Cluster Monte Carlo Algorithm Research Articles

Ising fracton spin liquid on the honeycomb lattice

We study a classical Ising model on the honeycomb lattice with local two-body interactions and present strong evidence that at low temperature it realizes a higher-rank Coulomb liquid with fracton excitations. We show that the excitations are (type-I) fractons, appearing at the corners of membranes of spin flips. Because of the threefold rotational symmetry of the honeycomb lattice, these membranes can be locally combined such that no excitations are created, giving rise to a set of ground states described as a liquid of membranes. We devise a cluster Monte Carlo algorithm purposefully designed for this problem that moves pairs of defects, and use it to study the finite-temperature behavior of the model. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase whose correlations precisely match those predicted for a higher-rank Coulomb phase. Published by the American Physical Society 2024

Canonical Monte Carlo multispin cluster method

We present a new Canonical Multispin-flip Cluster Monte Carlo algorithm for Ising model and describe efficient implementations for hybrid supercomputer. Our method takes advantage of the computing architecture for parallel and multi-threaded operations and uses a sequential cluster state update scheme. Due to the peculiarity of the implementation, the method is more effective for models with a restricted radius of interaction. It is based on combining a random selection of spin cluster by the Monte Carlo method with a complete enumeration of the all states of the selected cluster. To show how it works we applied our method to models of interacting magnetic Ising-moments: ferromagnetic Ising model, the Edwards–Anderson spin glass model, dipolar spin ice on hexagonal and pentagonal Cairo lattices.

Modern computational studies of the glass transition

The physics of the glass transition and amorphous materials continues to attract the attention of a wide research community after decades of effort. Supercooled liquids and glasses have been studied numerically since the advent of molecular dynamics and Monte Carlo simulations, and computer studies have greatly enhanced both experimental discoveries and theoretical developments. In this Review, we provide a modern perspective on this area. We describe the need to go beyond canonical methods when studying the glass transition — a problem that is notoriously difficult in terms of timescales, length scales and physical observables. We summarize recent algorithmic developments to achieve enhanced sampling and faster equilibration by using replica-exchange methods, cluster and swap Monte Carlo algorithms, and other techniques. We then review some major advances afforded by these tools regarding the statistical mechanical description of the liquid-to-glass transition, and the mechanical, vibrational and thermal properties of the glassy solid. Computer simulations may unlock crucial aspects of how a liquid transforms into a glass, but are hampered by rapidly growing relaxation times near the transition. This Review summarizes progress towards overcoming this problem and creating realistic in silico glasses, and discusses what understanding has been enabled.

Reexamining equationsof state of oblate hard ellipsoids of revolution: Numerical simulation utilizing a cluster Monte Carlo algorithm and comparison to virial theory.

We provide highly accurate equation-of-state data determined by means of cluster Monte Carlo simulations for the isotropic phase of oblate hard ellipsoids of revolution. Both equation-of-state data and phase boundaries of the isotropic phase are obtained from relatively large ensembles with typically 1000 particles. The comparison of simulation data with a virial approach gives evidence for the importance of high-order so-far-unknown virial coefficients and therewith many-particle interactions in dense, isotropic systems of anisotropic particles. While a virial approach with a rescaled Carnahan-Starling correction for the unknown, higher-order virial coefficients reproduces the simulation data of moderately anisotropic particles with high accuracy, we suggest for highly anisotropic shapes a simple, heuristic equationof state as a suitable approach.

Magnetization processes of fractal-like core shell nanoparticles

The paper refers to micromagnetic simulations of magnetization processes of fractal-like core–shell nanoparticles. The objects were generated using the 3D diffusion limited aggregation (DLA) algorithm for obtaining fractals with two kinds of magnetic phases – magnetically soft core and magnetically hard shell. The simulations were carried out using the cluster Monte Carlo algorithm designed for spin continuous and multiphase magnetic systems. The presented research includes different degrees of branch development, different strengths of the exchange coupling between the phases, as well as different soft phase contents. As was shown, the influence of microstructure on the coercivity mechanism is a complex phenomenon. The core–shell coupling and the magnetic properties of the entire system can be controlled by fractal development. The spring-exchange mechanism and high surface development of the fractals makes it possible to obtain a value of the |BH|max parameter higher than 300 kJ/m3. For K = 5 × 10−5 eV and K = 5 × 10−4 eV, it requires about 30% and 7% contribution of the hard magnetic phase, respectively.

Radial lattice quantization of 3D ϕ4 field theory

The quantum extension of classical finite elements, referred to as quantum finite elements ({\bf QFE})~\cite{Brower:2018szu,Brower:2016vsl}, is applied to the radial quantization of 3d $\phi^4$ theory on a simplicial lattice for the $\mathbb R \times \mathbb S^2$ manifold. Explicit counter terms to cancel the one- and two-loop ultraviolet defects are implemented to reach the quantum continuum theory. Using the Brower-Tamayo~\cite{Brower:1989mt} cluster Monte Carlo algorithm, numerical results support the QFE ansatz that the critical conformal field theory (CFT) is reached in the continuum with the full isometries of $\mathbb R \times \mathbb S^2$ restored. The Ricci curvature term, while technically irrelevant in the quantum theory, is shown to dramatically improve the convergence opening, the way for high precision Monte Carlo simulation to determine the CFT data: operator dimensions, trilinear OPE couplings and the central charge.

From Atomic Level to Large-Scale Monte Carlo Magnetic Simulations.

This paper refers to Monte Carlo magnetic simulations for large-scale systems. We propose scaling rules to facilitate analysis of mesoscopic objects using a relatively small amount of system nodes. In our model, each node represents a volume defined by an enlargement factor. As a consequence of this approach, the parameters describing magnetic interactions on the atomic level should also be re-scaled, taking into account the detailed thermodynamic balance as well as energetic equivalence between the real and re-scaled systems. Accuracy and efficiency of the model have been depicted through analysis of the size effects of magnetic moment configuration for various characteristic objects. As shown, the proposed scaling rules, applied to the disorder-based cluster Monte Carlo algorithm, can be considered suitable tools for designing new magnetic materials and a way to include low-level or first principle calculations in finite element Monte Carlo magnetic simulations.

Effect of quenched non-magnetic impurities on phase transitions in a two-dimensional Potts model

The Wolff cluster Monte Carlo algorithm is used to investigate the effect of frozen nonmagnetic canonically distributed impurities on the phase transitions in a two-dimensional Potts model with the spin state q = 5. Systems with linear dimensions L = 20–160 at spin concentrations p = 1.0 and 0.9 are considered. Using the fourth-order Binder cumulant method and the histogram analysis method it is shown that if a weak quenched disorder in the form of non-magnetic impurities (p = 0.9) is introduced in the system, the first-order phase transition changes to a second-order phase transition.

Magnetization processes of irregular dendrite structures - A Monte Carlo study

The paper refers to micromagnetic simulations of magnetization processes of dendrite-like object. The objects were generated by the DLA fractal algorithm that allows obtaining fractals with different ratio of the spins attributed to the surface to volume. The simulations were carried out using the cluster Monte Carlo algorithm designed for spin continuous and multiphase magnetic systems. The presented researches include different magnetic anisotropy of the surface and volume reflected magnetically soft, hard and ultra-high coercive phases. As it was shown, the influence of microstructure on the coercivity mechanism is a complex phenomenon. In the case of the fractals with magnetically soft volume the increasing surface contribution causes either increse or decrease of the coercive field for relatively high or low magnetic anisotropy of the surface, respectively. For the fractals with ultra-high coercive volume the occurrence of the surface anisotropy leads to the significant deterioration of their hard magnetic properties. The obtained spin configurations show that this effect is related to non-colinear directions of the surface anisoropy and strong enough exchange coupling between the surface and volume.

Optimization of hard magnetic properties of composites containing ultra-high coercive phases – Simulations

The paper refers to Monte Carlo simulation of multiphase magnetic systems. In order to study magnetization processes of the so-called spring-exchange magnets, a specific disorder-based cluster Monte Carlo algorithm was used. The examined systems were modeled by means of the Perlin noise generator. As it was shown, the coupling between the magnetically hard and soft phases results in the two dependent effects: increase of magnetic remanence and decrease of coercive field. Variation of these parameters constitutes a limit for optimization of hard magnetic properties (enhancement of the |BH|max energy product) which was estimated about 7 and 15 for the (Tb–Fe–B)/(Tb–Fe–B) and (Tb–Fe–B)/Fe type of composites, respectively. This effect is related to the observed changing coercivity mechanism of the hard magnetic component.

Dipolar spin glass transition in three dimensions

Dilute dipolar Ising magnets remain a notoriously hard problem to tackle both analytically and numerically because of long-ranged interactions between spins as well as rare region effects. We study a new type of anisotropic dilute dipolar Ising system in three dimensions [Phys. Rev. Lett. {\bf 114}, 247207 (2015)] that arises as an effective description of randomly diluted classical spin ice, a prototypical spin liquid in the disorder-free limit, with a small fraction $x$ of non-magnetic impurities. Metropolis algorithm within a parallel thermal tempering scheme fails to achieve equilibration for this problem already for small system sizes. Motivated by previous work [Phys. Rev. X {\bf 4}, 041016 (2014)] on uniaxial random dipoles, we present an improved cluster Monte Carlo algorithm that is tailor-made for removing the equilibration bottlenecks created by clusters of {\it effectively frozen} spins. By performing large-scale simulations down to $x=1/128$ and using finite size scaling, we show the existence of a finite-temperature spin glass transition and give strong evidence that the universality of the critical point is independent of $x$ when it is small. In this $x \ll 1$ limit, we also provide a first estimate of both the thermal exponent, $\nu=1.27(8)$, and the anomalous exponent, $\eta=0.228(35)$.

Theory of Ferroelectric ZrO2 Monolayers on Si

We use density functional theory and Monte Carlo lattice simulations to investigate the structure of ZrO$_{2}$ monolayers on Si(001). Recently, we have reported on the experimental growth of amorphous ZrO$_{2}$ monolayers on silicon and their ferroelectric properties, marking the achievement of the thinnest possible ferroelectric oxide [M. Dogan et al. Nano Lett., 18 (1) (2018)]. Here, we first describe the rich landscape of atomic configurations of monocrystalline ZrO$_{2}$ monolayers on Si and determine the local energy minima. Because of the multitude of low-energy configurations we find, we consider the coexistence of finite-sized regions of different configurations. We create a simple nearest-neighbor lattice model with parameters extracted from DFT calculations, and solve it numerically using a cluster Monte Carlo algorithm. Our results suggest that up to room temperature, the ZrO$_{2}$ monolayer consists of small domains of two low-energy configurations with opposite ferroelectric polarization. This explains the observed ferroelectric behavior in the experimental films as a collection of crystalline regions, which are a few nanometers in size, being switched with the application of an external electric field.

Diagrammatic Coupled Cluster Monte Carlo.

We propose a modified coupled cluster Monte Carlo algorithm that stochastically samples connected terms within the truncated Baker-Campbell-Hausdorff expansion of the similarity-transformed Hamiltonian by construction of coupled cluster diagrams on the fly. Our new approach-diagCCMC-allows propagation to be performed using only the connected components of the similarity-transformed Hamiltonian, greatly reducing the memory cost associated with the stochastic solution of the coupled cluster equations. We show that for perfectly local, noninteracting systems diagCCMC is able to represent the coupled cluster wavefunction with a memory cost that scales linearly with system size. The favorable memory cost is observed with the only assumption of fixed stochastic granularity and is valid for arbitrary levels of coupled cluster theory. Significant reduction in memory cost is also shown to smoothly appear with dissociation of a finite chain of helium atoms. This approach is also shown not to break down in the presence of strong correlation through the example of a stretched nitrogen molecule. Our novel methodology moves the theoretical basis of coupled cluster Monte Carlo closer to deterministic approaches.

Early stages of phase selection in MOF formation observed in molecular Monte Carlo simulations.

Metal–organic frameworks (MOF) comprising metal nodes bridged by organic linkers show great promise because of their guest-specific gas sorption, separation, drug-delivery, and catalytic properties. The selection of metal node, organic linker, and synthesis conditions in principle offers engineered control over both structure and function. For MOFs to realise their potential and to become more than just promising materials, a degree of predictability in the synthesis and a better understanding of the self-assembly or initial growth processes is of paramount importance. Using cobalt succinate, a MOF that exhibits a variety of phases depending on synthesis temperature and ligand to metal ratio, as proof of concept, we present a molecular Monte Carlo approach that allows us to simulate the early stage of MOF assembly. We introduce a new Contact Cluster Monte Carlo (CCMC) algorithm which uses a system of overlapping “virtual sites” to represent the coordination environment of the cobalt and both metal–metal and metal–ligand associations. Our simulations capture the experimentally observed synthesis phase distinction in cobalt succinate at 348 K. To the best of our knowledge this is the first case in which the formation of different MOF phases as a function of composition is captured by unbiased molecular simulations. The CCMC algorithm is equally applicable to any system in which short-range attractive interactions are a dominant feature, including hydrogen-bonding networks, metal–ligand coordination networks, or the assembly of particles with “sticky” patches, such as colloidal systems or the formation of protein complexes.

Disorder-based cluster Monte Carlo algorithm and its application in simulations of magnetization processes

The paper presents a modification of the so-called adding probability used in the Monte Carlo cluster algorithms. Certainly, the probability formula was supplemented by an entropy factor, depending on the local disorder of some selected properties of the studied system. The idea bases on the fact that disorder of some physical properties may affect clusterization of the system. The proposed modification enhances effectiveness of the simulations that is especially important in analysis of magnetization processes of irregular multiphase systems. In order to prove correctness of our approach a set of reverse magnetization curves was simulated for the systems composed of magnetically hard and soft phases.

Exploring cluster Monte Carlo updates with Boltzmann machines.

Boltzmann machines are physics informed generative models with broad applications in machine learning. They model the probability distribution of an input data set with latent variables and generate new samples accordingly. Applying the Boltzmann machines back to physics, they are ideal recommender systems to accelerate the Monte Carlo simulation of physical systems due to their flexibility and effectiveness. More intriguingly, we show that the generative sampling of the Boltzmann machines can even give different cluster Monte Carlo algorithms. The latent representation of the Boltzmann machines can be designed to mediate complex interactions and identify clusters of the physical system. We demonstrate these findings with concrete examples of the classical Ising model with and without four-spin plaquette interactions. In the future, automatic searches in the algorithm space parametrized by Boltzmann machines may discover more innovative Monte Carlo updates.

Many-body critical Casimir interactions in colloidal suspensions.

We study the fluctuation-induced Casimir interactions in colloidal suspensions, especially between colloids immersed in a binary liquid close to its critical demixing point. To simulate these systems, we present a highly efficient cluster Monte Carlo algorithm based on geometric symmetries of the Hamiltonian. Utilizing the principle of universality, the medium is represented by an Ising system while the colloids are areas of spins with fixed orientation. Our results for the Casimir interaction potential between two particles at the critical point in two dimensions perfectly agree with the exact predictions. However, we find that in finite systems the behavior strongly depends on whether the Z(2) symmetry of the system is broken by the particles. We present Monte Carlo results for the three-body Casimir interaction potential and take a close look onto the case of one particle in the vicinity of two adjacent particles, which can be calculated from the two-particle interaction by a conformal mapping. These results emphasize the failure of the common decomposition approach for many-particle critical Casimir interactions.

Rényi information flow in the Ising model with single-spin dynamics.

The n-index Rényi mutual information and transfer entropies for the two-dimensional kinetic Ising model with arbitrary single-spin dynamics in the thermodynamic limit are derived as functions of ensemble averages of observables and spin-flip probabilities. Cluster Monte Carlo algorithms with different dynamics from the single-spin dynamics are thus applicable to estimate the transfer entropies. By means of Monte Carlo simulations with the Wolff algorithm, we calculate the information flows in the Ising model with the Metropolis dynamics and the Glauber dynamics, respectively. We find that not only the global Rényi transfer entropy, but also the pairwise Rényi transfer entropy, peaks in the disorder phase.

On the non-ergodicity of the Swendsen–Wang–Kotecký algorithm on the kagomélattice

We study the properties of the Wang–Swendsen–Kotecký cluster Monte Carlo algorithm forsimulating the three-state kagomé-lattice Potts antiferromagnet at zero temperature. Weprove that this algorithm is not ergodic for symmetric subsets of the kagomé lattice withfully periodic boundary conditions: given an initial configuration, not all configurations areaccessible via Monte Carlo steps. The same conclusion holds for single-site dynamics.

Simulating Nanoscale Dielectric Response

We introduce a constrained energy functional to describe dielectric response. We demonstrate that the local functional is a generalization of the long-ranged Marcus energy. Our reformulation is used to implement a cluster Monte Carlo algorithm for the simulation of dielectric media. The algorithm avoids solving the Poisson equation and remains efficient in the presence of spatial heterogeneity, nonlinearity, and scale dependent dielectric properties.