Recently, the present authors proposed the nonequilibrium-to-equilibrium scaling (NE-ES) scheme for critical Monte Carlo relaxation process, which scales relaxation data in the whole simulation-time regions regardless of functional forms, namely both for the stretched-exponential critical relaxation in cluster algorithms and for the power-law critical relaxation in local-update algorithms. In the present study, we generalize this scheme to off-critical relaxation process, and scale relaxation data for various temperatures in the whole simulation-time regions. This is the first proposal of the off-critical scaling in cluster algorithms, which cannot be described by the dynamical finite-size scaling theory based on the power-law critical relaxation. As an example, we investigate the three-dimensional Heisenberg model previously analyzed with the NE-ES [Y. Nonomura and Y. Tomita, Phys. Rev. E 93, 012101 (2016)] in the Swendsen-Wang and Metropolis algorithms.
We study the zero-temperature Ising chain evolving according to the Swendsen-Wang dynamics. We determine analytically the domain length distribution and various ``historical'' characteristics, e.g., the density of unreacted domains is shown to scale with the average domain length as <l>^{-d} with d=3/2 (for the q-state Potts model, d=1+1/q). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics.
With a nonequilibrium relaxation method, we calculate the dynamic critical exponent z of the two-dimensional Ising model for the Swendsen-Wang and Wolff algorithms. We examine dynamic relaxation processes following a quench from a disordered or an ordered initial state to the critical temperature T_c, and measure the exponential relaxation time of the system energy. For the Swendsen-Wang algorithm with an ordered or a disordered initial state, and for the Wolff algorithm with an ordered initial state, the exponential relaxation time fits well to a logarithmic size dependence up to a lattice si…
Damage spreading for 2D Ising cluster dynamics is investigated numerically by using random numbers in a way that conforms with the notion of submitting the two evolving replicas to the same thermal noise. Two damage spreading transitions are found; damage does not spread either at low or high temperatures. We determine the critical exponents at the high-temperature transition point, which seem consistent with directed percolation.
We present the multiple GPU computing with the common unified device architecture (CUDA) for the Swendsen-Wang multi-cluster algorithm of two-dimensional (2D) q-state Potts model. Extending our algorithm for single GPU computing [Comp. Phys. Comm. 183 (2012) 1155], we realize the GPU computation of the Swendsen-Wang multi-cluster algorithm for multiple GPUs. We implement our code on the large-scale open science supercomputer TSUBAME 2.0, and test the performance and the scalability of the simulation of the 2D Potts model. The performance on Tesla M2050 using 256 GPUs is obtained as 37.3 spin f…
We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge α/νis close to but probably not sharp in d=2, and is far from sharp in d=3, for all q. The conjecture z \ge β/νis false (for some values of q) in both d=2 and d=3.
We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at zero temperature. We prove that this algorithm is not ergodic for symmetric subsets of the kagome lattice with fully periodic boundary conditions: given an initial configuration, not all configurations are accessible via Monte Carlo steps. The same conclusion holds for single-site dynamics.
We apply numerical simulations to study the criticality of the 3D Ising model with random site quenched dilution. The emphasis is given to the issues not being discussed in detail before. In particular, we attempt a comparison of different Monte Carlo techniques, discussing regions of their applicability and advantages/disadvantages depending on the aim of a particular simulation set. Moreover, besides evaluation of the critical indices we estimate the universal ratio $Γ^+/Γ^-$ for the magnetic susceptibility critical amplitudes. Our estimate $Γ^+/Γ^- = 1.67 \pm 0.15$ is in a good agreement wi…
We present sample CUDA programs for the GPU computing of the Swendsen-Wang multi-cluster spin flip algorithm. We deal with the classical spin models; the Ising model, the $q$-state Potts model, and the classical XY model. As for the lattice, both the 2D (square) lattice and the 3D (simple cubic) lattice are treated. We already reported the idea of the GPU implementation for 2D models [Comput. Phys. Commun. 183 (2012) 1155-1161]. We here explain the details of sample programs, and discuss the performance of the present GPU implementation for the 3D Ising and XY models. We also show the calculat…
We show that linear congruential pseudo-random-number generators can cause systematic errors in Monte Carlo simulations using the Swendsen-Wang algorithm, if the lattice size is a multiple of a very large power of 2 and one random number is used per bond. These systematic errors arise from correlations within a single bond-update half-sweep. The errors can be eliminated (or at least radically reduced) by updating the bonds in a random order or in an aperiodic manner. It also helps to use a generator of large modulus (e.g. 60 or more bits).
On directed Barabasi-Albert networks with two and seven neighbours selected by each added site, the Ising model does not seem to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation follows an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decays exponentially with time. On these networks the magnetisation behaviour of the Ising model, with Glauber, HeatBath, Metropolis, Wolf or Swendsen-Wang algorithm competing against Kawasaki dynamics, is studied by Monte Carlo simulations. We show that the model e…
We present the GPU calculation with the common unified device architecture (CUDA) for the Swendsen-Wang multi-cluster algorithm of two-dimensional classical spin systems. We adjust the two connected component labeling algorithms recently proposed with CUDA for the assignment of the cluster in the Swendsen-Wang algorithm. Starting with the q-state Potts model, we extend our implementation to the system of vector spins, the q-state clock model, with the idea of embedded cluster. We test the performance, and the calculation time on GTX580 is obtained as 2.51 nano sec per a spin flip for the q=2 P…
Recently Y. N. showed that the nonequilibrium critical relaxation of the 2D Ising model from the perfectly-ordered state in the Wolff algorithm is described by the stretched-exponential decay, and found a universal scaling scheme to connect nonequilibrium and equilibrium behaviors. In the present study we extend these findings to vector spin models, and the 3D Heisenberg model could be a typical example. In order to evaluate the critical temperature and critical exponents precisely with the above scaling scheme, we calculate the nonequilibrium ordering from the perfectly-disordered state in th…
We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.
We prove the ergodicity of the Wang--Swendsen--Kotecký (WSK) algorithm for the zero-temperature $q$-state Potts antiferromagnet on several classes of lattices on the torus. In particular, the WSK algorithm is ergodic for $q\ge 4$ on any quadrangulation of the torus of girth $\ge 4$. It is also ergodic for $q \ge 5$ (resp. $q \ge 3$) on any Eulerian triangulation of the torus such that one sublattice consists of degree-4 vertices while the other two sublattices induce a quadrangulation of girth $\ge 4$ (resp.~a bipartite quadrangulation) of the torus. These classes include many lattices of inte…
Recently we showed that the critical nonequilibrium relaxation in the Swendsen-Wang algorithm is widely described by the stretched-exponential relaxation of physical quantities in the Ising or Heisenberg models. Here we make a similar analysis in the Berezinsky-Kosterlitz-Thouless phase transition in the two-dimensional (2D) XY model and in the first-order phase transition in the 2D $q=5$ Potts model, and find that these phase transitions are described by the simple exponential relaxation and power-law relaxation of physical quantities, respectively. We compare the relaxation behaviors of thes…
We study the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case $q=2$, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) $Θ(1)$ for $β<β_c$, (ii) $Θ(n^{1/4})$ for $β=β…
We have performed a high-precision Monte Carlo study of the dynamic critical behavior of the Swendsen-Wang algorithm for the three-dimensional Ising model at the critical point. For the dynamic critical exponents associated to the integrated autocorrelation times of the "energy-like" observables, we find z_{int,N} = z_{int,E} = z_{int,E'} = 0.459 +- 0.005 +- 0.025, where the first error bar represents statistical error (68% confidence interval) and the second error bar represents possible systematic error due to corrections to scaling (68% subjective confidence interval). For the "susceptibili…
Metropolis algorithm has been extensively employed for simulating a canonical ensemble and estimating macroscopic properties of a closed system at any desired temperature. A mechanical property, like energy can be calculated by averaging over a large number of micro states of the stationary Markov chain generated by the Metropolis algorithm. However thermal properties like entropy, and free energies are not easily accessible. A method called umbrella sampling was proposed some forty years ago for this purpose. Ever since, umbrella sampling has undergone several metamorphoses and we have now mu…
We investigate the mixed-spin Blume-Capel model with spin-1/2 and spin-$S$ ($S=1$, $2$, and $3$) on the simple cubic and body-centered cubic lattices with single-ion-splitting crystal-field ($Δ$) by using the Metropolis and the Wang-Landau Monte Carlo methods. By numerical simulations, we prove that the tricritical point is spin-independent for both lattices. The positions of the tricritical point in the phase diagram are determined as ($Δ_t/J=2.978(1)$; $k_B T_t/J=0.439(1)$) and ($Δ_t/J=3.949(1)$; $k_B T_t/J=0.854(1)$) for the simple cubic and the body-centered cubic lattices, respectively. A…
Algorithmic overlaps as thermodynamic variables: from local to cluster Monte Carlo dynamics in critical phenomena
HSE University（高等经济大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）；Osipyan Institute for Solid State Physics（奥西皮扬固体物理研究所）
We study a two-dimensional kinetic Ising model with Swendsen-Wang dynamics, replacing the usual percolation on top of Ising clusters by explosive percolation. The model exhibits a reversible first-order phase transition with hysteresis. Surprisingly, at the transition flanks the global bond density seems to be equal to the percolation thresholds.
Dominant energy subspaces of statistical systems are defined with the help of restrictive conditions on various characteristics of the energy distribution, such as the probability density and the forth order Binder's cumulant. Our analysis generalizes the ideas of the critical minimum energy-subspace (CrMES) technique \cite{malakis04}, applied previously to study the specific heat's finite-size scaling. Here, we illustrate alternatives that are useful for the analysis of further finite-size anomalies and the behavior of the corresponding dominant subspaces is presented for the 2D Baxter-Wu, th…
We study the O(N) loop model on the Honeycomb lattice with real value $N \geq 1$ by means of a cluster algorithm. The formulation of the algorithm is based on the equivalence of the O(N) loop model and the low-temperature graphical representation of a $N$-color Ashkin-Teller model on the triangular lattice. The latter model with integer $N$ can be simulated by means of an embedding Swendsen-Wang-type cluster method. By taking into account the symmetry among loops of different colors, we develop another version of the Swendsen-Wang-type method. This version allows the number of colors $N$ to ta…
We study the dynamic critical behavior of the worm algorithm for the two- and three-dimensional Ising models, by Monte Carlo simulation. The autocorrelation functions exhibit an unusual three-time-scale behavior. As a practical matter, the worm algorithm is slightly more efficient than Swendsen-Wang for simulating the two-point function of the three-dimensional Ising model.
We present a method for estimating the density of states of a classical statistical model. The algorithm successfully combines the Wang-Landau flat histogram method with the N-fold way in order to improve efficiency of the original single spin flip version. We test our implementation of the Wang-Landau method with the two-dimensional nearest neighbor Ising model for which we determine the tunneling time and the density of states. Furthermore, we show that our new algorithm performs correctly at right edges of an energy interval over which the density of states is computed. This removes a disad…
We study the dynamic critical behavior of two algorithms: the Swendsen-Wang algorithm for the two-dimensional Potts model with q=2,3,4 and a Swendsen-Wang-type algorithm for the two-dimensional symmetric Ashkin-Teller model on the self-dual curve. We find that the Li--Sokal bound on the autocorrelation time τ_{{\rm int},{\cal E}} \geq const \times C_H is almost, but not quite sharp. The ratio τ_{{\rm int},{\cal E}}/C_H appears to tend to infinity either as a logarithm or as a small power (0.05 \ltapprox p \ltapprox 0.12). We also show that the exponential autocorrelation time τ_{{\rm exp},{\ca…
We introduce a two-temperature Ising model as a prototype of superstatistic critical phenomena. The model is described by two temperatures ($T_1,T_2$) in zero magnetic field. To predict the phase diagram and numerically estimate the exponents, we develop Metropolis and Swendsen-Wang Monte Carlo method. We observe that there is a non-trivial critical line, separating ordered and disordered phases. We propose an analytic equation for the critical line in the phase diagram. Our numerical estimation of the critical exponents illustrates that all points on the critical line belong to the ordinary I…
Monte Carlo simulations using Wang-Landau sampling are performed to study three-dimensional chains of homopolymers on a lattice. We confirm the accuracy of the method by calculating the thermodynamic properties of this system. Our results are in good agreement with those obtained using Metropolis importance sampling. This algorithm enables one to accurately simulate the usually hardly accessible low-temperature regions since it determines the density of states in a single simulation.
We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L=65536, we investigate the Kosterlitz-Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β_{KT} is 1.11996(6). We estimate the exponent to specify the multiplicative lo…
We investigate the critical behavior of the two-dimensional 8-state Potts model with an aperiodic distribution of the exchange interactions between nearest-neighbor rows. The model is studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm. The transition point is located through duality relations, and the critical behavior is investigated using FSS techniques at criticality. For strong enough fluctuations of the aperiodic sequence under consideration, a second order phase transition is found. The exponents $β/ν$ and $γ/ν$ are obtained at the new…
Recently, Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms based on canonical ensemble, which is aimed to overcome slow sampling problems associated with temperature-driven discontinuous phase transitions. We show in this work that Monte Carlo algorithms extended with this methodology also exhibit a remarkable efficiency near a critical point. Our study is performed for the particular case of 2D four-state Potts model on the square lattice with periodic boundary conditions. This analysis reveals that the extended version of Metropolis importance sample is more…
We propose a new effective cluster algorithm of tuning the critical point automatically, which is an extended version of Swendsen-Wang algorithm. We change the probability of connecting spins of the same type, $p = 1 - e^{- J/ k_BT}$, in the process of the Monte Carlo spin update. Since we approach the canonical ensemble asymptotically, we can use the finite-size scaling analysis for physical quantities near the critical point. Simulating the two-dimensional Potts models to demonstrate the validity of the algorithm, we have obtained the critical temperatures and critical exponents which are co…
We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation prepresentation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim (cond-mat/0201285), the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dyna…
In this work, we discuss the implications of a recently obtained equilibrium fluctuation-dissipation relation on the extension of the available Monte Carlo methods based on the consideration of the Gibbs canonical ensemble to account for the existence of an anomalous regime with negative heat capacities $C<0$. The resulting framework appears as a suitable generalization of the methodology associated with the so-called \textit{dynamical ensemble}, which is applied to the extension of two well-known Monte Carlo methods: the Metropolis importance sample and the Swendsen-Wang clusters algorithm. T…
We provide a new criterion based on graph duality to predict whether the 3-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge $c=1$. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the 3-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with ver…
We implement the Wang-Landau algorithm in the context of SU(N) lattice gauge theories. We study the quenched, reduced version of the lattice theory and calculate its density of states for N=20,30,40,50. We introduce a variant of the original algorithm in which the weight function used in the update does not asymptote to a fixed function, but rather continues to have small fluctuations which enhance tunneling. We formulate a method to evaluate the errors in the density of states, and use the result to calculate the dependence of the average action density and the specific heat on the `t Hooft c…
We present a dual geometrical worm algorithm for two-dimensional Ising models. The existence of such dual algorithms was first pointed out by Prokof'ev and Svistunov \cite{ProkofevClassical}. The algorithm is defined on the dual lattice and is formulated in terms of bond-variables and can therefore be generalized to other two-dimensional models that can be formulated in terms of bond-variables. We also discuss two related algorithms formulated on the direct lattice, applicable in any dimension. These latter algorithms turn out to be less efficient but of considerable intrinsic interest. We sho…
The thermodynamic properties of systems with long-range interactions is still an ongoing challenge, both from the point of view of theory as well as computer simulation. In this work we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states (CDOS), the thermodynamic entropy and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dyna…
Monte Carlo algorithms such as the Wang-Landau algorithm and similar `entropic' methods are able to accurately sample the density of states of model systems and thereby give access to thermal equilibrium properties at any temperature. Thermal equilibrium is however not achievable at low temperatures in glassy systems. Such systems are characterized by a multitude of metastable configurations, pictorially referred to as `valleys' of an energy landscape. Geometrical properties of the landscape, e.g. the local density of states describing the distribution in energy of the states belonging to a si…
The self-dual random-bond eight-state Potts model is studied numerically through large-scale Monte Carlo simulations using the Swendsen-Wang cluster flipping algorithm. We compute bulk and surface order parameters and susceptibilities and deduce the corresponding critical exponents at the random fixed point using standard finite-size scaling techniques. The scaling laws are suitably satisfied. We find that a belonging of the model to the 2D Ising model universality class can be conclusively ruled out, and the dimensions of the relevant bulk and surface scaling fields are found to take the valu…
We present results of the Monte-Carlo simulations for scaling of the free energy in dimers on the hexagonal lattice. The traditional Markov-chain Metropolis algorithm and more novel non-Markov Wang-Landau algorithm are applied. We compare the calculated results with the theoretical prediction for the equilateral hexagon and show that the latter algorithm gives more precise results for the dimer model. For a non-hexagonal domain the theoretical results are not available, so we present the numerical results for a certain geometry of the domain. We also study the two-point correlation function in…
We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang-Swendsen-Kotecky (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length $ξ\sim 5000$; the data are consistent with $ξ(β) = A e^{2β} β^p (1 + a_1 e^{-β} + ...)$ as $β\to\infty$, with $p \approx 1$. The staggered susceptibility behaves as $χ_{stagg} \sim ξ^{5/3}$. For q=4 the model is disordered ($ξ\ltapprox 2$) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagn…
We study the four-state antiferromagnetic Potts model on the triangular lattice. We show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector-valued topological charge. Using the properties of these defects, we deduce a 2+2-dimensional height representation for the model and hence show that the model is equivalent to the three-state Potts antiferromagnet on the Kagome lattice and to bond-coloring models on the triangular and hexagonal lattices. We also calculate critical exponents for the ground state …
The simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down. A number of cluster algorithms, usually based on the Fortuin-Kasteleyn representation of the Potts model, and suitable generalizations for continuous-spin models have been used to increase simulation efficiency. The first algorithm making use of this representation, suggested by Sweeny in 1983, has not found widespread adoption due to problems in its implementation. However, it has been recently shown that it is indeed more efficient in reducing …
Extending the Swendsen-Wang cluster algorithm to include both bulk (H) and surface fields (H_1) in L x L x D Ising films of thickness D and two free L x L surfaces, a Monte Carlo study of the capillary condensation critical point of the model is presented. Applying a finite-size scaling analysis where the lateral linear dimension L is varied over a wide range, the critical temperature T_c(D) and the associated critical field H_c(D) are estimated for 4 <= D <= 32 lattice spacings, for a choice of the surface field H_1 small enough that the dependence of H_c(D) on H_1 is still linear. It is show…
We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations $Γ$. Any quadrangulation can be built from a dual pair $(G,G^*)$. Based on the duality properties of $G$, we propose a new criterion to predict the phase diagram of this model. If $Γ$ is of self-dual type (i.e., if $G$ is isomorphic to its dual $G^*$), the model has a zero-temperature critical point with central charge $c=1$, and it is disordered at all positive temperatures. If $Γ$ is of non-self-dual type (i.e., if $G$ is not isomorphic to $G^*$), three ordered phases coexist at low temperature, a…
We review the recently developed critical minimum energy-subspace (CrMES) technique. This scheme produces an immense optimization of popular algorithms, such as the Wang-Landau (WL) and broad histogram methods, by predicting the essential part of the energy space necessary for the estimation of the critical behavior and provides a new route of critical exponent estimation. A powerful and efficient CrMES entropic sampling scheme is proposed as an alternative to the traditional importance sampling methods. Utilizing the WL random walk process in the dominant energy subspace (CrMES-WL sampling) a…
We review the development of update schemes for quantum lattice models simulated using world line quantum Monte Carlo algorithms. Starting from the Suzuki-Trotter mapping we discuss limitations of local update algorithms and highlight the main developments beyond Metropolis-style local updates: the development of cluster algorithms, their generalization to continuous time, the worm and directed-loop algorithms and finally a generalization of the flat histogram method of Wang and Landau to quantum systems.
We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts ferromagnet to non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95 the Ossola-Sokal conjecture z_{CM} \ge β/νis violated, though we also present plausible fits compatible with this conjecture. We show tha…
Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling
SISSA — International School for Advanced Studies and INFN（国际高等研究学院和意大利国家核物理研究所）；Laboratoire de Physique Statistique, École normale supérieure, PSL Research University（巴黎高等师范学院统计物理实验室，PSL研究大学）；Department of Physics, Duke University（杜克大学物理系）；Department of Chemistry, Duke University（杜克大学化学系）
We perform Monte Carlo simulations, combining both the Wang-Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume-Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo si…
