We smear quenched lattice QCD ensembles with lattice volume 32 3 × 8 by using Wilson flow. Six ensembles at temperature near the critical temperature T c corresponding to the critical inverse coupling β c ¼ 6.06173ð49Þ are used to investigate the localization of topological charge density. If the effective smearing radius of Wilson flow is large enough, the density, size and peak of Harrington-Shepard (HS) caloron-like topological lumps of ensembles are stable when β ≤ 6.050, but start to change significantly when β ≥ 6.055. The inverse participation ratio (IPR) of topological charge density shows similar results, it begins to increase when β ≥ 6.055 and is stable when β ≤ 6.050. The pseudoscalar glueball mass is extracted from the topological charge density correlator (TCDC) of ensembles at T ¼ 1.19T c , and 1.36T c , the masses are 1.915(98) and 1.829(123) GeV respectively, they are consistent with results from conventional methods.

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 , as well as lattice volume and lattice spacing a. It is found that the computational time for calculating the trace by employing the SMP sources is less dependent on than by using point sources. Therefore, the SMP method is very suitable for calculations on large lattices.

The topological charge density and topological susceptibility are determined by multi-probing approximation using overlap fermions in quenched SU(3) gauge theory. Then we investigate the topological structure of the quenched QCD vacuum, and compare it with results from the all-scale topological density, the results are consistent. Random permuted topological charge density is used to check whether these structures represent underlying ordered properties. Pseudoscalar glueball mass is extracted from the two-point correlation function of the topological charge density. We study 3 ensembles of different lattice spacing a with the same lattice volume 16 3 ×32, the results are compatible with the results of all-scale topological charge density, and the topological structures revealed by multi-probing are much closer to all-scale topological charge density than that by eigenmode expansion.