# Evaluating topological charge density with symmetric multi-probing method

## TL;DR

This paper demonstrates that the symmetric multi-probing (SMP) method efficiently and accurately computes topological charge density in lattice QCD, especially on large lattices, by reducing computational time and errors compared to point sources.

## Contribution

The paper introduces and validates the SMP method for calculating the trace of the overlap Dirac matrix, showing improved efficiency and accuracy over traditional point source techniques.

## Key findings

- SMP method achieves high accuracy in trace calculation.
- Computing time with SMP is less affected by lattice size.
- Errors correlate with probing parameters and lattice volume.

## Abstract

We evaluated the topological charge density of SU(3) gauge fields on lattice by calculating the trace of overlap Dirac matrix employing symmetric multi-probing(SMP) method with 3 modes. Since the topological charge $Q$ for a given lattice configuration must be an integer number, it's easy to estimate the systematic error (the deviation of $Q$ to nearest integer). The results showed high efficiency and accuracy in calculating the trace of the inverse of a large sparse matrix with locality by using SMP sources, compared with that using point sources. We also showed the correlation between the errors and probing scheme parameter $r_{\mathrm{min}}$ as well as lattice volume $N_{L}$ and lattice spacing $a$. It was found that the computing time of calculating the trace by employing SMP sources is less dependent on $N_{L}$ than that by using point sources. Therefore the SMP method is very suitable for calculations on large lattices.