Wang-Landau Algorithm Research Articles

Study of the Two-Dimensional Anisotropic Ising Model with Competing Interactions in the Region of a Transition from the Ferromagnetic to Paramagnetic State

The two-dimensional anisotropic Ising model with competing interactions in the region of a transition from the ferromagnetic to paramagnetic phase has been studied by the Monte Carlo method using the Wang–Landau algorithm. State density distributions have been obtained and the order parameters corresponding to different energies have been calculated. The temperature dependences of the thermodynamic parameters including the internal energy, magnetization, free energy, entropy, specific heat at constant magnetization, and Binder cumulants have been calculated. Using the Binder cumulant method and histogram analysis, it has been shown that a single second-order phase transition from the ferromagnetic to paramagnetic phase occurs at two exchange coupling constants ratios |J1/J| = 0.1 and 0.2 in the two-dimensional anisotropic next-nearest neighbor Ising model. The phase transition temperatures have been calculated. The main critical exponents have been determined using the finite-size scaling theory.

Исследование термодинамических параметров в модели магнитного дендримера алгоритмом Ванга–Ландау

Исследование термодинамических параметров в модели магнитного дендримера алгоритмом Ванга–Ландау

Фазовые переходы и термодинамические свойства модели Поттса с числом состояний спина q=4 на решетке кагоме

Phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a kagome lattice are investigated by the Mote Carlo method based on the Wang-Landau algorithm. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

Компьютерное моделирование модели Поттса с числом состояний спина q = 4 на треугольной решетке

Based on the Wang-Landau algorithm, the Monte Carlo method is used to study the thermodynamic properties of the two-dimensional Potts model with the number of spin states $q=4$ on a triangular lattice, taking into account the interactions of the first and second nearest neighbors. It is shown that taking into account antiferromagnetic interactions of the second nearest neighbors leads to frustration.

Исследование фазовых переходов и термодинамических свойств модели Поттса с C-=SUB=-q-=/SUB=-=4 на гексагональной решетке с взаимодействиями вторых ближайших соседей

Magnetic structures of the ground state, phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice with next-nearest neighbors interactions are investigated by the Monte Carlo method based on the Wang-Landau algorithm. It is shown that taking into account the antiferromagnetic interactions of the next-nearest neighbors leads to the appearance of frustration and a violation of magnetic ordering. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

Global Thermodynamic Properties of Complex Spin Systems Calculated from Density of States and Indirectly by Thermodynamic Integration Method

Evaluation of global thermodynamic properties such as the entropy or the free energy of complex systems featuring a high degree of frustration or disorder is often desirable. Nevertheless, they cannot be measured directly in standard Monte Carlo simulation. Therefore, they are either evaluated indirectly from the directly measured quantities, for example by the thermodynamic integration method (TIM), or by applying more sophisticated simulation methods, such as the Wang-Landau (WL) algorithm, which can directly sample density of states. In the present investigation we compare the performance of the WL and TIM methods for the calculation of the entropy of an Ising antiferromagnetic system on a Kagome lattice – a typical example of a complex spin system with high geometrical frustration resulting in a non-zero residual entropy the value of which is exactly known. It is found that the easier to implement TIM can yield results of comparable accuracy with that of the more involved WL method.

Исследование моделей магнитных дендримеров методом Ванга-Ландау

By the Wang-Landau algorithm of the entropy Monte Carlo method the thermodynamic properties of the magnetic dendrimers at different values of the exchange interaction are studied. The density of states of the system are calculated and the magnetic structure of the ground state defined. The temperature dependence of the various thermodynamic parameters, such as the magnetization m, entropy S, the internal energy E and the heat capacity C are calculated. It is shown that in the studied model dendrimer magnetic influence of surface units in the overall picture of the system behavior with increasing system size unabated.

Phase Transitions and the Thermodynamic Properties of the Potts Model with the Number of Spin States q = 4 on a Triangular Lattice

The phase transitions and the thermodynamic properties of the two-dimensional ferromagnetic Potts model with the number of spin states q = 4 on a triangular lattice are studied on the base of the Wang–Landau algorithm of the Monte Carlo method. The phase transition characters are analyzed using the method of the four-order Binder cumulants and the histogram analysis of the data. It is found that a first-order phase transition is observed in the model under study.

Monte Carlo simulations of Heisenberg magnets by the Wang-Landau method

In this work, we investigate the critical behavior of the three dimensional Heisenberg model by parallel Wang-Landau algorithm developed with using of the OpenMP technology.

Thermodynamic and Magnetic Properties of the Two-Dimensional Anisotropic Ising Model with Competing Interactions

The two-dimensional anisotropic Ising model with competing interactions is studied on a square lattice by Monte Carlo methods using the Wang–Landau algorithm. The temperature dependences of the main thermodynamic and magnetic parameters are calculated. The distinctive features of temperature dependences of these parameters for different values of the competing interaction are shown. The dependences of the wave number of modulated structures on the ratio of exchange interaction parameters are plotted. The phase diagram of the model is built. The coordinates of the Lifshits point and the localization region of the phases with different wave vector values are calculated by interpolating phase boundaries of the d-iagram.

Phase diagram of six-state clock model on rewired square lattices

The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T1 and T2, where T1 < T2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T1 and T2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.

Phase Transitions and Thermodynamic Properties of the Potts Model with Spin States Number q = 4 on a Hexagonal Lattice

Phase transitions and thermodynamic properties of the 2D ferromagnetic Potts model with the number of spin states q = 4 on a hexagonal lattice are investigated by the Mote Carlo method based on the Wang–Landau algorithm. The orders of the phase transitions are investigated using the Binder fourth-order cumulant method and histogram analysis of data. It is established that a first-order transition is observed in the model being investigated.

Monte Carlo Calculations of the Density of States for a Two-Dimensional Anisotropic Ising Model with Competing Interactions

A two-dimensional anisotropic Ising model with competing interactions on a square lattice is investigated using the Monte Carlo approach, based on the Wang–Landau algorithm. The curves of the density-of-states distribution and the order parameter are obtained. It is shown that the density-of-states distribution jumps sharply when |J1/J| = 0.6, due to strong degeneration of the modulated state. Sharp jumps are also observed on the distributions of the order parameter when |J1/J| > 0.2, testifying to the system’s transition from a homogeneous ordered state to a modulated phase.

On properties of the Wang–Landau algorithm

We review recent advances in the analysis of the Wang–Landau algorithm, which is designed for the direct Monte Carlo estimation of the density of states (DOS). In the case of a discrete energy spectrum, we present an approach based on introducing the transition matrix in the energy space (TMES). The TMES fully describes a random walk in the energy space biased with the Wang–Landau probability. Properties of the TMES can explain some features of the Wang–Landau algorithm, for example, the flatness of the histogram. We show that the Wang–Landau probability with the true DOS generates a Markov process in the energy space and the inverse spectral gap of the TMES can estimate the mixing time of this Markov process. We argue that an efficient implementation of the Wang–Landau algorithm consists of two simulation stages: the original Wang-Landau procedure for the first stage and a 1/t modification for the second stage. The mixing time determines the characteristic time for convergence to the true DOS in the second simulation stage. The parameter of the convergence of the estimated DOS to the true DOS is the difference of the largest TMES eigenvalue from unity. The characteristic time of the first stage is the tunneling time, i.e., the time needed for the system to visit all energy levels.

The application of Wang-Landau algorithm for simulation of ferroelectrics

Computer simulation of phase transitions in ferroelectrics was completed with considering changes of lattice sizes by entropic sampling. The model describes a ferroelectric with six possible directions of polarization and is based on the Potts model with dipolar interactions. The free energy was calculated for cubic and tetragonal lattices. The equilibrium lattice was determined for different temperatures. It was shown that the anisotropy decreases in the point of phase transition.

Термодинамические и магнитные свойства двумерной анизотропной модели Изинга с конкурирующими взаимодействиями

The two-dimensional anisotropic Ising model was studied with competing interactions on a square lattice using Monte-Carlo methods using the Wang-Landau algorithm. The temperature dependences of the main thermodynamic and magnetic parameters are calculated. The distinctive features of the temperature dependences of these parameters are shown for different values of the competing interaction. The graphs of the dependence of the wave number of modulated structures on the ratio of the parameters of exchange interactions are plotted. Built phase diagram of the model. Interpolating the phase boundaries of the diagram, the coordinates of the Lifshchits point and phase localization areas with different wave vector values are calculated.

On the Feasibility of 5G Slice Resource Allocation With Spectral Efficiency: A Probabilistic Characterization

An important concern that 5G networks face is supporting a wide range of services and use cases with heterogeneous requirements. Radio access network (RAN) slices, understood as isolated virtual networks that share a common infrastructure, are a possible answer to this very demanding scenario and enable virtual operators to provide differentiated services over independent logical entities. This article addresses the feasibility of forming 5G slices, answering the question of whether the available capacity (resources) is sufficient to satisfy slice requirements. As spectral efficiency is one of the key metrics in 5G networks, we introduce the minislot-based slicing allocation (MISA) model, a novel 5G slice resource allocation approach that combines the utilization of both complete slots (or physical resource blocks) and mini-slots with the adequate physical layer design and service requirement constraints. We advocate for a probabilistic characterization that allows to estimate feasibility and characterize the behavior of the constraints, while an exhaustive search is very computationally demanding and the methods to check feasibility provide no information on the constraints. In such a characterization, the concept of phase transition allows for the identification of a clear frontier between the feasible and infeasible regions. Our method relies on an adaptation of the Wang-Landau algorithm to determine the existence of, at least, one solution to the problem. The conducted simulations show a significant improvement in spectral efficiency and feasibility of the MISA approach compared to the slot-based formulation, the identification of the phase transition, and valuable results to characterize the satisfiability of the constraints.

Efficient implementation of the Wang-Landau algorithm for systems with length-scalable potential energy functions

We consider a class of systems where $N$ identical particles with positions ${\bf q}_1,...,{\bf q}_N$ and momenta ${\bf p}_1,...,{\bf p}_N$ are enclosed in a box of size $L$, and exhibit the scaling $\mathcal{U}(L{\bf r}_1,...,L{\bf r}_N)=\alpha(L)\, \mathcal{U}({\bf r}_1,...,{\bf r}_N)$ for the associated potential energy function $\mathcal{U}({\bf q}_1,...,{\bf q}_N)$. For these systems, we propose an efficient implementation of the Wang-Landau algorithm for evaluating thermodynamic observables involving energy and volume fluctuations in the microcanonical description, and temperature and volume fluctuations in the canonical description. This requires performing the Wang-Landau simulation in a scaled box of unit size and evaluating the density of states corresponding to the potential energy part only. To demonstrate the efficacy of our approach, as example systems, we consider Padmanabhan's binary star model and an ideal gas trapped in a harmonic potential within the box.

The parallel Wang-Landau algorithm for joint density of states calculation

The Wang-Landau algorithm is used for estimating density of states. The joint density of states (JDOS) with two parameters gives more information than conventional one but it requires a huge computational time. Recently, the developing of fast algorithms for the JDOS calculation is an actual problem. In this work, we employ the modified parallel WL algorithm for JDOS, which combines ideas of global updates and two level methods. This algorithm is checked the three-dimensional Ising and Potts models. The method of obtaining phase diagrams from JDOS is demonstrated for Ising model.

Finite size scaling in the dimer and six-vertex model

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 simulations of dimers and the six-vertex model. The logarithmic dependence of the correlation function on the distance, which is in accordance with the Gaussian free field description of fluctuations, is obtained.