Universality check of Abelian monopoles

We study the Abelian projected $SU(2)$ lattice gauge theory after gauge fixing to the maximally Abelian gauge (MAG). In order to check the universality of the Abelian dominance we employ the tadpole improved (TI) tree level action. We show that the density of monopoles in the largest cluster (the IR component) is finite in the continuum limit which is approximated already at relatively large lattice spacing. The value itself is smaller than in the case of Wilson action. We present results for the ratio of the Abelian to non-Abelian string tension for both Wilson and TI actions for a number of lattice spacings in the range $0.06\mathrm{fm}<a<0.35\mathrm{fm}$. These results show that the ratio is between 0.90 and 0.95 for all considered values of lattice couplings and both actions. We compare the properties of the monopole clusters in two gauges—in MAG and in the Laplacian Abelian gauge (LAG). Whereas in MAG the infrared component of the monopole density shows a good convergence to the continuum limit, we find that in LAG it is even not clear whether a finite limit exists.

Figure 1

Non-Abelian potentials for the TI action obtained without (circles, SM only) and after (squares, with HYP) hypercubic blocking (both with spatial smearing) vs $R/a$ for $T/a=5$ at ${\beta }_{\mathrm{imp}}=3.4$.

Figure 2

The monopole cluster length distribution $N(l)$ at $\beta =2.55$ for the Wilson action (left) and at ${\beta }_{\mathrm{imp}}=3.5$ for the TI action (right).

Figure 3

The monopole densities in MAG for the case of the TI action. The dashed line shows a quadratic fit.

Figure 4

Comparison of the monopole densities obtained with TI (full symbols) and Wilson actions (open symbols) in MAG: left—IR monopole density; right—UV monopole density.

Figure 5

The small (UV) clusters’ length distribution in MAG for the Wilson action at various $\beta$ (left) and for the TI action at various ${\beta }_{\mathrm{imp}}$ (right). Curves are fits to Eq. (14).

Figure 6

The constrained UV monopole density derived from the small clusters’ length distribution Eq. (15) for TI (left) and Wilson (right) actions.

Figure 7

The constrained UV monopole density derived from the small clusters’ length distribution and corrected for ambiguous clusters according to Eq. (16), for TI (left) and Wilson (right) actions. The curves show fits by Eq. (18). The data were lifted by a constant for better readability of the figures.

Figure 8

The monopole cluster length distribution $N(l)$ in LAG at $\beta =2.6$ for the Wilson action (left) and at ${\beta }_{\mathrm{imp}}=3.5$ for the TI action (right).

Figure 9

Various monopole densities in LAG for TI action (left) and for Wilson action (right). Lines are drawn to guide the eye.

Figure 10

The Abelian potential $V_{ab}(R,T)$ in MAG at ${\beta }_{\mathrm{imp}}=3.3$ vs $T/a$ for $R/a=4$ for 3 (open circles) and 100 (full circles) smearing sweeps.

Figure 11

Ratio between the Abelian string tension (in MAG) and the non-Abelian string tension vs lattice spacing for TI action (left) and for Wilson action (right).