Metropolis Updates Research Articles

First-principles quantum Monte Carlo study of charge-carrier mobility in organic molecular semiconductors

We present a first-principles numerical study of charge transport in a realistic two-dimensional tight-binding model of organic molecular semiconductors. We use the hybrid Monte Carlo (HMC) algorithm to simulate the full quantum dynamics of phonons and either single or multiple charge carriers without any tunable parameters. We introduce a number of algorithmic improvements, including efficient Metropolis updates for phonon fields based on analytical insights, which lead to negligible autocorrelation times and allow sub-per-mille precisions to be reached at a low computational cost of O(1) CPU hours. Our simulations produce charge-mobility estimates that are in good agreement with experiments and that also justify the phenomenological transient localization approach. Published by the American Physical Society 2024

Incommensurate phases in the two-dimensional XY model with Dzyaloshinskii-Moriya interactions.

The two-dimensional XY model with Dzyaloshinskii-Moriya interaction has been studied through extensive Monte Carlo simulations. A hybrid algorithm consisting of single-spin Metropolis and Swendsen-Wang cluster-spin updates has been employed. Single histogram techniques have been used to obtain the thermodynamic variables of interest and finite-size-scaling analysis has led to the phase transition behavior in the thermodynamic limit. Fluctuating boundary conditions have been utilized in order to match the incommensurability between the spin structures and the finite lattice sizes due to the Dzyaloshinskii-Moriya interaction. The effects of the fluctuating boundary conditions have been analyzed in detail in both commensurate and incommensurate cases. The Berezinskii-Kosterlitz-Thouless transition temperature has been obtained as a function of the Dzyaloshinskii-Moriya interaction and the results are in excellent agreement with the exact equationfor the transition line. The spin-spin correlation function critical exponent has been computed as a function of the Dzyaloshinskii-Moriya interaction and temperature. In the incommensurate cases, optimal sizes for the finite lattices and the distribution of the boundary shift angle have been extracted. Analysis of the low temperature configurations and the corresponding vortex-antivortex pairs have also been addressed in some regions of the phase diagram.

Monte Carlo study of a generalized icosahedral model on the simple cubic lattice

We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take one of twelve vectors of unit length, which are given by the normalized vertices of the icosahedron, as value. Similar to the Blume-Capel model, where in addition to $-1$ and $1$, as in the Ising model, the spin might take the value $0$, we add in the generalized model $(0,0,0)$ as allowed value. There is a parameter $D$ that controls the density of these voids. For a certain range of $D$, the model undergoes a second-order phase transition. On the critical line, $O(3)$ symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of $D$, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to $L=400$ by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with $O(3)$ symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\nu=0.71164(10)$, $\eta=0.03784(5)$, and $\omega=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the $O(3)$ symmetry is $y_{ico}=-2.19(2)$.

Revisiting the field-driven edge transition of the tricritical two-dimensional Blume-Capel model.

We reconsider the tricritical Blume-Capel model on the square lattice with a magnetic field acting on the open boundaries in one direction. Periodic boundary conditions are applied in the other direction. We apply three types of Monte Carlo algorithms, local Metropolis updates, and cluster algorithms of the Wolff and geometric type, adapted to the symmetry properties of the model. Statistical analyses of the bulk magnetization, the bulk Binder ratio, the edge magnetization, and the connected product of the edge and bulk magnetizations lead to new results confirming the presence of a singular edge transition at H_{sc}≈0.68, as we reported earlier [Phys. Rev. E 71, 026109 (2005)PLEEE81539-375510.1103/PhysRevE.71.026109]. We provide a plausible answer concerning a discrepancy between the behavior of the edge Binder ratio reported in that work and our new results.

Sampling Latent States for High-Dimensional Non-Linear State Space Models with the Embedded HMM Method

We propose a new scheme for selecting pool states for the embedded Hidden Markov Model (HMM) Markov Chain Monte Carlo (MCMC) method. This new scheme allows the embedded HMM method to be used for efficient sampling in state space models where the state can be high-dimensional. Previously, embedded HMM methods were only applicable to low-dimensional state-space models. We demonstrate that using our proposed pool state selection scheme, an embedded HMM sampler can have similar performance to a well-tuned sampler that uses a combination of Particle Gibbs with Backward Sampling (PGBS) and Metropolis updates. The scaling to higher dimensions is made possible by selecting pool states locally near the current value of the state sequence. The proposed pool state selection scheme also allows each iteration of the embedded HMM sampler to take time linear in the number of the pool states, as opposed to quadratic as in the original embedded HMM sampler.

SIMULATING SPIN MODELS ON GPU: A TOUR

The use of graphics processing units (GPUs) in scientific computing has gathered considerable momentum in the past five years. While GPUs in general promise high performance and excellent performance per Watt ratios, not every class of problems is equally well suitable for exploiting the massively parallel architecture they provide. Lattice spin models appear to be prototypic examples of problems suitable for this architecture, at least as long as local update algorithms are employed. In this review, I summarize our recent experience with the simulation of a wide range of spin models on GPU employing an equally wide range of update algorithms, ranging from Metropolis and heat bath updates, over cluster algorithms to generalized ensemble simulations.

Slice sampling

Markov chain sampling methods that adapt to characteristics of the distribution being sampled can be constructed using the principle that one can ample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal "slice" defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Such "slice sampling" methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially adapt to the dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done for univariate slice sampling by "overrelaxation," and for multivariate slice sampling by "reflection" from the edges of the slice.

Multigrid Monte Carlo algorithms for SU(2) lattice gauge theory: Two versus four dimensions.

We study a multigrid method for nonabelian lattice gauge theory, the time slice blocking, in two and four dimensions. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method. This result is in accordance with theoretical arguments based on the analysis of the scale dependence of acceptance rates for nonlocal Metropolis updates. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems.

Theoretical analysis of acceptance rates in multigrid Monte Carlo

We analyze the kinematics of multigrid Monte Carlo algorithms by investigating acceptance rates for nonclonal Metropolis updates. With the help of a simple criterion we can decide whether or not a multigrid algorithms will have a chance to overcome critical slowing down for a given model. Our method is introduced in the context of spin models. A multigrid Monte Carlo procedure for nonabelian lattice gauge theory is described, and its kinematics is analyzed in detail.

Acceptance rates in multigrid Monte Carlo simulations.

An approximation formula is derived for acceptance rates of nonlocal Metropolis updates in simulations of lattice field theories. The predictions of the formula agree quite well with Monte Carlo simulations of 2-dimensional Sine Gordon, XY and phi**4 models. The results are consistent with the following rule: For a critical model with a fundamental Hamiltonian H(phi) sufficiently high acceptance rates for a complete elimination of critical slowing down can only be expected if the expansion of < H(phi+psi) > in terms of the shift psi contains no relevant term (mass term).

Accelerating Abelian gauge dynamics

In this paper, we suggest a new acceleration method for Abelian gauge theories based on linear transformations to variables which weight all length scales equally. We measure the autocorrelation time for the Polyakov loop and the plaquette at β = 1.0 in the U(1) gauge theory in four dimensions, for the new method and for standard Metropolis updates. We find a dramatic improvement for the new method over the Metropolis method. Computing the critical exponent z for the new method remains an important open issue.

Investigation of the two-dimensional O(3) model using the overrelaxation algorithm.

We investigate the two-dimensional O(3) model with the standard action by Monte Carlo simulation at couplings $\ensuremath{\beta}$ up to 2.05. We measure the energy density, mass gap, and susceptibility of the model, and gather high statistics on lattices of size $L\ensuremath{\le}1024$ using the floating point systems T-series vector hypercube. Asymptotic scaling does not appear to set in for this action, even at $\ensuremath{\beta}=2.05$, where the correlation length is 304. We observe a 20% difference between our estimate $\frac{m}{{\ensuremath{\Lambda}}_{\stackrel{-}{\mathrm{MS}}}}=3.52(6)$ at this $\ensuremath{\beta}$ and the recent exact analytical result. We use the overrelaxation algorithm interleaved with Metropolis updates and show that the decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be ${z}^{\ensuremath{'}}=1.079(10)$; thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.

Accelerating Abelian gauge dynamics.

In this paper, we suggest a new acceleration method for Abelian gauge theories based on linear transformations to variables which weight all length scales equally. We measure the autocorrelation time for the Polyakov loop and the plaquette at \ensuremath{\beta}=1.0 in the U(1) gauge theory in four dimensions, for the new method and for standard Metropolis updates. We find a dramatic improvement for the new method over the Metropolis method. Computing the critical exponent z for the new method remains an important open issue.

Employing the ising representation to implement nonlocal Monte Carlo updating inO(N) models

O(N) ferromagnets, anisotropic or not, with standard nearestneighbour interaction are expressed asN Ising models coupled with certain auxiliary fields. A hybrid Monte Carlo algorithm is then developed. It consists of global updates of the Ising variables using the Fortuin-Kasteleyn representation, combined with ordinary heat-bath or Metropolis updates of the remaining variables. The method generalizes toZ(2N) models. A numerical test forZ(8) at β=1.1 is reported. It reveals a dramatic reduction in critical slowing-down.