Chiaki Yamaguchi

Three-dimensional antiferromagnetic q-state Potts models: application of the Wang-Landau algorithm

We apply a newly proposed Monte Carlo method, the Wang-Landau algorithm, to the study of the three-dimensional antiferromagnetic q-state Potts models on a simple cubic lattice. We systematically study the phase transition of the models with q = 3, 4, 5 and 6. We obtain the finite-temperature phase transition for q = 3 and 4, whereas the transition temperature is down to zero for q = 5. For q = 6 there exists no order for any temperature. We also study the ground-state properties. The size dependence of the ground-state entropy is investigated. We find that the ground-state entropy is larger than the contribution from the typical configurations of the broken-sublattice-symmetry state for q = 3. The same situations are found for q = 4, 5 and 6.

Combination of improved multibondic method and the Wang-Landau method.

We propose a method for Monte Carlo simulation of statistical physical models with discretized energy. The method is based on several ideas including the cluster algorithm, the multicanonical Monte Carlo method and its acceleration proposed recently by Wang and Landau. As in the multibondic ensemble method proposed by Janke and Kappler, the present algorithm performs a random walk in the space of the bond population to yield the state density as a function of the bond number. A test on the Ising model shows that the number of Monte Carlo sweeps required of the present method for obtaining the density of state with a given accuracy is proportional to the system size, whereas it is proportional to the system size squared for other conventional methods. In addition, the method shows a better performance than the original Wang-Landau method in measurement of physical quantities.

Application of new Monte Carlo algorithms to random spin systems

Abstract We explain the idea of the probability-changing cluster (PCC) algorithm, which is an extended version of the Swendsen–Wang algorithm. With this algorithm, we can tune the critical point automatically. We show the effectiveness of the PCC algorithm for the case of the three-dimensional (3D) Ising model. We also apply this new algorithm to the study of the 3D diluted Ising model. Since we tune the critical point of each random sample automatically with the PCC algorithm, we can investigate the sample-dependent critical temperature and the sample average of physical quantities at each critical temperature, systematically. We have also applied another newly proposed algorithm, the Wang–Landau algorithm, to the study of the spin glass problem.

Broad histogram relation for the bond number and its applications.

We discuss Monte Carlo methods based on the cluster (graph) representation for spin models. We derive a rigorous broad histogram relation (BHR) for the bond number; a counterpart for the energy was derived by Oliveira previously. A Monte Carlo dynamics based on the number of potential moves for the bond number is proposed. We show the efficiency of the BHR for the bond number in calculating the density of states and other physical quantities.

Transition-Matrix Monte Carlo Method for Quantum Systems

We propose an efficient method for Monte Carlo simulation of quantum lattice models by generalizing the method [Phys. Rev. E 66 (2002) 036704] proposed for classical models by the present authors. In particular, we derive an exact relation between the density of states and the microcanonical averages of some macroscopic quantities. The simulation method consists of the graph construction by the loop/cluster algorithm and the estimation of the density of states by the relation. While there may be a few variants for the graph construction, we consider the one proposed by Troyer et al. [Phys. Rev. Lett. 90 (2003) 120201]. The performance of the method is examined for S =1/2 antiferromagnetic Heisenberg chain, and compared with other algorithms.

Novel Monte Carlo algorithms and their applications

We describe a generalized scheme for the probability-changing cluster (PCC) algorithm, based on the study of the finite-size scaling property of the correlation ratio, the ratio of the correlation functions with different distances. We apply this generalized PCC algorithm to the two-dimensional 6-state clock model. We also discuss the combination of the cluster algorithm and the extended ensemble method. We derive a rigorous broad histogram relation for the bond number. A Monte Carlo dynamics based on the number of potential moves for the bond number is proposed, and applied to the three-dimensional Ising and 3-state Potts models.

Proposal of a checking parameter in the simulated annealing method applied to the spin glass model

Abstract We propose a checking parameter utilizing the breaking of the Jarzynski equality in the simulated annealing method using the Monte Carlo method. This parameter is based on the Jarzynski equality. By using this parameter, to detect that the system is in global minima of the free energy under gradual temperature reduction is possible. Thus, by using this parameter, one is able to investigate the efficiency of annealing schedules. We apply this parameter to the ± J Ising spin glass model. The application to the Gaussian Ising spin glass model is also mentioned. We discuss that the breaking of the Jarzynski equality is induced by the system being trapped in local minima of the free energy. By performing Monte Carlo simulations of the ± J Ising spin glass model and a glassy spin model proposed by Newman and Moore, we show the efficiency of the use of this parameter.

Conjectured exact percolation thresholds of the Fortuin–Kasteleyn cluster for the ±J Ising spin glass model

The conjectured exact percolation thresholds of the Fortuin–Kasteleyn cluster for the ±J Ising spin glass model are theoretically shown based on a conjecture. It is pointed out that the percolation transition of the Fortuin–Kasteleyn cluster for the spin glass model is related to a dynamical transition for the freezing of spins. The present results are obtained as locations of points on the so-called Nishimori line, which is a special line in the phase diagram. We obtain TFK=2/ln(z/z−2) and pFK=z/2(z−1) for the Bethe lattice, TFK→∞ and pFK→1/2 for the infinite-range model, TFK=2/ln3 and pFK=3/4 for the square lattice, TFK∼3.9347 and pFK∼0.62441 for the simple cubic lattice, TFK∼6.191 and pFK∼0.5801 for the 4-dimensional hypercubic lattice, and TFK=2/ln[1+2sin(π/18)/1−2sin(π/18)] and pFK=[1+2sin(π/18)]/2 for the triangular lattice, when J/kB=1, where z is the coordination number, J is the strength of the exchange interaction between spins, kB is the Boltzmann constant, TFK is the temperature at the percolation transition point, and pFK is the probability, that the interaction is ferromagnetic, at the percolation transition point.

New Cluster Algorithms Using the Broad Histogram Relation

We describe the Monte Carlo methods based on the cluster (graph) representation for spin models. We discuss a rigorous broad histogram relation (BHR) for the bond number; the BHR for the energy was previously derived by Oliveira. A Monte Carlo dynamics based on the number of potential moves for the bond number is presented. We also extend the BHR to the loop algorithm of the quantum simulation.

Applications of the Probability-Changing Cluster Algorithm and Related Problems

We describe the idea of the newly proposed efficient algorithpi of tuning the critical point automatically. This probability-changing cluster (PCC) algorithm is an extended version of the Swendsen-Wang algorithm. As an application of the PCC algorithm, we study the two-dimensional site-diluted Ising model, paying attention to the self-averaging property. We also use another newly proposed algorithm, the Wang-Landau algorithm, for the study of the three-dimensional antiferromagnetic q-state Potts models.