We report tests of the recently proposed multicanonical multigrid Monte Carlo method for the two-dimensional $Phi^4$ field theory. Defining an effective autocorrelation time we obtain real time improvement factors of about one order of magnitude compared with standard multicanonical simulations.
We present a novel Ensemble Monte Carlo Growth method to sample the equilibrium thermodynamic properties of random chains. The method is based on the multicanonical technique of computing the density of states in the energy space. Such a quantity is
temperature independent, and therefore microcanonical and canonical thermodynamic quantities, including the free energy, entropy, and thermal averages, can be obtained by re-weighting with a Boltzmann factor. The algorithm we present combines two approaches: the first is the Monte Carlo ensemble growth method, where a population of samples in the state space is considered, as opposed to traditional sampling by long random walks, or iterative single-chain growth. The second is the flat-histogram Monte Carlo, similar to the popular Wang-Landau sampling, or to multicanonical chain-growth sampling. We discuss the performance and relative simplicity of the proposed algorithm, and we apply it to known test cases.
We study constraint effective potentials for various strongly interacting $phi^4$ theories. Renormalization group (RG) equations for these quantities are discussed and a heuristic development of a commonly used RG approximation is presented which str
esses the relationships among the loop expansion, the Schwinger-Dyson method and the renormalization group approach. We extend the standard RG treatment to account explicitly for finite lattice effects. Constraint effective potentials are then evaluated using Monte Carlo (MC) techniques and careful comparisons are made with RG calculations. Explicit treatment of finite lattice effects is found to be essential in achieving quantitative agreement with the MC effective potentials. Excellent agreement is demonstrated for $d=3$ and $d=4$, O(1) and O(2) cases in both symmetric and broken phases.
We propose a new method for Hybrid Monte Carlo (HMC) simulations with odd numbers of dynamical fermions on the lattice. It employs a different approach from polynomial or rational HMC. In this method, gamma-five hermiticity of the lattice Dirac opera
tors is crucial and it can be applied to Wilson, domain-wall, and overlap fermions. We compare HMC simulations with two degenerate flavors and (1 + 1) degenerate flavors using optimal domain-wall fermions. The ratio of the efficiency, (number of accepted trajectories) / (simulation time), is about 3:2. The relation between pseudofermion action of chirally symmetric lattice fermions in four-dimensional(overlap) and five-dimensional(domain-wall) representation are also analyzed.
We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplicati
on is partially implemented numerically exactly and partially with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer.
The investigation of freezing transitions of single polymers is computationally demanding, since surface effects dominate the nucleation process. In recent studies we have systematically shown that the freezing properties of flexible, elastic polymer
s depend on the precise chain length. Performing multicanonical Monte Carlo simulations, we faced several computational challenges in connection with liquid-solid and solid-solid transitions. For this reason, we developed novel methods and update strategies to overcome the arising problems. We introduce novel Monte Carlo moves and two extensions to the multicanonical method.