Erratum to: Topological sampling through windings

Erratum to: Eur. Phys. J. C (2021) 81:873 https://doi.org/10.1140/epjc/s10052-021-09677-6

In this Erratum we do minor corrections to Table 1, Fig. 4 and Equations (28) and (29). These modifications do not modify any conclusion.

1 First modification

In Table 1, in page 5 of 12, the values of autocorrelation times $\tau _{P}$ and $\tau _{Q^{2}}$ for the HMC rows should be doubled. Also, the first sentence of the caption, which reads “Simulation parameters and results for the pure gauge model using $N_{conf}=5 \cdot 10^5$ configurations in each case” should read “Simulation parameters and results for the pure gauge model using $N_{\text {conf}}=5 \cdot 10^5$ configurations for wHMC and $10^6$ for HMC”. The correct table with the corrected caption can be found in this document.

Table 1 Simulation parameters and results for the pure gauge model using $N_{\text {conf}}=5 \cdot 10^5$ configurations for wHMC and $10^6$ for HMC. The column “# jumps” indicates the number of transitions in which the topological charge changes by at least one unit. The integrator of the HMC step is tuned such that the acceptance is $\sim 90\%$

Figure 4, in page 4 of 12, is affected by the modifications of Table 1 and should be changed with a new plot depicting the new data, but its interpretation in text is unchanged. The new figure can be found in this document.

2 Second modification

Equations (28) and (29), in page 5 of 12, are wrong. Their interpretation in the text is unchanged, but a modification of the paragraph prior to Equation (28) is necessary. The suggested modification of the paragraph prior to Equation (28) until Equation (29) follows:

One can understand the improvement of the acceptance rate of the winding step with $L_{w}$. The change in the action when a winding is performed is restricted to the plaquettes at the boundary of $S_{w}$, and is due to the change in the links at the boundary—see violet region in Fig. 2. The change in the phase of the plaquette, $\delta \theta _{p}$, due to the transformation $\Omega ^{\pm }$ is therefore $\mp \frac{\pi }{2L_{w}}$. We refer to $\delta S_{w}$ as the outer boundary of the winding region. For sufficiently large $L_{w}$, the change in the phase of the plaquette is small and we can approximate

$$\begin{aligned} \Delta S \approx - \beta \frac{\pi }{2L_{w}} \sum _{p\subset \partial S_{w}}^{} \sin (\theta _{p})+ \beta \frac{\pi ^2}{4L_{w}^2} \sum _{p \subset \partial S_{w}}^{} \cos (\theta _{p}). \end{aligned}$$

(28)

The average of the first term of $\Delta S$ vanishes, while the last term averages to

$$\begin{aligned} \langle \Delta S \rangle \simeq \frac{\beta \pi ^2}{L_{w}}. \end{aligned}$$

(29)

Fig. 4

Autocorrelation time of $Q^2$ as a function of $\beta \propto a^{-2}$, as obtained with the wHMC and HMC algorithms