Approaching $SU(2)$ gauge dynamics with smeared $Z(2)$ vortices

We present a method to smear (center projected) $Z(2)$ vortices in lattice gauge configurations such as to embed vortex physics into a full $SU(2)$ gauge configuration framework. In particular, we address the problem that using $Z(2)$ configurations in conjunction with overlap (or chirally improved) fermions is problematic due to their lack of smoothness. Our method allows us to regain this smoothness and simultaneously maintain the center vortex structure. We test our method with various gluonic and fermionic observables and investigate to what extent we are able to approach SU(2) gauge dynamics without destroying the original vortex structure.

Figure 1

Illustration of vortex smearing: the $Z(2)$ links of the thin vortex plaquette (indicated by red circle) and the link rotation profile of the smeared (thick) vortex. Using steady rotations of $\pi /3$ in the same $U(1)$ subgroup, the center vortex flux $-\mathbb{1}$ of the thin (dual) vortex plaquette is distributed uniformly within the original and neighboring plaquettes. This is indicated by the numbers below the plaquettes as multiples of $\pi$, indicating the composition of the total center vortex flux $\exp i\pi \equiv -\mathbb{1}$. The $U(1)$ subgroup for each vortex plaquette is chosen such as to minimize the affected plaquettes.

Figure 2

20 lowest (a) overlap and (b) asqtad staggered eigenvalues for original (full) $SU(2)$, Maximal Center Gauge (MCG) projected $Z(2)$, $U(1)$ and $SU(2)$ vortex smeared configurations with different numbers of HYP smearing steps.

Figure 3

Distribution of center vortex flux: the $Z(2)$ links of the initial vortex plaquette ($\exp \pm i\pi =-\mathbb{1}$) and the corresponding smeared links using $\pm \pi /8$ rotations away from the initial links. We distinguish four cases (a–d) according to the initial link configurations. The smeared links distribute the flux as shown in Fig. 4, for rotations in the same $U(1)$ subgroup.

Figure 4

Vortex flux distributions for smeared vortex plaquettes with link rotations shown in Fig. 3 (a–d). They add to a total flux of $\pm \pi$, consisting of $\pm \pi /2$ at the original vortex and $\pm \pi /8$ at neighboring plaquettes. These vortex flux distributions among plaquettes are valid for link rotations in the same $U(1)$ subgroup only, and do not necessarily correspond to the final vortex smeared configurations, where the individual rotations are performed in different $U(1)$ subgroups, such as to minimize the affected (including orthogonal) plaquettes, see text.

Figure 5

Refinement routine: Example of one $-\mathbb{1}$ (fat z-) link giving eight $-\mathbb{1}$ and eight $\mathbb{1}$ links after refinement.

Figure 6

Vortex refinement ($\mathrm{a}\to \mathrm{b}$), smearing ($\mathrm{b}\to \mathrm{c}$) and blocking ($\mathrm{c}\to \mathrm{a}$): Example of a partial vortex surface on the dual lattice including edges and corners (red circles). Smearing the central (red dotted) plaquette in the coarse lattice (left) influences all other attached plaquettes since we smear in the directions orthogonal to the plaquette (i.e., its dual plaquette on the original lattice) and the vortex corners (and edges) are deformed. For the refined lattice on the right, smearing individual plaquettes only affects its direct neighbors if they are connected via an edge and therefore plaquettes attached to the vortex corners do not affect each other directly during smearing. Further, deformations of the refined vortex surface during smearing within the original lattice spacing (as indicated in c) will reproduce the original vortex structure after blocking.

Figure 7

Refined link rotation smearing: The smearing rotation from $\mathbb{1}$ to $-\mathbb{1}$ can be extended to four links without disturbing any other vortex plaquettes. Center vortex plaquettes of the initial and refined (or MCG projected after smearing) configurations are again indicated by red circles. Link rotations in the top/bottom row are given by $\pi /8,\pi /4,3\pi /4,7\pi /8$ and $\pi /8,\pi /4,\pi /4,\pi /8$ and the individual $U(1)$ subgroups are chosen such that the affected plaquettes are minimized.

Figure 8

Refined link rotation smearing for $\mathbb{1}\to -\mathbb{1}$ and $-\mathbb{1}\to \mathbb{1}$ link pairs. The individual $U(1)$ rotations of the links from left to right in the top and bottom rows of the two cases are given by $\pi /8,\pi /4,5\pi /8,3\pi /4;\pi /8,3\pi /4,\pi /8,\pi /4$ and $5\pi /8,7\pi /8,\pi /8,\pi /8;\pi /8,3\pi /4,\pi /8,\pi /4$. The odd (lower) lattice slices also contain a certain number of smeared links close to $-\mathbb{1}$, introducing additional vortex plaquettes (red circles) which distort and therefore smear the vortex surface in link direction. The $U(1)$ subgroups are not the same for the individual rotations, but chosen such as to minimize the corresponding plaquettes.

Figure 9

Vortex flux distributions for refined link rotation smeared vortex plaquettes with $\mathbb{1}\to -\mathbb{1}$ and $-\mathbb{1}\to \mathbb{1}$ link pairs in terms of fractions of $\pm \pi$ adding up to a total flux of $\pm \pi$, with $\pm \pi /2$ at the original vortex and $\pm \pi /4$ for neighboring plaquettes. These vortex flux distributions among plaquettes are valid for link rotations in the same U(1) subgroups only, and do not necessarily correspond to the final vortex smeared configurations, where the individual rotations are chosen such as to minimize the corresponding plaquettes.

Figure 10

Examples of link configurations giving uniform center vortex flux distribution among the four refined plaquettes of the initial center vortex plaquette. The individual link rotations are given by $+\pi /2$, $\pm \pi /4$ and $\pm 3\pi /4$ in the same $U(1)$ subgroup. Each refined plaquette carries one fourth of the initial center vortex flux; (a), (c) $\exp i\pi =-\mathbb{1}$ and (b), (d) $\exp -i\pi =-\mathbb{1}$. In order to minimize the orthogonal plaquettes (the ones orthogonal to the displayed plane) we vary the $U(1)$ subgroup by applying “2D gauge transformations” (see text) at the central points of the plaquettes, indicated by the red circle.

Figure 11

Refinement and smeared blocking procedure: After refinement the blocking is not performed starting at (1,1,1,1), the lower, left corner, but half an original (i.e., one refined) lattice spacing forward in every space-time direction, i.e., (2,2,2,2) on the refined lattice, indicated by the red arrow (the time direction is indicated by the fine lines connecting the spacelike cubes). Without vortex smearing between steps 2 and 3 the lattices before and after the whole procedure are actually the same; with vortex smearing the reblocked lattice gives a smeared version of the original lattice.

Figure 12

20 lowest (a) overlap and (b) asqtad staggered eigenvalues for original (full) $SU(2)$, Maximal center gauge projected $Z(2)$ and vortex smeared configurations.

Figure 13

Scatter plot of (a) overlap and (b) asqtad staggered nonzero eigenvalues for original (full) $SU(2)$ and vortex smeared configurations (ensemble mean eigenvalues).

Figure 14

Scatter plot of overlap (a) nonzero eigenvalues and (b) number of zero modes for individual configurations.

Figure 15

Distribution of the “unfolded” level spacing $s$ for (a) overlap and staggered eigenvalues on original configurations vs RMT universality predictions and (b) overlap eigenvalues on original and smeared configurations. For better differentiation we attach the individual plots in Fig. 31 in Appendix pp1.

Figure 16

(a) Distribution of the “unfolded” level spacing $s$ for staggered eigenvalues on original and smeared configurations. (b) Distribution of the smallest staggered eigenvalue ${\lambda }_{\min }$ and RMT universality prediction $P(z)$ vs the rescaled variable $z=\mathrm{\Sigma }V{\lambda }_{\min }$, where $V$ is the volume and $\mathrm{\Sigma }$ is the infinite volume value of the chiral condensate. Fitted values for $\mathrm{\Sigma }$ are given in Table 1. For better differentiation we attach the individual plots in Figs. 34, 35 in Appendix pp2.

Figure 17

Distribution of the smallest overlap eigenvalue ${\lambda }_{\min }$ and RMT universality prediction $P(z)$ for topological sectors (a) $\nu =0$ and (b) $\nu =1$ vs the rescaled variable $z=\mathrm{\Sigma }V{\lambda }_{\min }$, where $V$ is the volume and $\mathrm{\Sigma }$ is the infinite volume value of the chiral condensate. Fitted values for $\mathrm{\Sigma }$ are given in Table 1. Find the individual plots in Fig. 32, 33 in Appendix pp1.

Figure 18

(a) Scatter plot of plaquette and hypercubic definition of topological charge $Q_T$ after cooling and LOG smearing. (b) Scatter plot of $Q_T$ after cooling and LOG smearing for plaquette and hypercube definitions. Both plots are for vortex smeared configurations, for original (full) configurations they look very similar.

Figure 19

Scatter plot of fermionic $Q_F$ and gluonic topological charge $Q_T$ after (a) cooling or (b) (LOG) smearing on original (full) $SU(2)$ and various vortex smeared configurations ($Q_{F/T}$ on same configs); Scatter plot of gluonic topological charge $Q_T$ after (c) cooling and (d) (LOG) smearing for original (full) vs vortex smeared configurations. In the blocked vortex flux distribution smeared case one can actually see a positive correlation, see also Fig. 20.

Figure 20

Scatter plot of (a) fermionic and gluonic topological charge correlations and (b) cross-correlations between original (full) vs blocked vortex flux distribution smeared configurations.

Figure 21

Scatter plot of the vortex vs gluonic topological charge for original (full) $SU(2)$ and vortex smeared configurations. Vortex topological charge determination after (a), (b) one and (c), (d) two blocking steps and (a), (c) no and (b), (d) maximal smoothing. In (e), (f) we show the combined results (a–d) for blocked flux distribution smeared configurations again.

Figure 22

Scatter plot of the vortex vs fermionic topological charge for original (full) $SU(2)$ and vortex smeared configurations. Vortex topological charge determination after (a), (b) one and (c), (d) two blocking steps and (a), (c) no and (b), (d) maximal smoothing. In (e), (f) we show the combined results (a–d) for blocked flux distribution smeared configurations again.

Figure 23

Topological susceptibility from fermionic $Q_F$, gluonic $Q_T$ and vortex topological charge $Q_V$ for original (full) $SU(2)$ (red) and refined (a), (c) link rotation (see Sec. 2d) or (b), (d) vortex flux distribution (see Sec. 2e) smeared (green) and blocked (blue, see Sec. 2f) configurations. In (a) and (b) we show all blocking and smearing steps for $Q_V$, i.e., we see four vertical groups of data points according to no, one, two and three blocking steps from left to right. Within these groups we plot the data points for zero to five smoothing steps from left to right again. In (c) and (d) we show 1-2 blocking steps for $8^4$ lattices, i.e., original and smeared blocked configurations, and 2-3 blocking steps for refined (${16}^4$) lattices, resulting in lattice spacings $a\approx 0.3-0.6\mathrm{fm}$, respectively. We also zoom into the interesting susceptibility region and therefore miss a few data points from vortex topological charge $Q_V$ [green dots in (a) and (c)] and the blue line for topological susceptibility from fermionic $Q_F$ for blocked link rotation smearing (blue solid line) in (c). Note that for original (full) $SU(2)$, the topological susceptibilities from fermionic (red solid line) and gluonic topological charge definitions (red dashed lines) are not consistent, $⟨Q_F^2⟩/V=(200\mathrm{MeV}{)}^4$ and $⟨Q_T^2⟩/V\approx (160\mathrm{MeV}{)}^4$ (averaging data from cooling and LOG smearing $Q_T$), presumably caused by our small original lattice volume of about $(1.2{\mathrm{fm}}^4)$. Vortex topological charge $Q_V$ (red dots) reproduces the two values after one resp. two blocking steps in (c) or (d). Refined smeared configurations (green) show very high susceptibilities for gluonic and vortex topological charge, caused by artificial vortex fluctuations introduced in the refined smearing methods. Blocking removes these fluctuations and we observe good agreement between the different topological charge definitions on original and vortex smeared configurations, especially for the vortex flux distribution smearing—see (d) at $a=0.6\mathrm{fm}$ (i.e., two blocking steps for $Q_V$).

Figure 24

Vortex correlation of (a) overlap and (b) asqtad staggered eigenmodes for original (full) $SU(2)$, maximal center gauge projected $Z(2)$ and vortex smeared configurations.

Figure 25

Correlation of vortex topological charge density with (a) overlap zero mode and (b) lowest asqtad staggered eigenmode densities for original (full) $SU(2)$, center projected $Z(2)$ and vortex smeared configurations.

Figure 26

Scatter plot of (a) $SU(2)$ and (b) $Z(2)$ Wilson loops during smearing for original and various vortex smeared configurations.

Figure 27

(a) Vortex limited Wilson loops and (b) Creutz ratios of original (full) $SU(2)$ and various vortex smeared configurations.

Figure 28

Two orthogonal pairs of planar vortices (a) on a $8^4$ lattice and after (b) refinement and (c) vortex smearing on ${16}^4$ lattices. For a more detailed view of the configuration in (c) see Fig. 29. Smoothing (c) gives (b) and blocking (c) [or (b)] gives (a). We plot the dual vortex plaquettes, which actually represent the closed vortex surfaces.

Figure 29

Dual vortex plaquettes of two orthogonal pairs of planar vortices after vortex smearing on ${16}^4$ lattices from Fig. 28. From left to right and top to bottom we plot the single time slices ($\mathrm{t}=1\text{−}16$).

Figure 30

The spherical vortex (a) on a $8^4$ lattice and after (b) refinement, (c) vortex smearing and (d) smoothing on ${16}^4$ lattices. Blocking the configuration in (c) [or (b) or (d)] gives (a). We plot the dual vortex plaquettes, representing the closed vortex surface.

Figure 31

“Unfolded” level spacing of overlap eigenvalues in the fundamental representation of SU(2) (orthogonal ensemble) for (a) all, (b) original and (c), (d) smeared configs: (c) link rotation smearing, (d) flux distribution smearing and their blocked versions in (e) and (f), respectively.

Figure 32

Distribution of lowest overlap eigenvalues in the fundamental representation of SU(2) (orthogonal ensemble) for (a), (b) original, (c), (d) link rotation and (e), (f) flux distribution smearing in topological sectors $\nu =0$ (left) and $\nu =1$ (right).

Figure 33

Distribution of lowest overlap eigenvalues in the fundamental representation of SU(2) (orthogonal ensemble) for (a), (b) combined configurations, (c), (d) link rotation and (e), (f) flux distribution smeared blocking in topological sectors $\nu =0$ (left) and $\nu =1$ (right).

Figure 34

“Unfolded” level spacing of staggered eigenvalues in the fundamental representation of SU(2) (symplectic ensemble) for (a) all, (b) original and (c)–(f) smeared configurations: (c) link rotation smearing, (d) flux distribution smearing and their blocked versions in (e) and (f), respectively.

Figure 35

Distribution of lowest staggered eigenvalues in the fundamental representation of SU(2) (symplectic ensemble) for (a) all, (b) original and (c), (f) smeared configurations: (c) link rotation and (d) flux distribution smearing and their blocked versions in (e) and (f), respectively.