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We present a new class of statistical error reduction techniques for Monte-Carlo simulations. Using covariant symmetries, we show that correlation functions can be constructed from inexpensive approximations without introducing any systematic bias in the final result. We introduce a new class of covariant approximation averaging techniques, known as all-mode averaging (AMA), in which the approximation takes account of contributions of all eigenmodes through the inverse of the Dirac operator computed from the conjugate gradient method with a relaxed stopping condition. In this paper we compare the performance and computational cost of our new method with traditional methods using correlation functions and masses of the pion, nucleon, and vector meson in $N_f=2+1$ lattice QCD using domain-wall fermions. This comparison indicates that AMA significantly reduces statistical errors in Monte-Carlo calculations over conventional methods for the same cost.
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We demonstrate the new class of variance reduction techniques for hadron propagator and nucleon isovector form factor in the realistic lattice of $N_f=2+1$ domain-wall fermion. All-mode averaging (AMA) is one of the powerful tools to reduce the statistical noise effectively for wider varieties of observables compared to existing techniques such as low-mode averaging (LMA). We adopt this technique to hadron two-point functions and three-point functions, and compare with LMA and traditional source-shift method in the same ensembles. We observe AMA is much more cost effective in reducing statistical error for these observables.
We examine the Dyson-Schwinger equation for the fermion propagator in quenched QED in three and four dimension based on spectral representation with vertex ansatz which preserves Ward-Takahashi Identity.An appropriate renormalization within dispersion integral smoothes the threshold behaviour of the fermion self energy in three dimension.Thus we avoid the infrared singurality in three dimension.The behaviour of the fermion propagator in three dimension near the threshold is then found to be similar to the four dimensional one.There exisit analytic solutions for arbitrary gauges and the full propagators are expressed in terms of hypergeometric function in four dimension.There is a possibility of dynamical chiral symmetry breaking in four dimension with vanishing bare mass.
We discuss possible definitions of the Faddeev-Popov matrix for the minimal linear covariant gauge on the lattice and present preliminary results for the ghost propagator.
We present a multigrid based eigensolver for computing low-modes of the Hermitian Wilson Dirac operator. For the non-Hermitian case multigrid methods have already replaced conventional Krylov subspace solvers in many lattice QCD computations. Since the $gamma_5$-preserving aggregation based interpolation used in our multigrid method is valid for both, the Hermitian and the non-Hermitian case,
In the continuum the definitions of the covariant Dirac operator and of the gauge covariant derivative operator are tightly intertwined. We point out that the naive discretization of the gauge covariant derivative operator is related to the existence of local unitary operators which allow the definition of a natural lattice gauge covariant derivative. The associated lattice Dirac operator has all the properties of the classical continuum Dirac operator, in particular antihermiticy and chiral invariance in the massless limit, but is of course non-local in accordance to the Nielsen-Ninomiya theorem. We show that this lattice Dirac operator coincides in the limit of an infinite lattice volume with the naive gauge covariant generalization of the SLAC derivative, but contains non-trivial boundary terms for finite-size lattices. Its numerical complexity compares pretty well on finite lattices with smeared lattice Dirac operators.
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