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

Abstract

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.

Comments: 11 pages, 5 EPS figures