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Progress of Theoretical Physics Supplement

Multigrid Monte Carlo Simulations

Fung-kiu Chow, P. Kevin MacKeown

Published Jan 1, 2000• 0 citations
Publisher

Abstract

With a view to eventually studying quantum effects in the thermal behaviour of Lennard-Jones clusters, we investigate the extent to which the limitations of quasi-ergodicity and critical slowing down can be overcome in the evaluation of imaginary time path integrals using the Multigrid method. We show the power of the method by looking at the correlation times for a 2 Ar atom Lennard Jones cluster along the lines described in Janke & Sauer's paper.1)

DOI
10.1143/ptps.138.456

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With a view to eventually studying quantum effects in the thermal behaviour of Lennard-Jones clusters, we investigate the extent to which the limitations of quasi-ergodicity and critical slowing down can be overcome in the evaluation of imaginary time path integrals using the Multigrid method. We show the power of the method by looking at the correlation times for a 2 Ar atom Lennard Jones cluster along the lines described in Janke & Sauer's paper.1)
We describe a novel simulation method that eliminates the slowing-down problem in the Monte Carlo (MC) simulations of imaginary-time path integrals near the continuum limit. This method combines a stochastic blocking procedure with the multigrid method to rapidly accelerate the sampling of paths in a quantum MC simulation, making its dynamics more ergodic. The effectiveness and efficiency of this method are demonstrated for several one-dimensional quantum systems and compared to other standard and accelerated methods.
We pose two questions about the dynamical critical behavior of multigrid Monte Carlo: Will a multigrid Monte Carlo simulation of the two dimensional Sine Gordon model exhibit critical slowing down, as expected by a theoretical analysis of Metropolis acceptance rates? Can we reduce critical slowing down caused by decreasing acceptance rates on large blocks by performing more updates on coarser lattices? To this end we simulate the model with a W-cycle (gamma = 2) and a higher cycle with gamma = 4 using piecewise constant interpolation. The answer to the first question is positive, the answer to the second one is negative.
Conventional Monte Carlo simulation algorithms for models in statistical mechanics and quantum field theory are afflicted by problems caused by their locality. They become highly inefficient if investigations of critical or nearly-critical systems, i.e., systems with important large scale phenomena, are undertaken. We present two types of multiscale approaches that alleveate problems of this kind: stochastic cluster algorithms and multigrid Monte Carlo simulation algorithms. Another formidable computational problem in simulations of phenomenologically relevant field theories with fermions is the need for frequently inverting the Dirac operator. This inversion can be accelerated considerably by means of deterministic multigrid methods, very similar to the ones used for the numerical solution of differential equations. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Statistically independent equilibrium configurations of interacting systems can easily be generated with little computing effort by multigrid coarse-to-fine transforms. The very slow long-wavelength motions close to critical points are thereby accelerated by many orders of magnitude. This is demonstrated with a novel updating algorithm from small to large two-dimensional Ising systems specifying effective interactions according to renormalisation group considerations, corrected for non-linearities and saturation effects.
We formulate a new quantum many body simulation method for a general quantum fluid at any given temperature. Unlike path integral Monte Carlo our method evolves in imaginary time the density matrix from its initial delta function condition to its final thermal form at time time equal to the inverse temperature. It does this with a molecular dynamics scheme on a classical Hamiltonian that has the same functional form as the one of the quantum mechanical Hamiltonian according to the properties of the continuous representation of John R. Klauder. We then end up with the thermal density matrix that can be used to extract thermal averages of observables using the Monte Carlo method equally well in any statistics.
Featuring detailed explanations of the major algorithms used in quantum Monte Carlo simulations, this is the first textbook of its kind to provide a pedagogical overview of the field and its applications. The book provides a comprehensive introduction to the Monte Carlo method, its use, and its foundations, and examines algorithms for the simulation of quantum many-body lattice problems at finite and zero temperature. These algorithms include continuous-time loop and cluster algorithms for quantum spins, determinant methods for simulating fermions, power methods for computing ground and excited states, and the variational Monte Carlo method. Also discussed are continuous-time algorithms for quantum impurity models and their use within dynamical mean-field theory, along with algorithms for analytically continuing imaginary-time quantum Monte Carlo data. The parallelization of Monte Carlo simulations is also addressed. This is an essential resource for graduate students, teachers, and researchers interested in quantum Monte Carlo techniques.