Cluster Monte Carlo Method Research Articles

Global scheme of sweeping cluster algorithm to sample among topological sectors

Local constraint is closely related to the gauge field, so constrained models are usually effective low energy descriptions and important in condensed matter physics. On the other hand, local restriction hinders the application of numerical algorithms. In addition to the computational difficulties of the constraints, the various topological sectors which cannot be connected through local operators are also one of the key computational difficulties. Taking quantum dimer model as an example in this paper, we construct a global scheme based on sweeping cluster Monte Carlo method, which can sample among different topological sectors. In principle, this method can be generalized to other models.

Surface Functionalization of 2D MXenes: Trends in Distribution, Composition, and Electronic Properties.

Using a multiscale computational scheme, we study the trends in distribution and composition of the surface functional groups −O, −OH, and −F on two-dimensional (2D) transition metal carbides and nitrides (MXenes). We consider Ti2N, Ti4N3, Nb2C, Nb4C3, Ti2C, and Ti3C2 to explore MXenes with different chemistry and different number of atomic layers. Using a combination of cluster expansion, Monte Carlo, and density functional theory methods, we study the distribution and composition of functional groups at experimentally relevant conditions. We show that mixtures of functional groups are favorable on all studied MXene surfaces. The distribution of functional groups appears to be largely independent of the type of metal, carbon, or nitrogen species and/or number of atomic layers in the MXene. We further show that some properties (e.g., the work function) strongly depend on the surface composition, while others, for example, the electric conductivity, exhibit only a weak dependence.

Quantum Boltzmann machine algorithm with dimension-expanded equivalent Hamiltonian

Quantum machine learning explores the devising and implementing of quantum software to enable the speedup for a classical machine learning counterpart, which has attracted extensive attention. The quantum Boltzmann machine is a promising quantum machine learning model and can be executed on near-term quantum devices to demonstrate quantum supremacy. We investigate an efficient algorithm for training a quantum Boltzmann machine based on a continuous-time quantum cluster Monte Carlo method realized with a dimension-expanded equivalent Hamiltonian. The proposed algorithm is capable of efficiently training various types of quantum Boltzmann machines, including a two-dimensional random transverse Ising Hamiltonian and other stochastic $\ensuremath{\kappa}$-local $(\ensuremath{\kappa}\ensuremath{\ge}2)$ Hamiltonians. We further apply a quantum mean-field method to speed up the training process through discarding the quantum fluctuations of visible-hidden interactions. The experiment results show that our algorithm-trained quantum Boltzmann machine is more efficient on a small and large data set in terms of network convergence and time complexity.

Cluster Monte Carlo method for the 3D Ashkin–Teller model

This paper presents a cluster Monte Carlo method suitable for both, the first- and the second-order phase transitions in the 3D Ashkin–Teller (AT) model. A cluster algorithm is necessary to verify correctness of the results obtained so far using Metropolis algorithm exhibiting significant critical slowing down. Moreover, metastable states have recently been investigated in this model which can affect the results, especially obtained using the Metropolis algorithm. Our Wolff type algorithm is described and its dynamic critical behavior is demonstrated. Our computer experiments exploit the properties of Binder and Challa cumulants and additionally the one proposed by Lee and Kosterlitz, the last two adapted by us to give clear results for the AT model. The energy distribution histogram method is also independently applied for the first time for the 3D AT model using the Wolff type algorithm. For validation of the previous results and of our algorithm, it is demonstrated that the results of our computations along the line between Ising and Potts points, which are rescaled to their thermodynamic limits, are consistent with those obtained using the Metropolis algorithm. It is also shown that the presented cluster algorithm of the Wolff type significantly reduces the problem of critical slowing down for the 3D AT model, and the dynamic critical exponent reaches values close to zero. As the best strategy, it is suggested to use the cluster algorithm in the critical region and the Metropolis one beyond.

Large scale parallelization in stochastic coupled cluster.

Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility, or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behavior of the algorithm is investigated on the uniform electron gas and the water dimer at coupled-cluster levels including up to quadruple excitations. We also describe two improvements to the original sampling algorithm, full non-composite, and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.

Cluster Monte Carlo methods for the FePt Hamiltonian

Cluster Monte Carlo methods for the classical spin Hamiltonian of FePt with long range exchange interactions are presented. We use a combination of the Swendsen–Wang (or Wolff) and Metropolis algorithms that satisfies the detailed balance condition and ergodicity. The algorithms are tested by calculating the temperature dependence of the magnetization, susceptibility and heat capacity of L10-FePt nanoparticles in a range including the critical region. The cluster models yield numerical results in good agreement within statistical error with the standard single-spin flipping Monte Carlo method. The variation of the spin autocorrelation time with grain size is used to deduce the dynamic exponent of the algorithms. Our cluster models do not provide a more accurate estimate of the magnetic properties at equilibrium.

Monte Carlo study of the phase transitions in 2D ferro- and antiferromagnetic Potts models on a triangular lattice

The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.

Adaptive Markov chain Monte Carlo for auxiliary variable method and its application to parallel tempering

Auxiliary variable methods such as the Parallel Tempering and the cluster Monte Carlo methods generate samples that follow a target distribution by using proposal and auxiliary distributions. In sampling from complex distributions, these algorithms are highly more efficient than the standard Markov chain Monte Carlo methods. However, their performance strongly depends on their parameters and determining the parameters is critical. In this paper, we proposed an algorithm for adapting the parameters during drawing samples and proved the convergence theorem of the adaptive algorithm. We applied our algorithm to the Parallel Tempering. That is, we developed an adaptive Parallel Tempering that tunes the parameters on the fly. We confirmed the effectiveness of our algorithm through the validation of the adaptive Parallel Tempering, comparing samples from the target distribution by the adaptive Parallel Tempering and samples by conventional algorithms.

Phase Diagram of XY Model with Nematic Coupling on the Simple Cubic Lattice

Using cluster Monte Carlo method, we numerically investigate the criticality in the XY model with nematic coupling on the simple cubic lattice. We determine critical lines belong to the three-dimensional XY universality class in variable of θ (2θ) between the XY-ferromagnetic (nematic) and disordered states. Furthermore, the phase transition between the XY-ferromagnetic and the nematic states is found to be in the three-dimensional Ising universality class. The critical points are determined from the intersections of Binder ratios for various system sizes. With two sets of critical points obtained, we finally construct the phase diagram on the Δ-J plane.

Variational ground states of the two-dimensional Hubbard model

Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.

Order- N cluster Monte Carlo method for spin systems with long-range interactions

An efficient O ( N ) cluster Monte Carlo method for Ising models with long-range interactions is presented. Our novel algorithm does not introduce any cutoff for interaction range and thus it strictly fulfills the detailed balance. The realized stochastic dynamics is equivalent to that of the conventional Swendsen–Wang algorithm, which requires O ( N 2 ) operations per Monte Carlo sweep if applied to long-range interacting models. In addition, it is shown that the total energy and the specific heat can also be measured in O ( N ) time. We demonstrate the efficiency of our algorithm over the conventional method and the O ( N log N ) algorithm by Luijten and Blöte. We also apply our algorithm to the classical and quantum Ising chains with inverse-square ferromagnetic interactions, and confirm in a high accuracy that a Kosterlitz–Thouless phase transition, associated with a universal jump in the magnetization, occurs in both cases.

Bose–Fermi Kondo model with Ising anisotropy: Cluster-Monte Carlo approach

The Bose–Fermi Kondo model (BFKM) captures the physics of the destruction of Kondo screening, which is of extensive current interest to the understanding of quantum critical heavy fermion metals. There are presently limited theoretical methods to study the finite temperature properties of the BFKM. Here we provide some of the consistency checks on the cluster-Monte Carlo method, which we have recently applied to the Ising-anisotropic BFKM. We show that the method correctly captures the scaling properties of the Kondo phase, as well as those on approach to the Kondo-destroying quantum critical point. We establish that comparable results are obtained when the Kondo couplings are placed at or away from a Toulouse point.

Universal reduction of effective coordination number in the quasi-one-dimensional Ising model

Critical temperature of quasi-one-dimensional general-spin Ising ferromagnets is investigated by means of the cluster Monte Carlo method performed on infinite-length strips, $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ or $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$. We find that in the weak interchain coupling regime the critical temperature as a function of the interchain coupling is well-described by a chain mean-field formula with a reduced effective coordination number, as the quantum Heisenberg antiferromagnets recently reported by Yasuda et al. [Phys. Rev. Lett. 94, 217201 (2005)]. It is also confirmed that the effective coordination number is independent of the spin size. We show that in the weak interchain coupling limit the effective coordination number is, irrespective of the spin size, rigorously given by the quantum critical point of a spin-$1∕2$ transverse-field Ising model.

Hybrid cluster expansions for local structural relaxations

A model is constructed in which pair potentials are combined with the cluster expansion method in order to better describe the energetics of structurally relaxed substitutional alloys. The effect of structural relaxations away from the ideal crystal positions, and the effect of ordering is described by interatomic-distance dependent pair potentials, while more subtle configurational aspects associated with correlations of three- and more sites are described purely within the cluster expansion formalism. Implementation of such a hybrid expansion in the context of the cluster variation method or Monte Carlo method gives improved ability to model phase stability in alloys from first-principles.

Nonmetal ordering inTiC1−xNx: Ground-state structure and the effects of finite temperature

The $\mathrm{Ti}{\mathrm{C}}_{1\ensuremath{-}x}{\mathrm{N}}_{x}$ system has long been prized in industry because of its desirable thermodynamic and hardness characteristics. Previous studies have not produced comprehensive results describing the ordering tendencies of $\mathrm{Ti}{\mathrm{C}}_{1\ensuremath{-}x}{\mathrm{N}}_{x}$ at any temperature. We apply the mixed-basis cluster expansion and Monte Carlo methods to the problem and find a fascinating array of ground-state structures occurring at precise nitrogen-concentration intervals of $\ensuremath{\Delta}x=1∕16$ and related to each other by simple (201) quasisuperlattice motifs. Thermodynamic Monte Carlo results indicate that the critical ordering temperatures at all concentrations are well below room temperature. Short-range ordering develops at $T\ensuremath{\approx}800\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and exhibits the characteristic motifs of the predicted ground-state structures. Bulk modulus optimization is not feasible since this quantity shows little sensitivity to atomic configuration.

Constrained tricritical Blume-Capel model in three dimensions

Using the Wolff and geometric cluster Monte Carlo methods, we investigate the tricritical Blume-Capel model in three dimensions. Since these simulations conserve the number of vacancies and thus effectively introduce a constraint, we generalize the Fisher renormalization for constrained critical behavior to tricritical systems. We observe that, indeed, the tricritical behavior is significantly modified under this constraint. For instance, at tricriticality, the specific heat has only a finite cusp and the Binder ratio assumes a different value from that in unconstrained systems. Since 3 is the upper tricritical dimensionality of Ising systems, we expect that the mean-field theory correctly predicts a number of universal parameters in three dimensions. Therefore, we calculate the partition sum of the mean-field tricritical Blume-Capel model, and accordingly obtain the exact value of the Binder ratio. Under the constraint, we show that this mean-field tricritical system reduces to the mean-field critical Ising model. However, our three-dimensional data do not agree with this mean-field prediction. Instead, they are successfully explained by the generalized Fisher renormalization mechanism.

Magnetic relaxation and dipole-coupling-induced magnetization in nanostructured thin films during growth: A cluster Monte Carlo study

For growing inhomogeneous thin films with an island nanostructure similar to that observed in experiment we determine the nonequilibrium and the equilibrium remanent magnetization. The single-island magnetic anisotropy, the dipole coupling, and the exchange interaction between magnetic islands are taken into account within a micromagnetic model. A cluster Monte Carlo method is developed which includes coherent magnetization changes of connected islands. This causes a fast relaxation towards equilibrium for irregularly connected systems. We analyze the transition from dipole coupled islands at low coverages to a strongly connected ferromagnetic film at high coverages during film growth. For coverages below the percolation threshold the dipole interaction induces a collective magnetic order with ordering temperatures of 1--10 K for the assumed model parameters. Anisotropy causes blocking temperatures of 10--100 K and thus pronounced nonequilibrium effects. The dipole coupling leads to a somewhat slower magnetic relaxation.

Conformal invariance and the Ising model on a spheroid.

We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.

Cluster Monte Carlo simulation of the transverse Ising model.

We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.

Binary tree summation Monte Carlo simulation for Potts models

In this talk, we briefly comment on Sweeny and Gliozzi methods, cluster Monte Carlo method, and recent transition matrix Monte Carlo for Potts models. We mostly concentrate on a new algorithm known as ‘binary tree summation’. Some of the most interesting features of this method will be highlighted—such as simulating fractional number of Potts states, as well as offering the partition function and thermodynamic quantities as functions of temperature in a single run.