wang-Landau sampling algorithm to calculate the density of states for two discrete spin models and then extract their phase diagrams. The first system is an asymmetric Ising model on a triangular lattice with two-and three-body interactions in an external field. An accurate density of states allows us to locate the critical endpoint accurately in a two-dimensional parameter space. We observe a divergence of the spectator phase boundary and of the temperature derivative of the magnetization coexistence diameter at the critical endpoint in quantitative agreement with theoretical predictions. The second model is a Q-state Potts model in an external field H. We map the phase diagram of this model for Q ≥ 8 and observe a first-order phase transition line that starts at the H = 0 phase transition point and ends at a critical point (T c , H c ), which must be located in a two-dimensional parameter space. The critical field H c (Q) is positive and increases with Q, in qualitative agreement with previous theoretical predictions.

We use the Wang-Landau algorithm to calculate a density of states for an asymmetric Ising model on a triangular lattice with two-and three-body interactions in an external field. An accurate density of states allows us to determine the phase diagram and to study the critical behavior of this model at and near the critical endpoint. We observe a divergence of the curvature of the spectator phase boundary at the critical endpoint in accordance with theoretical predictions.