The numerical simulation of strongly first-order phase transitions has remained a notoriously difficult problem even for classical systems due to the exponentially suppressed (thermal) equilibration in the vicinity of such a transition. In the absence of efficient update techniques, a common approach to improve equilibration in Monte Carlo simulations is to broaden the sampled statistical ensemble beyond the bimodal distribution of the canonical ensemble. Here we show how a recently developed feedback algorithm can systematically optimize such broad-histogram ensembles and significantly speed up equilibration in comparison with other extended ensemble techniques such as flat-histogram, multicanonical or Wang-Landau sampling. As a prototypical example of a strong first-order transition we simulate the two-dimensional Potts model with up to Q=250 different states on large systems. The optimized histogram develops a distinct multipeak structure, thereby resolving entropic barriers and their associated phase transitions in the phase coexistence region such as droplet nucleation and annihilation or droplet-strip transitions for systems with periodic boundary conditions. We characterize the efficiency of the optimized histogram sampling by measuring round-trip times tau(N,Q) across the phase transition for samples of size N spins. While we find power-law scaling of tau vs. N for small Q \lesssim 50 and N \lesssim 40^2, we observe a crossover to exponential scaling for larger Q. These results demonstrate that despite the ensemble optimization broad-histogram simulations cannot fully eliminate the supercritical slowing down at strongly first-order transitions.
Using the random-cluster representation of the $q$-state Potts models we consider the pooling of data from cluster-update Monte Carlo simulations for different thermal couplings $K$ and number of states per spin $q$. Proper combination of histograms allows for the evaluation of thermal averages in a broad range of $K$ and $q$ values, including non-integer values of $q$. Due to restrictions in the sampling process proper normalization of the combined histogram data is non-trivial. We discuss the different possibilities and analyze their respective ranges of applicability.
The existence and limits of metastable droplets have been calculated using finite-system renormalization-group theory, for q-state Potts models in spatial dimension d=3. The dependence of the droplet critical sizes on magnetic field, temperature, and number of Potts states q has been calculated. The same method has also been used for the calculation of hysteresis loops across first-order phase transitions in these systems. The hysteresis loop sizes and shapes have been deduced as a function of magnetic field, temperature, and number of Potts states q. The uneven appearance of asymmetry in the …
We study the q-state Potts model on the simple cubic lattice with ferromagnetic interactions in one lattice direction, and antiferromagnetic interactions in the two other directions. As the temperature T decreases, the system undergoes a second-order phase transition that fits in the universality class of the 3D O(n) model with n=q-1. This conclusion is based on the estimated critical exponents, and histograms of the order parameter. At even smaller T we find, for q=4 and 5, a first-order transition to a phase with a different type of long-range order. This long-range order dissolves at T=0, a…
Nucleation in the two-dimensional q-state Potts model has been studied by means of Monte-Carlo simulations using the heat-bath dynamics. The initial metastable state has been prepared by magnetic quench of the ordered low-temperature phase. The magnetic field dependence of the nucleation time has been measured as the function of the magnetic field for different q and lattice sizes at T=0.5 Tc. A size-dependent crossover from the coalescence to nucleation region is observed at all q. The magnetic field dependence of the nucleation time is roughly described by the classical nucleation theory. Ou…
This study present a Monte Carlo investigations of low-temperature magnetic ordering and phase transitions in three-state Potts model on triangular lattice with various exchange interactions between nearest (J1) and next-nearest (J2) neighbors. The density of states for varying J1 and J2 are calculated. The magnetic structure of the ground state for various J1 and J2 are obtained. The critical temperature are calculated and the order of the phase transition determined. The density of states difference (DOSD) and histogram analysis method are used to investigate the order of the phase transitio…
We present a phenomenological theory describing the finite-size evaporation-condensation transition of the $q$-state Potts model in the microcanonical ensemble. Our arguments rely on the existence of an exponent $σ$, relating the surface and the volume of the condensed phase droplet. The evaporation-condensation transition temperature and energy converge to their infinite-size values with the same power, $a=(1-σ)/(2-σ)$, of the inverse of the system size. For the 2D Potts model we show, by means of efficient simulations up to $q=24$ and $1024^2$ sites, that the exponent $a$ is compatible with …
We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time τscales with the number of spins N=L^d as τ\propto N^2L^z. The critical exponent z is fo…
We describe the link between the Zipf law and statistical distributions for the Fortuin-Kasteleyn clusters in Ising as well as Potts models. From these results it is seen that Zipf's law can be a criterion of a phase transition, but it does not determine its order. We present the corresponding histograms for fixed domain configurations.
Wang-Landau sampling (WLS) of large systems requires dividing the energy range into "windows" and joining the results of simulations in each window. The resulting density of states (and associated thermodynamic functions) are shown to suffer from boundary effects in simulations of lattice polymers and the five-state Potts model. Here, we implement WLS using adaptive windows. Instead of defining fixed energy windows (or windows in the energy-magnetization plane for the Potts model), the boundary positions depend on the set of energy values on which the histogram is flat at a given stage of the …
For the first order transition of the Ising model below $T_c$, Isakov has proven that the free energy possesses an essential singularity in the applied field. Such a singularity in the control parameter, anticipated by condensation theory, is believed to be a generic feature of first order transitions, but too weak to be observable. We study these issues for the temperature driven transition of the $q$ states 2D Potts model at $q>q_c=4$. Adapting the droplet model to this case, we relate its parameters to the critical properties at $q_c$ and confront the free energy to the many informations br…
We report on Monte Carlo simulations for the two-dimensional frustrated $J_1$-$J_2$ Ising model on the square lattice. Recent analysis has shown that for the phase transition from the paramagnetic state to the antiferromagnetic collinear state different phase-transition scenarios apply depending on the value of the frustration $J_2 / J_1$. In particular a region with critical Ashkin-Teller-like behavior, i.e., a second-order phase transition with varying critical exponents, and a noncritical region with first-order indications were verified. However, the exact transition point $[J_2/J_1]_C$ be…
A generalization of the microcanonical ensemble suggests a simple strategy for the simulation of first order phase transitions. At variance with flat-histogram methods, there is no iterative parameters optimization, nor long waits for tunneling between the ordered and the disordered phases. We test the method in the standard benchmark: the Q-states Potts model (Q=10 in 2 dimensions and Q=4 in 3 dimensions), where we develop a cluster algorithm. We obtain accurate results for systems with more than one million of spins, outperforming flat-histogram methods that handle up to tens of thousands of…
We investigate in this paper the ground state and the nature of the transition from an orientational ordered phase at low temperature to the disordered state at high temperature in a molecular crystal. Our model is a Potts model which takes into account the exchange interaction $J$ between nearest-neighbor molecules and a dipolar interaction between molecular axes in three dimensions. The dipolar interaction is characterized by two parameters: its amplitude $D$ and the cutoff distance $r_c$. If the molecular axis at a lattice site has three orientations, say the $x$, $y$ or $z$ axes, then when…
Spinodal-like scaling behavior after a temperature quench across the first-order phase transition in three-dimensional $q$-state Potts models
Dipartimento di Fisica dell’Università di Roma La Sapienza and INFN, Sezione di Roma I（罗马第一大学物理系和INFN罗马第一分部）；Dipartimento di Fisica dell’Università di Pisa and INFN（比萨大学物理系和INFN）；Dipartimento di Fisica dell’Università di Pisa（比萨大学物理系）
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. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. In this paper,…
We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blöte algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N^2) problem for systems of size N. We show how this can be reduced to an O(N logN) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In pa…
The investigation of phase coexistence in systems with multi-component order parameters in finite systems is discussed, and as a generic example, Monte Carlo simulations of the two-dimensional q-state Potts model (q=30) on LxL square lattices (40<=L<=100) are presented. It is shown that the microcanonical ensemble is well-suited both to find the precise location of the first order phase transition and to obtain an accurate estimate for the interfacial free energy between coexisting ordered and disordered phases. For this purpose, a microcanonical version of the heatbath algorithm is implemente…
Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application a re-weighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multi-histograms method of Ferrenberg and Swendsen. As example of application, we reconsider the stu…
We demonstrate, by means of a convolutional neural network, that the features learned in the two-dimensional Ising model are sufficiently universal to predict the structure of symmetry-breaking phase transitions in considered systems irrespective of the universality class, order, and the presence of discrete or continuous degrees of freedom. No prior knowledge about the existence of a phase transition is required in the target system and its entire parameter space can be scanned with multiple histogram reweighting to discover one. We establish our approach in q-state Potts models and perform a…
