Topological order and the deconfinement transition in the $(2+1)$-dimensional compact Abelian Higgs model

We study an Abelian compact gauge theory minimally coupled to bosonic matter with charge $q$, which may undergo a confinement-deconfinement transition in $2+1$ dimensions. The transition is analyzed using a nonlocal order parameter $\tilde{W}$, which is related to large Wilson loops for fractional charges. We map the model to a dual representation with no gauge field but only a global $q$-state clock symmetry and show that $\tilde{W}$ corresponds to the domain wall energy of that model. $\tilde{W}$ is also directly connected to the concept of topological order. We exploit these facts in Monte Carlo simulations to study the detailed nature of the deconfinement transition.

Figure 1

(Color online) Possible phases of the system. At the top the topological-ordered, deconfined phase, where vortices cost much energy. At the bottom the percolating vortex tangle in the confined phase. The large Wilson loop $\tilde{W}$ surrounds the shaded area as indicated in the figures.

Figure 2

A flux quantum tunnels out from the hole of a torus.

Figure 3

The large Wilson-loop order parameter $\tilde{W}$ as a function of coupling constant for different values of the screening length $\lambda$. Upper left: $\tilde{W}$ vs $g∕{\Phi }_0^2$ for $\lambda =0$. Upper right, lower left, and lower right: $\tilde{W}$ vs $J^{-1}$ for $\lambda =0.5$, 1, and 2, respectively. The critical point is where the curves for different system sizes cross. Note that only system sizes much larger than $\lambda$ scale well (solid lines); smaller sizes (dashed lines) suffer from finite-size corrections.

Figure 4

Phase diagram of the model with $q=2$. Circles indicate the points which have been simulated. The parameters have been rescaled in order to fit the phase diagram into the figure, so that $J\to \infty$ occurs at the upper side and $g\to 0$ at the right side. The phase diagram connects the 3D $XY$ point at $g=0$ and the $Z_2$ transition at strong coupling $J\to \infty$. The deconfined phase is topologically ordered.

Figure 5

Area law and perimeter law demonstrated for $\lambda \to 0$, for a Wilson loop $W\left(\mathcal{C}\right)$ which covers a quarter of a cross section of the system and therefore has a finite perimeter and area. Dashed and solid lines are exact area and perimeter laws, inserted for comparison.

Figure 6

$\tilde{W}$ vs $g∕{\Phi }_0^2$ for a model with $q=3$ and $\lambda =0$. Note that this model has a first-order transition and no scaling behavior is expected.

Figure 7

Finite-size scaling plot of the $\tilde{W}$ order parameter for $\lambda =0.5$ (top) with $\nu =0.64$ and $\lambda =1$ (bottom) with $\nu =0.62$. $J_c^{-1}$ is determined from Fig. 3 to 1.25 and 2.29, respectively. The inset shows fitting error as a function of $\nu$.

Figure 8

The energy gap to vortex excitations as a function of temperature in the deconfined phase, for $q=2$ and $\lambda =0$.

Figure 9

The scaling combination $2\Delta {\xi }^2∕L$ plotted against $L∕\xi$ for $q=2$, $\lambda =1$. In the topologically ordered deconfinement phase the data follow the exponential dependence given in Eq. (30), as indicated by the solid line. The dashed line indicates the area law obeyed in the confinement phase.