Practical all-to-all propagators for lattice QCD

TL;DR

This paper introduces a practical all-to-all propagator for lattice QCD by combining exact low-lying eigenmodes of the hermitian Dirac operator $Q=oldsymbol{ ho}_5 M$ with diluted stochastic estimators to reconstruct $M^{-1}$. The method partitions $Q$ into $Q_0$ (low modes) and $Q_1$ (remainder), treating $Q_0^{-1}$ exactly and estimating $Q_1^{-1}$ with noise vectors that are diluted across time, spin, color, and space to dramatically reduce variance. The hybrid estimator uses a structured set of vectors $(w^{(i)},u^{(i)})$, enabling efficient construction of meson two-point functions with nonlocal operators and allowing variance reduction via noise recycling and optimized contractions. Results on a quenched $12^3 imes24$ lattice demonstrate substantial improvements over point-to-all propagators, enabling precise extraction of P-wave and static-light meson states and showing feasibility for isoscalar, disconnected, and heavy-light correlators. The approach generalizes to baryons and thermodynamic quantities and offers a practical path toward exploiting limited gauge configurations with high-precision all-to-all information.

Abstract

A new method for computing all elements of the lattice quark propagator is proposed. The method combines the spectral decomposition of the propagator, computing the lowest eigenmodes exactly, with noisy estimators which are 'diluted', i.e. taken to have support only on a subset of time, space, spin or colour. We find that the errors are dramatically reduced compared to traditional noisy estimator techniques.

Figures (18)

• Figure 1: The pseudoscalar propagator computed with and without time dilution.
• Figure 2: A cartoon of possible deviations of the stochastic estimates of the exact solution (at $N_{dil}=N_{\text{max}}$) for different dilution paths. Simply adding noise vectors will give a $1/\sqrt{N}$ behaviour. We have found that simple dilutions typically follow the behaviour exhibited by the bottom curve.
• Figure 3: The pion effective mass from 50, 100 eigenvectors and from the hybrid method with 100 and a time-diluted noise vector.
• Figure 4: Effective masses for isovector mesons with 100 eigenvectors, time, colour, spin and space-even-odd dilution and 75 configurations.
• Figure 5: Comparison of spectrum from all-to-all and point-to-all propagators. The open symbols are point-to-all, the closed symbols all-to-all with 100 eigenvectors and time, colour, spin and space-even-odd dilution.