International Journal of Modern Physics C

This paper is divided into three parts: an introduction to the Hybrid Monte Carlo (HMC) algorithm; a detailed discussion of its implementation for Quantum Chromodynamics (QCD) in CMIS on the Connection Machine (CM-2); and a summary of recent analytic results on the behavior of HMC for free field theory. We present a detailed description of the architecture of the CM-2 and explain how it can execute the Conjugate Gradient (CG) solver for HMC at 6.35 GFlops/s.

Performance of the Intel iPSC/860 parallel processor for Quantum Chromodynamics codes with dynamical fermions is described. After reviewing the hardware and software environments provided by the manufacturer, the data structures appropriate for the QCD code are described. Techniques for maximum performance are briefly discussed. We achieve a speed of 10–15 Mŕlops per node depending upon how many lattice sites are located on each node.

The acceptance probability in Hybrid Monte Carlo simulations of QCD, in particular, its dependence on the lattice size, quark mass and the coupling, is discussed. Results on the tuning of parameters required in order to achieve low autocorrelations in the 2-d XY model are presented. In both phases of this model, the dynamical critical exponent is close to 2 for runs with trajectory lengths between 1 and 3 molecular dynamics units.

In this talk we present results of a recent investigation of the topological vacuum structure in SU(2) lattice gauge theory with Wilson fermions. An effective action given by the hopping parameter expansion up to the 6-th order is used. To check the reliability of this approximation we compare it with the outcomes of an exact algorithm. The topological charge is measured according to a geometric prescription.

The worldline quantum Monte Carlo method is a computational technique for studying the properties of many-electron and quantum-spin systems. In this paper, we describe our efforts in developing an efficient implementation of this method for the massively-parallel Connection Machine CM-2. We discuss why one must look beyond the obvious parallelism in the method in order to reduce interprocessor communication and increase processor utilization, and how these goals may be achieved using a plaquette-based data representation. We also present performance statistics for our implementation and sample calculations for the spinless fermion model.

We present in this article results of Quantum Monte Carlo simulations for the three-band Hubbard model obtained both with the grand canonical approach as well as with the T=0 formalism. A direct comparison of Quantum Monte Carlo results with experimental values for the charge-transfer gap leads to a set of parameters for the three-band Hubbard model that consistently describes salient features of the high-T_c superconductors. We find long-range antiferromagnetic order in the undoped case as well as an incommensurate structure for a doping concentration of δ≈0.20. Moreover the magnetic susceptibility as a function of doping δ is in a qualitative agreement with experiments. Concerning dynamical properties, we have evaluated the spectral density . The dispersion of the state with the lowest excitation energy is in very good agreement with experimental photoemission and inverse photoemission experiments for a doping concentration of δ=0.25. In the superconducting regime, we find that the interaction vertex for the pairing correlation function in the extended s-channel shows a maximum for δ~0.20, resembling the dependence of T_c on doping. This feature is not present for other symmetries of the order parameter or in other parameter regions.

All known Quantum-Monte-Carlo algorithms for fermions suffer from the so-called “minus-sign-problem” which is detrimental to the application of these simulation methods to fermion systems at very low temperatures and/or of very many lattice sites. We identify the origin of this fundamental problem, demonstrate that it is a very general feature, not necessarily related to the presence of fermionic degrees of freedom. We describe an novel algorithm which does not suffer from the minus-sign problem. Illustrative results for the two-dimensional Hubbard model are presented

We study a model for high-T_c superconductors describing both the strong electronic correlations and a coupling of the carriers to anharmonic phonons. The Projector Quantum Monte Carlo Scheme has been employed to investigate ground state properties. We analyze the superconducting properties of the model by directly addressing to the existence of Off-Diagonal Long Range Order (ODLRO). We review the numerical algorithm and discuss the possibility to observe ODLRO. In the presented model, we find ODLRO pointing towards superconductivity instead of the presence of strong electronic correlations. We discuss several features of the superconducting condensate.

We discuss blockspins for staggered fermions, i. e. averaging and interpolation procedures which are needed in a real space renormalization group approach to gauge theories with staggered fermions and in a multigrid approach to the computation of gauge covariant propagators. The discussion starts from the requirement that the symmetries of the free action should be preserved by the blocking procedure in the limit of a pure gauge. A definition of an averaging kernel as a solution of a gauge covariant eigenvalue equation is proposed, and the properties of a corresponding interpolation kernel are examined in the light of general criteria for good choices of blockspins. Some results of multigrid computations of bosonic propagators in an SU(2) gauge field in 4 dimensions are also presented.

The convergence rates for inverting the Dirac matrix in the SU(2) gauge theory on a square two-dimensional lattice are compared for different lattice sizes in the physical region. The Parallel-Transported Multigrid algorithm (PTMG) turns out to be superior to the Conjugate-Gradient (CG) approach. As the lattice size L increases (together with the correlation length ξ) the PTMG becomes more and more efficient. On a 256×256 lattice with the bare quark mass m_bare=0.0005 and ξ=20, the PTMG is at least ten times faster than the CG. It seems reasonable to expect similar results in four dimensions.

We describe a ground state decomposition multigrid method for the inversion of fermion matrices and we test it for staggered fermions coupled to U(1) and SU(2) gauge fields on a two dimensional lattice.

I discuss a multigrid algorithm for calculating fermionic propagators in the lattice gauge theories. In order to preserve the infrared physics on the coarse lattice the coarse grid operators are obtained by an approximate real-space renormalization-group transformation, namely the Migdal-Kadanoff transformation. The resulting projection and the interpolation operators have a simple interpretation in terms a path integral for a fermionic particle on the lattice. The method was tested in two-dimensions in the presence of U(1) gauge-field, where it reduces the critical slowing down for the large values of β. The algorithm was implemented on the Connection-Machine.

We study the finite temperature chiral phase transition of SU(3) lattice QCD in the strong coupling limit using the monomer dimer polymer MDP representation of the partition function. By Monte Carlo simulation we measured the chiral condensate and the susceptibility as a function of temperature for various values of the quark mass. The effect of a global symmetry of the lattice action is discussed in terms of critical exponents.

It is shown that the sign of the fermionic determinant in the projector Monte Carlo method is directly related to a topological invariant. A key ingredient to obtain this result is the identification of the appropriate manifolds to describe the evolution of a fermionic trial wavefunction. They allow for a purely geometrical consideration of the minus-sign problem.

Kernel control of the complex Langevin simulation is discussed in detail in a simple model with one degree of freedom. Some comments are made on the so-called segregation phenomenon. We report about results obtained in close collaboration with K. Okano (Univ. of Tokuyama).

The cluster Monte Carlo method, where variables are updated in groups, is very efficient at second order phase transitions. Much better results can be obtained with less computer time. This article reviews the method of Swendsen and Wang and some of its applications.

Percolation cluster Monte Carlo algorithms for nonlinear σ-models on the lattice are reviewed with special emphasis on their possible generalizations. While they have been found to practically eliminate critical slowing down for the standard O(n) invariant vector models, their extension to other physically similar models — like RP_n−1 and SU(n)×SU(n) chiral models — is less straight forward than one might have thought. I outline the present situation in this area of research. In the second part of my talk I described a numerical calculation of a physical running coupling constant in the O(3) model. This represents an application of the cluster technique in a preparatory study for a later lattice gauge theory calculation. This material can be found in Ref. 11.

An efficient algorithm for identifying the independent cluster flips in a Z(2) lattice gauge theory with stochastic percolation is presented. It applies Gaussian elimination to the incidence matrix, with special attention payed to the pivoting strategy and appropriate linked list structures. At the critical point of the 3-dimensional pure gauge model, storage and cpu-time scale like L³ and L³ log L, respectively. The algorithm is also applied to the 3-D Z(2) gauge-Higgs model along the self-dual line. A second order critical line is found, endpoints and critical indices are determined. The cluster update is superior to the heat bath in the region near the triple point.

We discuss a new cluster algorithm that completely eliminates critical slowing down for surface models of the SOS (solid-on-solid) type.