Visualisations of Centre Vortices
VVisualisations of Centre Vortices
James
Biddle , ∗ , Waseem
Kamleh , ∗∗ , and Derek
Leinweber , ∗∗∗ Centre for the Subatomic Structure of Matter, Department of Physics, The University of Adelaide, SA5005, Australia
Abstract.
The centre vortex structure of the vacuum is visualised throughthe use of novel 3D visualisation techniques. These visualisations allow for ahands-on examination of the centre-vortex matter present in the QCD vacuum,and highlights some of the key features of the centre-vortex model. The con-nection between topological charge and singular points is also explored. Thiswork highlights the useful role visualisations play in the exploration of the QCDvacuum.
Our current understanding of the strong interaction is encapsulated in the gauge field theory ofQuantum Chromodynamics (QCD). Because the gauge bosons of QCD, the gluons, can self-interact, the QCD vacuum is populated by highly non-trivial gluon and quark condensates.However, it is not yet analytically determined what feature of the non-trivial QCD groundstate fields is fundamental to the distinctive properties of QCD, namely the • Confinement of quarks, and • Dynamical chiral symmetry breaking leading to dynamical mass generation.The most promising candidate supported by numerical studies is the centre vortex pic-ture [1, 2], which postulates that these two features are caused by sheets of chromo-magneticflux carrying charge associated with the centre of the
S U (3) gauge group, given by the threevalues of √
1. The centre vortex picture has already had much success in reproducing manydistinctive QCD properties, such as the linear static quark potential [3–7], enhancement of theinfrared gluon propagator [8–11], enhancement of the infrared quark mass function [12, 13]and mass splitting in the low-lying hadron spectrum [13–15].This work seeks to visualise these centre vortices on the lattice through the use of 3Dmodelling techniques, allowing us to explore the vortex vacuum in a never-before-seen way.These visualisations are presented as interactive 3D models embedded in the document. Tointeract with these models, it is necessary to open the document in Adobe Reader or AdobeAcrobat (requires version 9 or newer). Linux users should install Adobe Acroread version9.4.1, the last edition to have full 3D support. Note that 3D content must also be enabledfor the interactive content to be available, and for proper rendering it is necessary to enable ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] ∗∗∗ e-mail: [email protected] a r X i v : . [ h e p - l a t ] S e p x ( x ) Z † y ( x ) Z y ( x + ˆ x ) Z † x ( x + ˆ y ) xz y Z x ( x ) Z † y ( x ) Z y ( x + ˆ x ) Z † x ( x + ˆ y ) Figure 2.
An example of the plotting convention for vortices located within a 3D time slice.
Left: A + + ˆ z direction. Right: A − − ˆ z direction. double-sided rendering in the preferences menu. To view the models, click on the figuresmarked as Interactive in the caption. To rotate the model, click and hold the left mousebutton and move the mouse. Use the scroll wheel or shift-click to zoom. Some pre-set viewsof the model are also provided to highlight areas of interest. To reset the model back toits original orientation and zoom, press the ‘home’ icon in the toolbar or change the view to‘Default view’. In addition, the 3D models presented here can be viewed in augmented realitythrough Josh Charvetto’s Android application. See Fig 1 for a QR code link to download theapp.
Figure 1.
QR code to download the aug-mented reality app for Android devices.
When projected onto 3D space, vortices appear asclosed lines carrying centre charge. They are iden-tified on the lattice by projecting the gluon fieldlinks onto their nearest centre element in maximalcentre gauge. Each 1 × P µν ( x ), willthen take one of three possible values P µν ( x ) = exp (cid:32) m π i (cid:33) I , m ∈ {− , , + } . (1)If P µν ( x ) takes one of the two complex phases,we say it is pierced by a vortex. We refer to the vortex by it’s centre charge parameter, m = ± m = + m = − In 4D space-time, centre vortices map out a 2D wold sheet. To visualise these vortices, weproject onto 3D space where the sheets map out lines that vary with time. As we have taken igure 3.
The first time-slice of spatially-oriented vortices. (
Interactive online ) Figure 4. Left:
Vortices form continuous lines, highlighted with orange arrows in this diagram.
Middle:
Vortices must form closed loops to conserve the vortex flux.
Right:
S U (3) vortices arecapable of forming monopoles or branching points where three vortices emerge or converge at a singlepoint.
3D slices, we have suppressed all vortex information in the time direction. In each 3D slicewe only have access to one link belonging to the plaquettes associated with vortices in theforwards and backwards x i − t planes. As such we plot an ‘indicator’ link, to signify thepresence of a vortex in the suppressed direction. This follows the convention, • + = ⇒ cyan arrow, positively oriented • + = ⇒ cyan arrow, negatively oriented • − = ⇒ orange arrow, positively oriented • − = ⇒ orange arrow, negatively oriented.An example of these conventions is shown in Fig. 5. Adding these space-time indicator links t y Figure 5. Left: A + x − t plane (shaded blue) will be indicated by a cyan arrowin the + ˆ x direction. Right: A − x − t plane (shaded red) will be indicated by anorange arrow in the + ˆ x direction. to our previous visualisation, the first time slice now appears as Fig. 6. Figure 6.
The first time-slice of now containing both spatially-oriented and space-time vortices.(
Interactive online ) We can see how these space-time oriented indicator links predict the motion of spatially-oriented vortices by looking at Fig. 7. Here we see a line of m = − Using our vortex illustrations, we can identify locations where vortex surfaces span all fourspace-time dimensions. These locations are known as singular points, and can be identifiedas shown below by an indicator link running parallel to a spatially-oriented jet, as shown inFig. 8. These points are significant because they necessarily generate regions of topologicalcharge density. A visualisation of these singular points is presented in Fig. 9. a) t = t = Figure 7.
An example of a sheet of space-time oriented vortices predicting the motion of spatially-oriented vortices over multiple lattice sites from t = t = xt y Figure 8.
The signature of a singular point, in which the tangentvectors of the vortex surface span all four dimensions. In this case,the blue jet is associated with field strength in the x − y plane, andthe orange space-time vortex indicator link is associated with a vor-tex generating field strength in the z − t plane. Hence, the vortexsurface spans all four dimensions at the front lower left corner of theillustration. Figure 9.
The singular points on the t = Interactive online ) Conclusion
Visualisations of centre vortices provide valuable insight into the nature of the QCD vacuum.Through these visualisations we can identify structures of interest such as branching pointsand singular points, and study their relationship with topological charge density. For a moredetailed description of these visualisations, and an analysis of the correlation between vorticesand topological charge, see Ref. [17]. Work such as this allows one to explore QCD andcentre vortices in a novel manner, and provides exciting new perspectives on the centre vortexvacuum.
The authors wish to thank Amalie Trewartha for her contributions to the gauge ensemblesunderlying this investigation. We also thank Ian Curington of Visual Technology ServicesLtd. for his support of the PDF3D software used in creating the interactive 3D visualisa-tions presented herein. This research is supported with supercomputing resources providedby the Phoenix HPC service at the University of Adelaide and the National ComputationalInfrastructure (NCI) supported by the Australian Government. This research is supportedby the Australian Research Council through Grants No. DP190102215, DP150103164,DP190100297 and LE190100021.
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