J.R. Heringa

Ising universality in three dimensions: a Monte Carlo study

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as yt=1.587 (2), yh=2.4815 (15) and yi=-0.82 (6). The universal ratio Q=(m2)2/(m4) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is Kc=0.2216546 (10).

NEW PRIMITIVE TRINOMIALS OF MERSENNE-EXPONENT DEGREES FOR RANDOM-NUMBER GENERATION

The list of primitive binary trinomials with a degree equal to a Mersenne exponent is extended. The newly found primitive trinomials have a degree equal to the 29th and 30th Mersenne exponent. These trinomials enable the construction of new, high-performance random-number generators for use in large-scale Monte Carlo simulations.

Monte Carlo renormalization of the three-dimensional Ising model

Abstract We have investigated the simple cubic Ising model by means of the Monte Carlo renormalization technique. The emphasis of our study concerns the influence of truncation, i.e. the dimensionality of the coupling subspace in which the analysis of the correlation functions generated by the Monte Carlo and spin blocking algorithms is performed. To this purpose we have included up to 36 even and 21 odd couplings in our analysis. We find that the increase in the number of couplings has a significant influence on the largest eigenvalues of the linearized renormalization transformation matrices. These eigenvalues serve to estimate the renormalization exponents yI and yH. Remarkably, we find no significant finite-size effect on these eigenvalues when the maximum number of couplings is included in the analysis, except for the smallest system sizes (83 → 43). After a suitable extrapolation to the fixed point, we find good agreement with existing results for the critical exponents. We have determined the critical point of the simple cubic Ising model as K = 0.221652(6), also in agreement with existing results.

The simple-cubic lattice gas with nearest-neighbour exclusion: Ising universality

The lattice gas with nearest neighbour-exclusion on the simple cubic lattice is studied by means of statistically accurate Monte Carlo simulations with an efficient cluster algorithm. Our results for critical exponents are yh = 2.47(1) and yt = 1.60(2). These results agree well with the three-dimensional Ising universality class. We explain the discrepancy with an earlier study. The critical activity is zc = 1.0559 (1). The size distribution of the clusters indicates that the percolation threshold of the cluster formation process of the Monte Carlo algorithm coincides with the critical point.

Simulation of a directed random-walk model: the effect of pseudo-random-number correlations

We investigate the mechanism that leads to systematic deviations in cluster Monte Carlo simulations when correlated pseudo-random numbers are used. We present a simple model which enables an analysis of the effects due to correlations in several types of pseudo-random-number sequences. This model provides qualitative understanding of the bias mechanism in a class of cluster Monte Carlo algorithms.

Monte Carlo renormalization of the 3D ising model: Analyticity and convergence

We review the assumptions on which the Monte Carlo renormalization technique is based, in particular the analyticity of the block spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved analyticity properties, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are y_H=2.481(1) and y_T=1.585(3).

Cluster Monte Carlo: Extending the range

Abstract Monte Carlo simulations with local updates tend to become time-consuming when large-scale correlations exist, such as in critical systems. For a limited, but increasing number of model systems, nonlocal ‘cluster’ algorithms are available that are orders of magnitude more efficient than algorithms with local updates. Cluster algorithms can be defined on the basis of the symmetry properties of the Hamiltonian; different symmetries can thus lead to different cluster algorithms. We review a number of existing cluster algorithms, and describe new ones for an Ising-like model with two- and three-spin interactions, and for the chiral Potts model. New simulation data for the Ising-like model allow an accurate determination of its specific-heat exponent; this result confirms existing ideas that the model belongs to the 4-state Potts universality class.

Monte Carlo renormalization of the three-dimensional Ising model: the influence of truncation

We have applied the Monte Carlo renormalization technique to the simple cubic Ising model. In particular we have investigated the influence of truncation, i.e. the number of coupling used in the analysis of the conjugate correlation functions generated by the Monte Carlo and spin blocking algorithms. To this purpose we have included up to 36 even and 21 odd couplings in our analysis, which is considerably more than used so far. We find that the addition of extra couplings does significantly influence the largest eigenvalues of the stability matrices. However, after correction for the finite-size effect and extrapolation to the fixed point, the effects on the estimated critical exponents are no longer significant.

High-dimensional lattice gases

We investigate the critical behaviour of hard-core lattice gases in four, five and six dimensions by means of Monte Carlo simulations. In order to suppress critical slowing down, we use a geometrical cluster Monte Carlo algorithm. In particular, nearest-neighbour-exclusion lattice gases on simple hypercubic lattices are investigated. These models undergo Ising-like ordering transitions where the majority of the lattice-gas particles settle on one of two sublattices. A finite-size-scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour. The results agree with the renormalization predictions at and above the upper critical dimensionality. In particular, the predicted value of the Binder cumulant is confirmed.

Understanding the Electrochemical Formation and Decomposition of Li2O2 and LiOH with Operando X-ray Diffraction

The lithium air, or Li–O2, battery system is a promising electrochemical energy storage system because of its very high theoretical specific energy, as required by automotive applications. Fundamental research has resulted in much progress in mitigating detrimental (electro)chemical processes; however, the detailed structural evolution of the crystalline Li2O2 and LiOH discharge products, held at least partially responsible for the limited reversibility and poor rate performance, is hard to measure operando under realistic electrochemical conditions. This study uses Rietveld refinement of operando X-ray diffraction data during a complete discharge–charge cycle to reveal the detailed structural evolution of Li2O2 and LiOH crystallites in 1,2-dimethoxyethane (DME) and DME/LiI electrolytes, respectively. The anisotropic broadened reflections confirm and quantify the platelet crystallite shape of Li2O2 and LiOH and show how the average crystallite shape evolves during discharge and charge. Li2O2 is shown to form via a nucleation and growth mechanism, whereas the decomposition appears to start at the smallest Li2O2 crystallite sizes because of their larger exposed surface. In the presence of LiI, platelet LiOH crystallites are formed by a particle-by-particle nucleation and growth process, and at the end of discharge, H2O depletion is suggested to result in substoichiometric Li(OH)1–x, which appears to be preferentially decomposed during charging. Operando X-ray diffraction proves the cyclic formation and decomposition of the LiOH crystallites in the presence of LiI over multiple cycles, and the structural evolution provides key information for understanding and improving these highly relevant electrochemical systems.