Topological susceptibilty in lattice QCD with exact chiral symmetry -- the index of overlap-Dirac operator versus the clover topological charge in Wilson flow

Using an ensemble of 535 gauge configurations (on the $24^4 \times 48 $ lattice with $ a \simeq 0.06 $~fm and $ M_{\pi} \simeq 260 $~MeV) which are generated by hybrid Monte Carlo (HMC) simulation of $N_f=2$ lattice QCD with the optimal domain-wall quark, we compute the index of the overlap-Dirac operator, and also measure the clover topological charge in the Wilson flow, $Q_{\text{clover}}(t) $, by integrating the flow equation from $ t = 0 $ to $ t = 128 $ with $\delta t = 0.01 $. We observe that $Q_{\text{clover}}(t) $ of each configuration converges to a value close to an integer, and its nearest integer $Q_c(t) = \text{round} [Q_{\text{clover}}(t)] $ becomes invariant for $ t \ge t_c $, with the $ \max\{t_c \} \sim 77 $ for all 535 configurations. For each configuration, we compare the asymptotically-invariant $ Q_c $ with the index of overlap-Dirac operator at $t=0$. It turns out that there are 167 configurations with $Q_c \ne \text{index}(D_{o}) $, amounting to $31.2\%$ of the total 535 configurations. However, the histograms of $ Q_c $ and $ \text{index}(D_o) $ are almost identical. Consequently, the topological susceptibility using the asymptotically-invariant $ Q_c $ agrees with that using the index of overlap-Dirac operator at $ t=0 $. This implies that the topological susceptibility in lattice QCD with exact chiral symmetry can be obtained from the asymptotically-invariant $ Q_c $ in the Wilson flow.