${\chi }_{\mathrm{top}}(T\gg T_{\mathrm{c}})$ in pure-glue QCD through reweighting

We calculate the topological susceptibility at $2.5T_{\mathrm{c}}$ and $4.1T_{\mathrm{c}}$ in SU(3) pure Yang-Mills theory. We define topology with the help of gradient flow and we largely overcome the problem of poor statistics at high temperatures by applying a reweighting technique in terms of the topological charge, measured after a specific small amount of gradient flow. This allows us to obtain a sample of configurations which compares topological sectors with good statistics, with enhanced tunneling between topologies. We quote continuum extrapolated results at these two temperatures and conclude that our method is viable and can be extended without new conceptual problems to the case of full QCD with fermions.

Figure 1

Schematic depiction of how $W$ is built with $N_{\mathrm{int}}=5$ and $s=0.4$. The red dot indicates the measured $Q^{\prime }$, while the orange points and arrows show the change in the $W$ function. The solid black line shows the updated $W$ function, the dashed line is the updated part of $W$ before the update. Top: $Q^{\prime }=0.3$. Bottom: $Q^{\prime }=0.64$.

Figure 2

$W$ function of $6\times 16^3$ lattice at $2.5T_{\mathrm{c}}$ (Lattice ${\mathrm{A}}_1$).

Figure 3

$W$ functions of $8\times 16^3$ lattice. Left: $2.5T_{\mathrm{c}}$ (Lattice ${\mathrm{A}}_2$). Right: $4.1T_{\mathrm{c}}$ (Lattice ${\mathrm{B}}_{2\mathrm{c}}$).

Figure 4

Finite volume dependence of the susceptibility at $N_{\tau }=8$ and $T=4.1T_{\mathrm{c}}$ (Lattices ${\mathrm{B}}_{2\mathrm{a}}$ through ${\mathrm{B}}_{2\mathrm{e}}$), using Zeuthen flow with $t=2.4a^2$ and three different values of $Q_{\text{thresh}}$ (points have been displaced for reasons of visibility).

Figure 5

Continuum extrapolation at $T=2.5T_{\mathrm{c}}$ based on lattices ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$, carried out in terms of $\chi (T)$ directly (right) and $\ln (\chi (T))$ (left) using Zeuthen flow with $t=2.4a^2$ and $Q_{\text{thresh}}=0.7$.

Figure 6

Same as Fig. 5 except for $T=4.1T_{\mathrm{c}}$, using lattices ${\mathrm{B}}_1$, ${\mathrm{B}}_{2\mathrm{d}}$, ${\mathrm{B}}_3$. In addition we have shown separately the measured values for two amounts of flow $t=1.2a^2$ and $t=2.4a^2$ and for two flow actions (Wilson and Zeuthen). Note that the linear-extrapolated continuum limit is negative for all choices.

Figure 7

Piece of a Markov-chain history of $Q^{\prime }$ against measurement number, for lattice ${\mathrm{B}}_{2\mathrm{d}}$. The reweighting allows efficient sampling in three regions, $0<Q^{\prime }<0.27$, $0.20<Q^{\prime }<0.85$, and $0.8<Q^{\prime }<1.05$, but has difficulty moving between these regions.