Compact $Q=2$ Abelian Higgs model in the London limit: Vortex-monopole chains and the photon propagator

The confining and topological properties of the compact Abelian Higgs model with doubly-charged Higgs field in three space-time dimensions are studied. We consider the London limit of the model. We show that the monopoles are forming chainlike structures (kept together by Abrikosov-Nielsen-Olesen vortices), the presence of which is essential for getting simultaneously permanent confinement of singly-charged particles and breaking of the string spanned between doubly-charged particles. In the confinement phase, the chains are forming percolating clusters, while in the deconfinement (Higgs) phase, the chains are of finite size. The described picture is in close analogy with the synthesis of the Abelian monopole and the center vortex pictures in confining non-Abelian gauge models. The screening properties of the vacuum are studied by means of the photon propagator in the Landau gauge.

Figure 1

The phase diagram of the 3D $Q=2$ Abelian Higgs model in the London limit.

Figure 2

A schematic view of simplest vortex-monopole and pure vortex configurations in the ${\mathrm{cAHM}}_3$ with (a) $Q=2$ and (b) $Q=3$ dynamical Higgs field.

Figure 3

The $q=1,2$ Polyakov loops (a) and their susceptibilities (b) vs. $\kappa$ at $\beta =1.2$ on a ${32}^2\times 8$ lattice. The inset in (b) shows the behavior of the $q=1$ susceptibility in the vicinity of the phase transition with the fit according to Eq. (30).

Figure 4

The $q=1,2$ potentials in the confinement ($\kappa =1.275$) and the Higgs ($\kappa =1.325$) phases at $\beta =1.2$ (from Ref. 6).

Figure 5

(a) The monopole and vortex densities (from Ref. 6) and (b) their susceptibilities, vs. $\kappa$ at $\beta =1.2$ on ${32}^2\times 8$ lattice. The inset in (b) shows the behavior of the susceptibilities in the vicinity of the phase transition. The solid lines show the fits by Eq. (30).

Figure 6

Examples of the monopole configurations in the confinement phase at $\kappa =1.275$ (upper row) and in the deconfinement (Higgs) phase at $\kappa =1.325$ (lower row). The (anti)monopoles are shown by the dots, while the vortices are shown by the solid lines.

Figure 7

(a) The expectation values of the Polyakov loop for $q=1$ and $q=2$ and (b) their susceptibilities vs. $\kappa$ for a lattice ${16}^3$. The inset in (b) shows the behavior of the $q=1$ susceptibility in the vicinity of the phase transition and the fit according to Eq. (30).

Figure 8

(a) The densities of the topological defects and (b) their susceptibilities vs. $\kappa$. The inset in (b) shows the behavior of the vortex susceptibility in the vicinity of the phase transition. The solid line shows the fit by Eq. (30).

Figure 9

Histograms of the (a) vortex and (b) monopole distributions.

Figure 10

Fits of (a) the vortex and (b) the monopole histograms by Eqs. (38, 39).

Figure 11

The densities of topological defects in infrared clusters as given by fits (38, 39). The fits by Eq. (40) are shown by dashed lines.

Figure 12

The multiplicity distributions of vortex clusters and the corresponding fits by the Gaussian distribution (42) for various $\kappa$ values.

Figure 13

(a) The average number ${\overline{N}}_{\mathrm{cl}}$ of clusters and (b) the width $\Lambda$ of the multiplicity distribution as functions of $\kappa$.

Figure 14

(a) The expectation values of the Polyakov loop and (b) their susceptibilities vs. $\kappa$. The inset in (b) shows the behavior of the $q=1$ susceptibility in the vicinity of the phase transition together with its fit.

Figure 15

(a) The densities of the topological defects and (b) their susceptibilities vs. $\beta$. The inset in (b) shows the behavior of the monopole and vortex susceptibilities in the vicinity of the phase transition with their fits and critical $\beta$’s.

Figure 16

Histograms of the (a) monopole and (b) vortex distributions.

Figure 17

Fits of (a) the monopole and (b) the vortex histograms by Eqs. (38, 39).

Figure 18

The densities of the topological defects in infrared clusters as given by fits (38, 39). The fit curves using ${\rho }_{\mathrm{IR}}^{M,V}(\beta )$ similar to Eq. (40) are shown by dashed lines.

Figure 19

The fitting parameters for the propagator measured on ${40}^3$ at $\beta =2.0$ vs. $\kappa$.