Hybrid Monte Carlo Algorithm Research Articles

Parallelizing QCD with dynamical fermions on a CRAY multiprocessor system

Quantum Chromodynamics, the theory of the nuclear forces, is simulated on a space-time lattice by means of a Hybrid Monte Carlo algorithm which incorporates dynamical quarks exactly. The large computational needs of the program necessitate to use modern supercomputers like the CRAY Y-MP that achieve their high computing speed by using both vector and parallel hardware. This paper gives a short introduction into the CRAY Y-MP system architecture and the autotasking concept which offers a way for finding automatically parallelism in existing FORTRAN programs. In detail, the paper describes the application as well as the optimization and transformation process which had to be done to improve the performance. After some code modifications a speedup of nearly 7 on a CRAY Y-MP8/832 could be achieved, leading to a performance of about 1.65 GFLOPS.

Some recent results on lattice fermions

I discuss some recent work on the Hybrid Monte Carlo algorithm as applied to the simulation of dynamical fermions as well as some interesting aspects of the nature of the phase transition at κ c in the Wilson fermion formulation.

A comparison of numerical algorithms for dynamical fermions

Abstract The Langevin algorithm with bilinear noise and the hybrid Monte Carlo algorithm for dynamical fermions are applied to the simulation of the two-dimensional Gross-Neveu model with Wilson fermions. The hybrid Monte Carlo algorithm is implemented both with the classical “leap-frog” and with a new improved discretization scheme. The results are compared to theoretical predictions obtained by 1/N expansion and to previous simulations based on the pseudo-fermion algorithm. In the determination of the chiral transition point the three algorithms behave quite differently, and only the hybrid Monte Carlo method gives satisfactory results. The three methods are essentially equivalent in the evaluation of vacuum expectation values in a definite physical phase.

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.

QCD beta function with dynamical Wilson fermions.

We present data for the QCD \ensuremath{\beta} function with two flavors of light Wilson fermions. We use the hybrid Monte Carlo algorithm and the method of matching of ratios of products of Wilson loops. At \ensuremath{\beta}=6/${\mathrm{g}}^{2}$=6.0 perturbative scaling has not yet set in, whereas at \ensuremath{\beta}=6.8 it clearly has. Since the matching is done for the integral of the \ensuremath{\beta} function this scaling is expected above \ensuremath{\beta}=6.0. .AE

Classical mechanics on group manifolds and applications to hybrid Monte Carlo

We formulate classical mechanics on a general group manifold, defining the natural symplectic structure as the 2-form ω ≏ -Σ i d(p iθ i) , with p being the momentum and θ the left-invariant coframe. We verify that our formulation has the desired invariance properties and satisfies Lioville's theorem, and thus satisfies the conditions required for use in the hybrid Monte Carlo algorithm. We show that the equations of motion reduce to those previously hypothesized when written in terms of the constrained variables usually used to formulae lattice gauge theories, and discuss their discretization.

The finite temperature phase transition in four flavour QCD on an 8×12 3 lattice

We present results of a numerical study of lattice QCD with four dynamical flavours of staggered fermions, performed by using a hybrid Monte Carlo algorithm on an 8×12 3 lattice. We find a rapid change in the average value of the Polyakov loop at β c=5.25±0.025 for a quark mass ma=0.025; at this mass value, the behaviour of the chiral order parameter, 〈 ΨΨ〉 does not yet allow an independent determination of the transition point. Using existing hadron mass calculations, the value of β c we have obtained here would lead to a transition temperature T∼100 MeV.

QCD with dynamical Wilson fermions.

We present results for the QCD spectrum and the matrix elements of scalar and axial-vector densities at β=6/g2=5.4, 5.5, 5.6. The lattice update was done using the hybrid Monte Carlo algorithm to include two flavors of dynamical Wilson fermions. We have explored quark masses in the range ms≤mq≤3ms. The results for the spectrum are similar to quenched simulations and mass ratios are consistent with phenomenological heavy-quark models. The results for matrix elements of the scalar density show that the contribution of sea quarks is comparable to that of the valence quarks. This has important implications for the pion-nucleon σ term.

An improved hybrid Monte Carlo method

The hybrid Monte Carlo algorithm does not completely remove problems with ergodicity in the molecular dynamics trajectories unless the length of each trajectory, τ 0, is kept shorter than the period of the fastest mode of the system, 2 π/ ω max . The correlations which remain when larger values of τ 0 are used may be eliminated by randomizing not only the molecular dynamics velocities but also the length of the trajectory, τ 0, at the beginning of each new trajectory. This allows τ 0 to be increased to the time scale of the slowest modes in the system, 1/ ω min , and reduces the correlation time (and therefore the computer time used) by a factor of around ω max /2 ω min .

Renormalized yukawa couplings from the lattice

Renormalized Yukawa couplings, y R , are evaluated numerically in the single component lattice λϕ 4 theory coupled to staggered fermions. For a wide range of bare couplings, the renormalized y R is found to be “small” (⪅1.5) whenever physical scalar and fermion masses have dropped below about one half the cutoff scale. In these cases Yukawa coupling generated fermion masses are never much larger than the symmetry breaking scale ν = 〈 ϕ R 〉. The simulations have been performed using the hybrid Monte Carlo algorithm for dynamical fermions.

Hybrid Monte Carlo algorithm as applied to dynamical Wilson fermion

The hybrid Monte Carlo algorithm is tested for lattice QCD with Wilson quark action. A variety of technical points are examined. As an application the finite temperature behavior of QCD with Wilson quark action is studied.

The lattice λφ4 model with Yukawa couplings to staggered fermions

An investigation of the λφ 4 theory coupled via Yukawa couplings to fermions has been initiated on the lattice. Several algorithms for dynamical fermions have been tested on this model including hybrid molecular dynamic and hybrid Monte Carlo algorithms. Some preliminary results for renormalized Yukawa couplings are presented.

The beta function for QCD with dynamical Wilson fermions

We present preliminary data for a calculation of the QCD beta function with two flavors of Wilson fermions. For the simulation of QCD we use the Hybrid Monte Carlo algorithm and for the calculation of the beta function (or rather its integral over a scale change of 2) we use the method of matching of ratios of plaquettes and ratios of products of plaquettes with the same perimeter. We find that at β = 6.0 perturbative scaling has not set in where as at β = 6.8 it clearly has. The transition region is being probed in detail and early indications are that it is near β = 6.25. Since the matching is done for the integral of the beta function this implies that perturbative scaling is expected above β = 6.0.

Simulating QCD with dynamical Wilson and staggered fermions

I present a summary of the work done during the last year by the Los Alamos lattice collaboration (C. Baillie, R. Gupta, G. Kilcup, G. Guralnik, A. Patel and S. Sharpe) to study QCD with the Hybrid Monte Carlo Algorithm. The specific questions we have addressed are the finite temperature transition for both staggered and Wilson fermions and screening in the heavy q q potential with Wilson fermions. I also present a discussion of the efficiency of the fermion update algorithm.

Chiral transition in the SU(2) Higgs model with dynamical fermions

We studied the phase diagram of the lattice formulation of the SU(2)-Higgs model with dynamical fermions. The Yukawa coupling of the fermion fields to the Higgs field is set to zero. The dynamical fermions are included via the hybrid Monte Carlo algorithm. Extensive comparison is made with previous results obtained by Aoki, Lee and Shrock simulating the same model by the Langevin method. We argue that the current status of lattice investigations does not eliminate the possibility of the Abbott-Farhi phase.

Performance of new algorithms for self-avoiding walks with fixed endpoints

We discuss the dynamic critical behaviour of a hybrid Monte Carlo algorithm for the self-avoidings walks of variable length and fixed endpoints. A local algorithm is augmented by non-local cut-and-paste moves, which speed up equilibration within the subspace of walks with fixed length. The percentage of non-local moves can be optimized to get the lowest dynamic critical exponent for the autocorrelation time in CPU units.

Higher order hybrid Monte Carlo

The Hybrid Monte Carlo algorithm requires a mapping on phase space which is area-preserving, reversible, and at least approximately conserves energy. We are thus naturally lead to investigate schemes for the numerical integration of Hamilton's equations. First we generalize the “leapfrog” scheme to higher order in the time step δτ, and show why we cannot find an initial “half-step” which satisfies the constraints required for Hybrid Monte Carlo. We then introduce a new symmetric integration scheme which can be used to any order in δτ, and we prove that it too is exactly reversible and area-preserving order-by-order in δτ. Finally, we show that the symmetric integration scheme conserves energy exactly for the simple harmonic oscillator, and we give numerical results for the anharmonic oscillator.

Study of the φ4-model with Yukawa coupling and dynamical fermions

We study the one-component φ 4-model with Yukawa coupling and dynamical fermions. The unbiased Hybrid Monte Carlo algorithm is applied to this model for the first time. Our results show that inside the broken symmetry phase all the measured quantities change continuously when the bare Yukawa coupling Y is increased and the fermion sector behaves approximately like a free fermion with mass m f related to the scalar field vacuum expectation value 〈 φ〉 by m f = Y 〈 φ〉 even for large Y. We do not find evidence for a phase transition induced by large Y inside the broken symmetry phase. We also present a method to simulate the effective potential in the presence of dynamical fermions. For small Y, we demonstrate that the measured effective potential agrees well with lowest order perturbative calculation in Y.

Hybrid Monte Carlo and quantum chromodynamics

In this paper we study the properties of the hybrid Monte Carlo algorithm applied to lattice quantum chromodynamics including the dynamical effects of fermions. We find that this “exact” method performs very well on 4 4 lattices, and shows considerable promise for being a practicable means with which to undertake realistic Monte Carlo computations of fermionic systems.

COMPARISON OF UPDATE ALGORITHMS FOR PURE GAUGE SU(3)

We show that the overrelaxed algorithm of Creutz and of Brown and Woch is the optimal local update algorithm for simulation of pure gauge SU(3). Our comparison criterion includes computer efficiency and decorrelation times. We also investigate the rate of decorrelation for the Hybrid Monte Carlo algorithm.