Visualization of center vortex structure

The center vortex structure of the $SU(3)$ gauge field vacuum is explored through the use of novel visualization techniques. The lattice is partitioned into 3D time slices, and vortices are identified by locating plaquettes with nontrivial center phases. Vortices are illustrated by rendering vortex lines that pierce these nontrivial plaquettes. Nontrivial plaquettes with one dimension in the suppressed time direction are rendered by identifying the visible spatial link. These visualizations highlight the frequent presence of singular points and reveal an important role for branching points in $SU(3)$ gauge theory in creating high topological charge density regimes. Visualizations of the topological charge density are presented, and an investigation into the correlation between vortex structures and topological charge density is conducted. The results provide new insight into the mechanisms by which center vortices generate nontrivial gauge field topology. This work demonstrates the utility of visualizations in conducting center vortex studies, presenting new avenues with which to investigate this perspective of the QCD vacuum.

Figure 1

An example of the plotting convention for vortices located within a 3D time slice. Left: A $+1$ vortex in the $+\widehat{z}$ direction. Right: A $-1$ vortex in the $-\widehat{z}$ direction.

Figure 2

The $t=1$ slice with all spatially oriented vortices plotted. The flow of $m=+1$ center charge is illustrated by the jets as described in the text. (interactive. See Sec. 1 for more information about interacting with the 3D models available in the Supplemental Material [21].)

Figure 3

The $t=2$ slice with all spatially oriented vortices plotted. Only a small subset of jets are stationary between $t=1$ and $t=2$. Symbols are as in Fig. 2. (interactive).

Figure 4

Left: Vortices form directed continuous lines, highlighted with orange arrows in this diagram. Note that because of the lattice periodicity, these lines may wrap around to the opposite edge of the lattice. Middle: Vortices must form closed loops to conserve the vortex flux. Right: $SU(3)$ vortices are capable of forming monopoles or branching points where three or five vortices emerge or converge at a single point.

Figure 5

A vortex branching point with center charge $+2$ flowing into a vertex (left) is equivalent to a vortex monopole with charge $+1$ flowing out of the vertex (right). The arrows indicate the direction of flow for the labeled charge. Our illustrations adopt $m_k\in \{-1,0,+1\}$ and jets denoting the oriented flow of a positive unit of charge ($m=+1$).

Figure 6

The ensemble average of the number of vortices piercing each 3D cube. As it is necessary to preserve continuity of the vortex flux, we see that there are no cubes with one vortex piercing them. The largest vortex contribution is from $N_{\mathrm{cube}}=2$, arising from vortices propagating without branching or touching. We also see that $N_{\mathrm{cube}}=3$ branching points dominate the $N_{\mathrm{cube}}=5$ branching points.

Figure 7

Top: A $+1$ vortex in the forward (left)/backward (right) $x-t$ plane (shaded blue) will be plotted as a cyan arrow in the $\pm \widehat{x}$ direction respectively. Bottom: A $-1$ vortex in the forward (left)/backward (right) $x-t$ plane (shaded red) will be plotted as an orange arrow in the $\pm \widehat{x}$ direction respectively.

Figure 8

On the $t=1$ time slice, the flow of $m=+1$ center charge is illustrated by the jets, and the spatial links indicate the presence of center vortices in the suppressed time direction. These indicator links show how the jets will evolve through the suppressed Euclidean time direction. Rendering conventions are described in the text.

Figure 9

Space-time oriented vortices changing as we step through time. We observe the space-time indicator links change direction as we transition from (a) $t=1$ to (b) $t=2$; however the phase (color) of the vortex remains the same.

Figure 10

An example of a spatially oriented vortex at the space-time coordinate $x$ moving one plaquette between time slices. The solid red line indicates how the flow of vortex charge pierces between time slices. By our visualization conventions, the shaded red plaquettes would have a spatially oriented jet plotted in the suppressed $\widehat{x}$ direction. Space-time vortices are illustrated by the orange indicator links belonging to the space-time plaquette. We observe that spatially oriented vortices move in the time direction (hidden in our 3D models), perpendicular to the indicator link.

Figure 11

An example of space-time oriented vortices predicting the motion of the spatially oriented vortices. Here we see the $m=+1$ (blue) vortex line transition one lattice spacing down as we step from (a) $t=1$ to (b) $t=2$. Note that the orange space-time vortex indicator links have changed direction.

Figure 12

An second example of space-time oriented vortices predicting the motion of the spatially oriented vortices. We observe the $-1$ (red) vortex line with no associated space-time vortex indicator links remains stationary as we transition from (a) $t=1$ to (b) $t=2$. However, the branching point with associated space-time vortex indicator link moves down and to the left during the transition.

Figure 13

An example of a sheet of space-time oriented vortices predicting the motion of spatially oriented vortices over multiple lattice sites from (a) $t=1$ to (b) V. The highlighted line of red vortices flows along the sheet of cyan time-oriented indicator links.

Figure 14

A demonstration of how spatially oriented vortices can transition multiple lattice spacings in a single time step. Conventions are the same as in Fig. 10.

Figure 15

Regions of high topological charge density are rendered as translucent blue [$q(x)<0$) and yellow ($q(x)>0$] volumes, overlaying the $t=1$ slice.

Figure 16

A histogram of total topological charge $Q$ (a) using the gluonic definition after five sweeps of overimproved stout-link smearing and (b) direct center projection from the original gauge fields. It is apparent that singular points following center projection do not preserve the total topological charge [note also the scale change between (a) and (b)]. Panel (c) shows whether the sign matches between plots (a) and (b); again it is apparent that there is little correlation.

Figure 17

The center vortex structure and topological charge density after eight sweeps of cooling, for $t=1$. (interactive).

Figure 18

The signature of a singular point, in which the tangent vectors of the vortex surface span all four dimensions. In this case, the blue jet is associated with field strength in the $x-y$ plane, and the orange space-time vortex indicator link is associated with a vortex generating field strength in the $z-t$ plane. Hence, the vortex surface spans all four dimensions.

Figure 19

All points on the $t=1$ time slice in which a singular point occurs; i.e., a spatially oriented vortex jet runs parallel to a space-time oriented vortex indicator link as shown in Fig. 18. The color indicates the multiplicity $M$ observed on this slice, with the lowest value in blue ($M=1$) and the highest in red ($M=12$). (interactive).

Figure 20

A singular point (green sphere) resembling the structure of Fig. 18. This singular point is generated by the red jet and the orange indicator link running in parallel.

Figure 21

Topological charge density from singular points (shown as dots) is compared with topological charge calculated from vortex-only configurations for $t=1$. (interactive).

Figure 22

Points with two or more vortices piercing a 3D cube are shown on the $t=1$ time slice. The number of vortices piercing a cube is denoted by the color: $\mathrm{blue}=3$, $\mathrm{green}=4$, $\mathrm{orange}=5$, $\mathrm{red}=6$. Whilst there are no red points present in this slice, they occur rarely on other slices. (interactive).

Figure 23

An example of vortex branching generating a region of high topological charge by piercing two out of the four plaquettes that make up a clover.

Figure 24

Branching points (dots) plotted alongside the topological charge density from the projected vortex configurations. It can be observed that the branching points are almost always neighboring topological charge density. (interactive).

Figure 25

An example of two branching points and their associated topological charge density.

Figure 26

Summary of the processes used to obtain vortex and topological charge density configurations. “MCG” denotes gauge fixing to maximal center gauge and “smear” denotes application of five sweeps of overimproved stout-link smearing as described in the text. From these methods we obtain the vortex only (VO), smeared (S) and preconditioned smeared (PS) topological charge and vortex configurations. As the topological charge density is gauge invariant, it could equivalently be calculated following gauge fixing to maximal center gauge.

Figure 27

Correlation values for vortices (V), singular points (SP) and branching points (BP) obtained from centre projected configurations, $Z_{\mu }(x)$, with topological charge density, $q(x)$, obtained via various means described in the text and shown diagrammatically in Fig. 26. Data points indicate results for the normalised correlation values, $\overline{C/C_{\text{Ideal}}}$. (a) Smeared $q(x)$ vs projected $Z_{\mu }(x)$. (b) Preconditioned smeared $q(x)$ vs projected $Z_{\mu }(x)$. (c) Smeared $q(x)$ vs smeared $Z_{\mu }(x)$. (d) Preconditioned smeared $q(x)$ vs preconditioned smeared $Z_{\mu }(x)$. (e) Projected $q(x)$ vs projected $Z_{\mu }(x)$.

Figure 28

The vortex structure and topological charge present after five sweeps of overimproved stout-link smearing, preconditioned with maximal center gauge [$Z_{\mu }^{\mathrm{PS}}(x)$ and $q^{\mathrm{PS}}(x)$]. (interactive).