# Evaluating the topological charge density with the symmetric multi-probing method

• We evaluate the topological charge density of SU(3) gauge fields on a lattice by calculating the trace of the overlap Dirac matrix employing the symmetric multi-probing (SMP) method in 3 modes. Since the topological charge Q for a given lattice configuration must be an integer number, it is easy to estimate the systematic error (the deviation of Q to the nearest integer). The results demonstrate a high efficiency and accuracy in calculating the trace of the inverse of a large sparse matrix with locality by using the SMP sources when compared to using point sources. We also show the correlation between the errors and probing scheme parameter $r_{\min}$ , as well as lattice volume $N_{L}$ and lattice spacing a. It is found that the computational time for calculating the trace by employing the SMP sources is less dependent on $N_{L}$ than by using point sources. Therefore, the SMP method is very suitable for calculations on large lattices.
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Guang-Yi Xiong, Jian-Bo Zhang and You-Hao Zou. Evaluating the topological charge density with the symmetric multi-probing method[J]. Chinese Physics C. doi: 10.1088/1674-1137/43/3/033102
Guang-Yi Xiong, Jian-Bo Zhang and You-Hao Zou. Evaluating the topological charge density with the symmetric multi-probing method[J]. Chinese Physics C.
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## Evaluating the topological charge density with the symmetric multi-probing method

###### Corresponding author: You-Hao Zou, 11006067@zju.edu.cn
• Department of Physics, Zhejiang University, Zhejiang 310027, China

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

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