Investigating a two-level algorithm for fermionic observables

TL;DR

This work demonstrates that a two-level sampling strategy, when combined with distillation and a propagator factorisation in temporal domains, can drastically reduce the variance of disconnected fermionic observables in quenched QCD. By decomposing the quark propagator into domain-local contributions and applying independent submeasurements, the authors achieve a variance scaling of $1/N_1^2$ up to boundary effects, with substantial improvements at large operator separations. They validate the approach on pure SU(3) gauge theory at $\beta=6.0$, using thick frozen regions and high statistics, and show that including the leading corrections computed with two-level sampling further enhances accuracy, enabling robust extraction of singlet effective masses. The results establish a rigorous, low-cost benchmark for future dynamical QCD studies of glueballs and isosinglet mesons, where the combination of distillation and two-level sampling can address the long-standing signal-to-noise challenges. These findings pave the way for applying such multilevel strategies to full QCD and multi-hadron analyses requiring extensive Wick contractions and all-to-all propagators.

Abstract

We investigate the combination of a two-level sampling algorithm with distillation techniques to compute disconnected fermionic correlation functions. The method relies on a factorization of the quark propagator into domain-local contributions that depend only on the gauge fields within overlapping temporal regions, enabling independent submeasurements of each term through a two-level sampling strategy. The two-level estimators exhibit the expected $1/N_1^2$ scaling of the variance, up to exponential boundary effects, and achieve an exponential reduction of statistical errors at nearly the same computational cost as standard sampling. The method is tested on pure gauge ensembles, providing a controlled benchmark for its forthcoming application to dynamical QCD studies of glueball and isosinglet meson correlation functions.

Figures (7)

• Figure 1: Domain decomposition of the temporal lattice extent used in this analysis. The temporal extent is factorised in four regions: $\Lambda=\Lambda_0 \oplus \Lambda_1 \oplus \Lambda_2 \oplus \Lambda_3$, and we consider the two overlapping domains $\Omega_0= \Lambda_3 \oplus \Lambda_0 \oplus \Lambda_1$, and $\Omega_1= \Lambda_1 \oplus \Lambda_2 \oplus \Lambda_3$.
• Figure 2: Scalar one-point functions obtained with the full propagator (green), the leading approximation (blue), and the leading approximation plus first correction (orange). Blue vertical bands mark the frozen regions $\Lambda_1$ and $\Lambda_3$. Estimates follow Eqs. \ref{['1pt_std']}–\ref{['1pt_approx_omega1']}.
• Figure 3: Statistical errors of the two-level estimators for the disconnected scalar two-point function before (open circles) and after (filled circles) adding the global correction of Equation \ref{['correction']}. Open (filled) circles correspond to Equations \ref{['c2pt_2lvl_approx']} (\ref{['c2pt_2lvl_corrected']}). Blue vertical bands indicate the frozen regions $\Lambda_1$ and $\Lambda_3$. One loop is fixed at $t_0=40a$ while the other is varied. Dashed lines show ideal $1/N_1$ error scaling normalised to $\sigma_C^{(0)}(N_1\!=\!1)$ for $N_1=1,10,200,1000$. Yellow points: statistical error of the one-level estimate of the first correction (Eq. \ref{['firstcorr_2pt']}) with $N_0=101$, $N_1=200$.
• Figure 4: (left) Signals and errors of the scalar disconnected two-point functions with the full solver (black) from standard sampling, with first approximation (blue), and with second approximation (orange) of the quark propagator from two-level sampling. The source is fixed at $y_0=24a$ and the sink position $x_0$ is varied. (right) Statistical errors: empty markers are approximated results and full markers are corrected results. The grey crosses show the signal (left plot) and error (right plot) of the third term in Eq. \ref{['firstcorr_2pt']} with one-level sampling using $N_0=101$ and $N_1=200$ configurations.
• Figure 5: Noise-to-signal ratios of weighted disconnected two-point functions for $\mathrm{O}_\Gamma=\bar{\psi}\Gamma\psi$ and $\Gamma=I,\,\gamma_5,\,\gamma_\mu,\,\gamma_i\gamma_j\epsilon_{ijk},\,\gamma_5\gamma_k$. One-level: $N_0=101$, $N_1=200$; two-level: $N_0=101$, $N_1=200$.