Effective noise reduction techniques for disconnected loops in Lattice QCD

TL;DR

The paper tackles the computational challenge of all-to-all propagators in Lattice QCD by introducing and benchmarking the Truncated Solver Method (TSM), in combination with partitioning, hopping parameter expansion (HPE), and truncated eigenmode acceleration (TEA). The authors demonstrate substantial variance reductions for disconnected loops, with gains up to factors of 6–30 depending on the observable and quark mass, and they apply these improvements to extract nucleon strangeness spin Δs and scalar content ⟨N|ss|N⟩. Their results indicate Δs is consistent with zero within errors, while scalar disconnected contributions remain noisy but informative for future, higher-precision studies. The work provides a practical, broadly applicable toolkit for efficient all-to-all propagator calculations in Lattice QCD and outlines directions for reducing gauge-noise-dominated errors in future simulations.

Abstract

Many Lattice QCD observables of phenomenological interest include so-called all-to-all propagators. The computation of these requires prohibitively large computational resources, unless they are estimated stochastically. This is usually done. However, the computational demand can often be further reduced by one order of magnitude by implementing sophisticated unbiased noise reduction techniques. We combine both well known and novel methods that can be applied to a wide range of problems. We concentrate on calculating disconnected contributions to nucleon structure functions, as one realistic benchmark example. In particular we determine the strangeness contributions to the nucleon, <N|ss|N>, and to the spin of the nucleon, Delta s.

Figures (11)

• Figure 1: Connected and quark-line disconnected current insertion into the nucleon.
• Figure 2: Truncated estimates of the zero momentum projected $\mathrm{Tr}\,(M^{-1}\Gamma)$, obtained after $n_{\rm t}$ CG solver iterations for $\Gamma={\mathbb 1}$ (left) and for $\Gamma=\gamma_3\gamma_5$ (right) at $\kappa_{\rm loop}=0.166$. The results are averaged over $300$ stochastic sources. Horizontal lines indicate the result with statistical errors at convergence.
• Figure 3: Estimates of the zero momentum projected $\mathrm{Tr}\,[(\kappa D)^kM^{-1}\Gamma]$ at $\kappa_{\rm loop}=0.166$ for $\Gamma = \mathbb{1}$ (left) and $\Gamma=\gamma_3\gamma_5$ (right). The errors are obtained from 300 estimates on one gauge configuration and the zero-order HPE contribution for $\Gamma=\mathbb{1}$ was calculated explicitly.
• Figure 4: Squared residual of the solver, as a function of the number of CG iterations. $\mathbb{P}_{n_{\rm ev}}$ denote the outcomes, after deflating the $n_{\rm ev}$ lowest modes.
• Figure 5: The numbers of iterations to convergence. For the heavier two $\kappa$ values (left) the comparison is with 70 sources per configuration, for $\kappa=0.1684$ (right) we only utilize 10 sources.