Monte Carlo Renormalization of the 3-D Ising model: Analyticity and Convergence

We review the assumptions on which the Monte Carlo renormalization technique is based, in particular the analyticity of the block spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved analyticity properties, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are y_H=2.481(1) and y_T=1.585(3).Comment: RevTeX, 4 PostScript file

Dual Monte Carlo and Cluster Algorithms

We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. By using the perspective of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum $S=$ 1/2 model, and proposing a free cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed.Comment: 25 pages, 4 figures, to appear in Phys.Rev.

Generalization of the Fortuin-Kasteleyn transformation and its application to quantum spin simulations,

We generalize the Fortuin-Kasteleyn (FK) cluster representation of the partition function of the Ising model to represent the partition function of quantum spin models with an arbitrary spin magnitude in arbitrary dimensions. This generalized representation enables us to develop a new cluster algorithm for the simulation of quantum spin systems by the worldline Monte Carlo method. Because the Swendsen-Wang algorithm is based on the FK representation, the new cluster algorithm naturally includes it as a special case. As well as the general description of the new representation, we present an illustration of our new algorithm for some special interesting cases: the Ising model, the antiferromagnetic Heisenberg model with $S=1$, and a general Heisenberg model. The new algorithm is applicable to models with any range of the exchange interaction, any lattice geometry, and any dimensions.Comment: 46 pages, 10 figures, to appear in J.Stat.Phy

Graphical representations and cluster algorithms for critical points with fields

A two-replica graphical representation and associated cluster algorithm is described that is applicable to ferromagnetic Ising systems with arbitrary fields. Critical points are associated with the percolation threshold of the graphical representation. Results from numerical simulations of the Ising model in a staggered field are presented. The dynamic exponent for the algorithm is measured to be less than 0.5.Comment: Revtex, 12 pages with 2 figure

Transition Matrix Monte Carlo Reweighting and Dynamics

We study an induced dynamics in the space of energy of single-spin-flip Monte Carlo algorithm. The method gives an efficient reweighting technique. This dynamics is shown to have relaxation times proportional to the specific heat. Thus, it is plausible for a logarithmic factor in the correlation time of the standard 2D Ising local dynamics.Comment: RevTeX, 5 pages, 3 figure

The Block Spin Renormalization Group Approach and Two-Dimensional Quantum Gravity

A block spin renormalization group approach is proposed for the dynamical triangulation formulation of two-dimensional quantum gravity. The idea is to update link flips on the block lattice in response to link flips on the original lattice. Just as the connectivity of the original lattice is meant to be a lattice representation of the metric, the block links are determined in such a way that the connectivity of the block lattice represents a block metric. As an illustration, this approach is applied to the Ising model coupled to two-dimensional quantum gravity. The correct critical coupling is reproduced, but the critical exponent is obscured by unusually large finite size effects.Comment: 10 page

Comments on Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation prepresentation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim (cond-mat/0201285), the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49(1); the three-dimensional Ising model has z = 0.37(2).Comment: RevTeX, 4 pages, 5 figure

Reconstructing the Density of States by History-Dependent Metadynamics

We present a novel method for the calculation of the energy density of states D(E) for systems described by classical statistical mechanics. The method builds on an extension of a recently proposed strategy that allows the free energy profile of a canonical system to be recovered within a pre-assigned accuracy,[A. Laio and M. Parrinello, PNAS 2002]. The method allows a good control over the error on the recovered system entropy. This fact is exploited to obtain D(E) more efficiently by combining measurements at different temperatures. The accuracy and efficiency of the method are tested for the two-dimensional Ising model (up to size 50x50) by comparison with both exact results and previous studies. This method is a general one and should be applicable to more realistic model systems

An Almost Perfect Quantum Lattice Action for Low-energy SU(2) Gluodynamics

We study various representations of infrared effective theory of SU(2) Gluodynamics as a (quantum) perfect lattice action. In particular we derive a monopole action and a string model of hadrons from SU(2) Gluodynamics. These are lattice actions which give almost cut-off independent physical quantities even on coarse lattices. The monopole action is determined by numerical simulations in the infrared region of SU(2) Gluodynamics. The string model of hadrons is derived from the monopole action by using BKT transformation. We illustrate the method and evaluate physical quantities such as the string tension and the mass of the lowest state of the glueball analytically using the string model of hadrons. It turns out that the classical results in the string model is near to the one in quantum SU(2) Gluodynamics.Comment: 39 pages, 10 figure