New Abelian-like monopoles and the dual Meissner effect

Violation of non-Abelian Bianchi identity can be regarded as $N^2-1$ Abelian-like monopole currents in the continuum SU(N) QCD. Three Abelian-like monopoles, when defined in SU(2) gluodynamics on the lattice à la DeGrand–Toussaint, are shown to have the continuum limit with respect to the color-invariant monopole density and the effective monopole action. Since each Abelian-like monopole is not gauge invariant, we have introduced various partial gauge fixing for the purpose of reducing lattice artifact monopoles in the thermalized vacuum. Here we investigate Abelian and monopole dominances and the Abelian dual Meissner effects adopting the same gauges like the maximal center gauge (MCG) in comparison with the maximal Abelian gauge (MAG). Abelian and monopole contributions to the string tension in these gauges are observed to be a little smaller than the non-Abelian string tension. However, we find that the monopole dominance is improved well when use is made of the block-spin transformations with respect to Abelian-like monopoles. We find each electric field is squeezed by the corresponding colored Abelian-like monopole in such gauges and the Abelian dual Meissner effect is observed independently for each color. Moreover, we confirm the dual Ampère’s law in these new gauges as well as in MAG. The SU(2) vacuum is shown to be near the border between the type 1 and type 2 dual superconductors. The penetration length is almost equal for the four gauge fixings, and the vacuum type in MCG is almost the same value as the previous results. These results are consistent with the previous results suggesting the continuum limit and the gauge-independence of Abelian monopoles.

Figure 1

The potential between quark and antiquark in the MCG. Only the data at $\beta =3.9$ on the ${48}^4$ lattice is shown as an example.

Figure 2

The static-quark potentials from monopole Wilson loops on a blocked reduced lattice with the spacing $b=na$ in the MCG. The data at $\beta =3.0$ is without the block-spin transformation. The data at $\beta =3.4$ ($\beta =3.6$) are taken from $n=2$ ($n=3$) blocked monopoles.

Figure 3

Note that (a) is the schematic figure of the disconnect correlation between an Abelian Wilson loop and an Abelian operator. Note that (b) is the definition of the cylindrical coordinate $(r,\varphi ,z)$ along the $q-\overline{q}$ axis.

Figure 4

The color distribution of electric fields $E_z$ on the ${24}^4$ lattice in the MCG. Only the $\beta =3.3$ case is plotted.

Figure 5

The distribution of electric-field components $E_z$, $E_r$, and $E_{\varphi }$ on the ${24}^4$ lattice in the MCG. Only the $\beta =3.5$ case is shown.

Figure 6

The profile of the color electric field $E_z$ at $\beta =3.5$ on the ${24}^4$ lattice for four smooth gauge fixings.

Figure 7

The profile of monopole current $k_{\varphi }$ distributions at $\beta =3.5$ on the ${24}^4$ lattice in the MCG. There is no correlation between different colors.

Figure 8

The dual Ampère’s law at $\beta =3.5$ on the ${24}^4$ lattice in the MCG.

Figure 9

The behaviors of the electric field squeezing and the monopole density at $\beta =1.40$ on the ${24}^4$ lattice in the MCG.