Model of random center vortex lines in continuous $2+1$-dimensional spacetime

A picture of confinement in QCD based on a condensate of thick vortices with fluxes in the center of the gauge group (center vortices) is studied. Previous concrete model realizations of this picture utilized a hypercubic space-time scaffolding, which, together with many advantages, also has some disadvantages, e.g., in the treatment of vortex topological charge. In the present work, we explore a center vortex model which does not rely on such a scaffolding. Vortices are represented by closed random lines in continuous $2+1$-dimensional space-time. These random lines are modeled as being piecewise linear, and an ensemble is generated by Monte Carlo methods. The physical space in which the vortex lines are defined is a torus with periodic boundary conditions. Besides moving, growing, and shrinking of the vortex configurations, also reconnections are allowed. Our ensemble therefore contains not a fixed but a variable number of closed vortex lines. This is expected to be important for realizing the deconfining phase transition. We study both vortex percolation and the potential $V(R)$ between the quark and antiquark as a function of distance $R$ at different vortex densities, vortex segment lengths, reconnection conditions, and at different temperatures. We find three deconfinement phase transitions, as a function of density, as a function of vortex segment length, and as a function of temperature.

Figure 1

(a) The action $S=\alpha L+\gamma {\phi }^2$ of the current node (c) is given by the vortex length $L$ to the previous node (p) and the angle $\phi$ between the vortex lines to the previous (p) and the next node (n). Three node clusters of shape (b) are more likely accepted by the Metropolis algorithm than of shape (c).

Figure 2

The movement of the current node (c) within a certain range $r_m$ is affecting the connected vortex lines and the angles at the current (c), the previous (p), and the next node (n).

Figure 3

The add update adds a node after the current node (c) within a range $r_a$ around the midpoint of the vortex line to the next node (n). Two vortex angles before and three after the update are affected, and one vortex line is split into two.

Figure 4

The delete update deletes the current node (c), joining the connected vortex lines into one between the previous (p) and the next node (n). It affects three and two vortex angles before and after the update, respectively. The new vortex length $L$ has to lie in the range $L_{\min }<L<L_{\max }$, like for all other updates.

Figure 5

The reconnection update deletes the vortex lines between nodes 1-2 and 3-4 and reconnects nodes 1-3 and 4-2. The plot shows the affected vortex angles, the reconnection angle $\epsilon$ and distance $r_r$. The plotted vortex lines/nodes might belong to the same or different clusters. Note that the orientation of some vortex lines was reversed in this example, cf. the main text.

Figure 6

(a) Cluster size histogram, elucidating the percolation properties of the vortex structure: at high temperatures (small temporal lattice size, e.g., $16^2\mathrm{x}1$, 2 or 3), we find mainly small vortex clusters (histogram peaks on the left side), whereas for lower temperatures, most of the vortex material is found in clusters of maximal size (right peak), i.e., clusters percolating through the whole lattice. (b) quark-antiquark potentials and (c) maximal cluster fraction (i.e., the fraction of vortex material in clusters of maximal size, the right peak of the histogram plot), temporal and spatial string tensions $\sigma$ and ${\sigma }_s$ for $16^2\times L_T$ volumes, density cutoff $\rho =4$, and vortex lengths $L_{\max }=1.7$, $L_{\min }=0.3$.

Figure 7

(a) String tension $\sigma$ and maximal cluster fraction vs. time extent $L_T$ for different vortex densities $\rho$ on $16^2\times L_T$ volumes with vortex lengths $L_{\max }=1.7$ and $L_{\min }=0.3$.

Figure 8

Cluster size histogram for density cutoff (a) $\rho =6$, (b) $\rho =8$, (c) $\rho =10$, (d) $\rho =12$ on $16^2\times L_T$ volumes with vortex lengths $L_{\max }=1.7$ and $L_{\min }=0.3$.

Figure 9

Maximal cluster fraction, temporal and spatial string tensions $\sigma$ and ${\sigma }_s$ for density cutoff (a) $\rho =6$, (b) $\rho =8$, (c) $\rho =10$, (d) $\rho =12$ on $16^2\times L_T$ volumes with vortex lengths $L_{\max }=1.7$ and $L_{\min }=0.3$.

Figure 10

(a) Total number of vortex clusters and (b) windings around the temporal dimension, for $16^2\times L_T$ volumes, vortex lengths $L_{\max }=1.7$ and $L_{\min }=0.3$.

Figure 11

Sample configurations on $16^2\times L_T$ volumes for (a) $L_T=2$, (b) $L_T=4$, (c) $L_T=8$, (d) $L_T=16$ with density cutoff $\rho =4$ and vortex lengths $L_{\max }=1.7$ and $L_{\min }=0.3$.

Figure 12

(a) Quark-antiquark potentials on a $16^2\times 2$ volume, string tensions (b) $\sigma$, (c) ${\sigma }_s$, and (d) maximal vortex cluster fraction as a function of density cutoff $\rho$ for $16^2\times L_T$ volumes and vortex lengths $L_{\max }=1.7$, $L_{\min }=0.3$.

Figure 13

Cluster size histograms on $16^2\times L_T$ volumes for (a) $L_T=2$, (b) $L_T=3$, (c) $L_T=4$ and different density cutoffs $\rho =4-13$ with vortex lengths $L_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.7$ and $L_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.3$.

Figure 14

(a) Cluster size histogram, (b) quark-antiquark potential, and (c) string tensions $\sigma$ and ${\sigma }_s$ as well as maximal cluster fraction as a function of maximal vortex segment length $L_{\max }$, for a $16^2\times 2$ volume, density cutoff $\rho =8$, and $L_{\min }=0.3$.

Figure 15

(a) Cluster size histogram, (b) quark-antiquark potential, (c) string tensions and maximal vortex cluster fraction, average (d) node action, and (e) vortex line density ($=\mathrm{avg}\mathrm{.}\text{node}\text{density}\times \mathrm{avg}\mathrm{.}\text{vortex}\text{length}$) for $16^2\times 2$ volumes, density cutoff $\rho =8$, and different vortex/reconnection lengths $L_{\min }$ at $L_{\max }=1.7$.

Figure 16

(a) Quark-antiquark potential $V(R,T=1.5)$ and (b) string tension $\sigma$ saturating with temporal extent of Wilson loops $T$, for $16^2\times 8$ and $32^2\times 8$ volumes, density cutoff $\rho =8$, $L_{\max }=1.7$, and $L_{\min }=0.3$.