Deconfinement, gradient, and cooling scales for pure SU(2) lattice gauge theory

We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its continuum limit using the deconfining phase transition, the gradient flow and the cooling flow to set the scale. For the gradient and cooling scales we explore three different energy observables and two distinct reference values for the flow time. When the aim is to follow scaling towards the continuum limit, one gains at least a factor of 100 in computational efficiency by relying on the gradient instead of the deconfinement scale. Using cooling instead of the gradient flow one gains another factor of at least 34 in computational efficiency on the gradient flow part without any significant loss in the accuracy of scale setting. Concerning our observables, the message is to keep it simple. The Wilson action itself performs as well as or even better than the other two observables explored. Two distinct fitting forms for scaling are compared, of which one connects to asymptotic scaling. Differences of the obtained estimates show that systematic errors of length ratios, though only about 1%, can be considerably larger than statistical errors of the same observables.

Figure 1

Reweighting of the Polyakov loop susceptibility on our ${64}^{3}10$ lattice.

Figure 2

Ratios of the Polyakov loop susceptibilities around the ${\beta }_{\max }$ value of our previous figure.

Figure 3

Cooling of the topological charge on a $44^4$ lattice at $\beta =2.816$.

Figure 4

Gradient flows $y_i(t)$ for the energy densities $E_0$, $E_1$ and $E_4$ at $\beta =2.3$ on an $8^4$ lattice ($t$ on lower abscissa) and at $\beta =2.574$ on a $40^4$ lattice ($t$ on upper abscissa). The up-down order in the legend agrees on the right-hand side with that of the curves.

Figure 5

Gradient flow ratios as functions of $y$. Outer curves: $s_i(12,2.43)/s_i(8,2.3)$. Inner curves: $s_i(24,2.43)/s_i(16,2.3)$.

Figure 6

Cooling flows $y_i(t)$ for the energy densities $E_0$, $E_1$ and $E_4$ at $\beta =2.3$ on an $8^4$ lattice ($t$ on lower abscissa) and at $\beta =2.574$ on a $40^4$ lattice ($t$ on upper abscissa).

Figure 7

Scaling corrections of order $a^2$ for ratios $L_i/L_{10}$. Here and in the next figures some data are slightly shifted for better visibility. To label all fits, some labels are attached to the lines and others put into the legend. The up-down order in the legend mirrors the up-down order in the plot.

Figure 8

Enlargement of the continuum approach of Fig. 7 for the $E_0$, the $L_{11}$ and the deconfinement scales.

Figure 9

Enlargement of the continuum approach of Fig. 7 for the $E_4$ scales.

Figure 10

Asymptotic scaling.

Figure 11

Scaling corrections of the $E_0$, the $L_{11}$ and the deconfinement scale ratios with respect to $L_{10}$.

Figure 12

Figure 11 plotted in the $(1/L_{10}{)}^2$ range of Fig. 8.

Figure 13

Three fits of the deconfinement length scale $L_0$ versus $(1/L_{10}{)}^2$.