We analyze properties of local Polyakov loops using quenched as well as
dynamical SU(3) gauge configurations for a wide range of temperatures. It is
demonstrated that for both, the confined and the deconfined regime, the local
Polyakov loop prefers phase values near the center elements 1, exp(i 2 pi/3),
exp(-i 2 pi/3). We divide the lattice sites into three sectors according to
these phases and show that the sectors give rise to the formation of clusters.
For a suitable definition of these clusters we find that in the quenched case
deconfinement manifests itself as the onset of percolation of the clusters. A
possible continuum limit of the center clusters is discussed

A Schmidt

Aoki Y

C Gattringer

Coniglio A

Fortunato S

J Danzer

S Borsanyi

Schmidt A

Z Fodor

Journal of Physics : Conference Series

English

arXiv

Crossref

Journal of Physics Conference Series

Coherent center domains from local Polyakov loops

We analyze properties of local Polyakov loops using quenched as well as dynamical SU(3) gauge configurations for a wide range of temperatures. It is demonstrated that for both, the confined and the deconfined regime, the local Polyakov loop prefers phase values near the center elements 1,e ±i2π/3. We divide the lattice sites into three sectors according to these phases and show that the sectors give rise to the formation of clusters. For a suitable definition of these clusters we find that in the quenched case deconfinement manifests itself as the onset of percolation of the clusters. A possible continuum limit of the center clusters is discussed

Borsanyi, S

Danzer, J

Fodor, Z

Gattringer, C

Schmidt, A

ELTE Digital Institutional Repository (EDIT)

arXiv:1007.5403v1  [hep-lat]  30 Jul 2010Coherent center domains from local Polyakov loopsS. Borsanyi 1, J. Danzer 2,†, Z. Fodor 1, C. Gattringer 2, A. Schmidt 21 University of Wuppertal, Department of Physics, 42097 Wuppertal, Germany2 Karl-Franzens University Graz, Institute of Physics, 8010 Graz, Austria† Speaker. E-mail: julia.danzer@uni-graz.atAbstract. We analyze properties of local Polyakov loops using quenched as well as dynamicalSU(3) gauge configurations for a wide range of temperatures. It is demonstrated that for both,the confined and the deconfined regime, the local Polyakov loop prefers phase values near thecenter elements 1, e±i2π/3. We divide the lattice sites into three sectors according to thesephases and show that the sectors give rise to the formation of clusters. For a suitable definitionof these clusters we find that in the quenched case deconfinement manifests itself as the onsetof percolation of the clusters. A possible continuum limit of the center clusters is discussed.1. Motivation and general frameworkWith the running and upcoming experiments at LHC, RHIC and GSI the analysis of the QCDphase diagram has become an important focus of research. Not only the transition curveswhich separate different phases are of interest, but one would also like to understand from firstprinciples the physical mechanisms that drive the various transitions.In this project we probe the finite temperature transition of QCD using static quark sources.In the framework of lattice QCD these can be implemented using local Polyakov loops. Thelocal Polyakov loop L(~x) is given by the trace of the product of temporal gauge links U4(~x, t) ata fixed spatial position ~x (Nt is the number of lattice points in time direction):L(~x) = TrNt−1∏t=0U4(~x, t) , (1)i.e., the Polyakov loop is a gauge transporter that propagates a static quark at position ~x forwardin time. For later use we also define the spatially averaged Polyakov loop P as P = V −1∑~x L(~x),where V is the spatial volume. Due to translation invariance P and L(~x) have the same vacuumexpectation value. After suitable renormalization the Polyakov loop may be related to the freeenergy Fq of a single quark, i.e., 〈P 〉 ∝ exp (−Fq/T ), where T is the temperature. Below Tc thefree energy is infinite, thus 〈P 〉 = 0, and the quarks are confined. Above Tc we have a finite freeenergy and thus 〈P 〉 6= 0, signaling deconfinement. Hence the Polyakov loop acts as an orderparameter for deconfinement.In the deconfined phase of pure SU(3) gauge theory the phases of the summed Polyakov loopsP assume values in the vicinity of the three center elements 1, e±i2π/3 of SU(3). This behaviorshows the underlying center symmetry, a symmetry of the action and the path integral measure,which in pure gauge theory becomes broken spontaneously above Tc. The Polyakov loop Ptransforms non-trivially under the center transformation and thus is also an order parameterfor the symmetry breaking. Above Tc the Polyakov loop 〈P 〉 is non-vanishing with phases near1, e±i2π/3. For full QCD the fermion determinant breaks the center symmetry explicitly andacts as an external magnetic field favoring the real sector.Except for the interchange of low and high temperature the situation is equivalent to simplespin systems with spins s(~x). Without external field the corresponding Hamiltonian has asymmetry which may be broken spontaneously. The symmetry breaking can be analyzed withobservables that transform non-trivially under the symmetry group, e.g., the magnetizationM = V −1∑~x s(~x). To obtain the equivalence between the spin system and the gauge theoryone has to identify the local loops L(~x) with the spins s(~x) and the spatially averaged Polyakovloop P with the magnetization M . In the absence of an external field (the fermion determinant)the phase of the magnetization 〈M〉 (the spatially averaged Polyakov loop 〈P 〉) spontaneouslyselects one of the phases according to the underlying symmetry. If the external field (the fermiondeterminant) is turned on, the previously sharp transition becomes a crossover, and the phaseof the magnetization (the Polyakov loop) is determined by the symmetry breaking term.For the case of pure gauge theory these arguments are the basis of the Svetitsky-Yaffeconjecture [1] which states that at Tc pure SU(3) gauge theory in 4 space-time dimensionscan be described by a 3-d effective spin system with an effective action which is symmetricunder the center group Z3. The spin degrees of freedom are related to the local loops L(~x). Theleading term of the effective action is given by (τ and κ are real non-negative couplings)S[s] = −τ∑〈x,y〉[s(~x)s(~y)∗ + s(~y)s(~x)∗]− κ∑~x[s(~x) + s(~x)∗], (2)where for illustration purposes we also included a symmetry breaking term which can be turnedoff when κ = 0. In the simplest version the effective spins s(~x) have values s(~x) ∈ {1, e±i2π/3}.Magnetic finite temperature transitions for spin systems are well understood phenomena.For discrete symmetry groups the transition is accompanied by the formation of locally spincoherent Weiss domains near Tc, before one spin orientation wins out in the symmetry brokenphase. Even if a modest external magnetic field is applied one can still observe local clusters ofaligned spins different from the direction preferred by the magnetic field.One can go a step further and analyze the connectedness properties of the Weiss domains.One may define clusters of aligned spins and switch to a cluster description of magnetic systems[2, 3, 4]. For several spin systems one finds that the magnetic transition may be characterizedas a percolation phenomenon of suitably defined clusters.With the Svetitsky-Yaffe conjecture in mind, which describes the deconfinement transitionwith an effective spin system, we can formulate the central questions we explore in our project:• Can one identify (at least near Tc) characteristic properties of spin-like behavior in anab-initio lattice simulation of pure gauge theory and/or full QCD?• Is it possible to identify spatial structures (clusters) that correspond to Weiss domains?• How do the domains behave near Tc? Do suitably defined clusters percolate?• What is the role of the fermion determinant which breaks the underlying center symmetry?• Can the clusters (Weiss domains) be given a physical meaning also in the continuum limit?For SU(2) gauge theory similar questions were addressed in [5] – [8] (see also [9]). First resultsfor SU(3) gauge theory were presented in [10].2. Distribution properties of local Polyakov loopsWe study the Polyakov loop (1) using quenched configurations as well as configurations fromfull QCD. For our quenched analysis we use the Lüscher-Weisz gauge action with lattice sizesfrom 203×6 to 403×12 and temperatures ranging from T = 0.63Tc to 1.32Tc [10]. In full QCDwe use configurations generated with a Symanzik improved gauge action and 2 + 1 flavors ofstout-link improved staggered quarks at physical masses [11, 12]. We study lattices with sizes183 × 6, 363 × 6 and 243 × 8 in a temperature range from T = 110 MeV to 320 MeV.To analyze the properties of local Polyakov loops we evaluate the L(~x) and write them asL(~x) = ρ(~x) eiϕ(~x) , (3)i.e., we decompose the local loops into modulus and phase. We begin with studying histogramsH[ρ(~x)] and H[ϕ(~x)] for the distribution of the local modulus ρ(~x) and the local phase ϕ(~x),shown in Figs. 1 and 2. In both figures the lhs. is for the quenched case, while the rhs. is forfull QCD and we compare results in the confining (low T ) and the deconfining (high T ) phase.Fig. 1 shows that the distribution of the modulus is essentially independent of T , andfurthermore is the same for both quenched and full QCD. The distribution of the modulus almostperfectly follows the Haar measure distribution (full curves in Fig. 1), P [ρ] =∫dUδ(ρ− |TrU |),where dU is the SU(3) Haar measure. Comparing the distributions below and above thetransition/crossover temperature clearly shows that the local modulus is not involved in thetransition, which in the case of pure SU(3) gauge theory is even manifest as a first order jumpof 〈P 〉. Obviously it must be the local phases ϕ(~x) which drive the transition.The histograms for the local phase in Fig. 2 show a pronounced peak structure, with maximaat the three center angles 0, 2π/3 and −2π/3. In the confining phase (top plots) for allthree maxima have the same height and we again observe no difference between the quenchedand the dynamical case, both of which can be described by the Haar measure distributionP [ϕ] =∫dUδ(ϕ − arg TrU) (full curves in the top plots). In the deconfined phase (bottomplots) the situation is different. One of the three center phases is more populated. For fullQCD, where the fermion determinant acts like an external field, it is always the real center sector(phase values near 0) which becomes enhanced. In the quenched case any of the three centersectors may be selected spontaneously (similar to, e.g., the Ising system where the magnetizationhas two signs to choose from in the symmetry broken phase). In the quenched distribution weshow here we have chosen configurations where the phase angle of the summed Polyakov loopP is near −2π/3 and obviously the local phases are enhanced for that value. If one switches toany of the other equivalent center sectors, the distribution is shifted by ±2π/3.3. Cluster propertiesIn the previous section we have seen that the transition to the deconfined phase is accompaniedby an increasing population of the histograms for the phase ϕ(~x) of the local Polyakov loop L(~x)0.0 1.0 2.0 3.0ρ(x)0.00.20.40.60.81.0H[ρ(x)]0.0 1.0 2.0 3.0ρ(x)T = 0.63Tc T = 1.32TcQuenched SU(3) gauge theory0.0 1.0 2.0 3.0ρ(x) 0.0 1.0 2.0 3.0ρ(x)T = 110 MeV T = 200 MeVFull QCDFigure 1. Histograms for the distribution of the local modulus ρ(~x). We compare quenched(lhs.) and full QCD (rhs.) at low and high T . The full curve is the Haar measure distribution.00.20.40.6H[ϕ(x)]-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 ϕ(x)00.20.40.6H[ϕ(x)]T = 0.63 TcT = 1.32 TcQuenched SU(3) gauge theory 00.20.40.6H[ϕ(x)]-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 ϕ(x)00.20.40.6H[ϕ(x)]T = 110 MeVT = 200 MeVFull QCDFigure 2. Histograms for the distribution of the local phase ϕ(~x). We compare quenched (lhs.)and full QCD (rhs.) at low and high T . The full curve is the Haar measure distribution.near one of the center angles. This accumulation of the local phases in one sector drives theincrease of the expectation value 〈P 〉, while the modulus of the local loops plays no role. It isinteresting to note that at high temperatures (bottom plots in Fig. 2) there are still pronouncedpeaks also in the subdominant sectors. The question is whether the phase values at differentpositions ~x are distributed independently, or if there are spatial domains where the phases of thelocal Polyakov loops tend to align in the same center sector. The latter case is what is suggestedby the effective action (2), where the first term favors parallel spins.In order to study the formation of domains we assign sector numbers n(~x) to the sites ~x,n(~x) =−1 for ϕ(~x) ∈ [−π + δ , −π/3− δ ] ,0 for ϕ(~x) ∈ [−π/3 + δ , π/3 − δ ] ,+1 for ϕ(~x) ∈ [π/3 + δ , π − δ ] .(4)Here δ is a free real and positive parameter which allows to cut lattice points ~x where thecorresponding phase ϕ(~x) is near a minimum of the distributions in Fig. 2. These points do nothave a clear preference for one of the center sectors and the parameter δ allows one to removethem from the cluster analysis. The remaining lattice points ~x can now be organized in clustersaccording to the sector numbers: We put two neighboring points ~x, ~y into the same cluster ifn(~x) = n(~y). For illustration purposes in Fig. 3 we show the largest cluster for two quenchedconfigurations, one below Tc (lhs.), the other one above (rhs.). The plot is for lattice size 303×6and a value of δ chosen such that 39 % of the lattices points are cut. It is obvious, that aboveTc the largest cluster percolates (stretches over all of the lattice), while it is finite below Tc.Of course the cutoff parameter δ will influence the size of the clusters, since with increasing δless sites are available. We stress at this point that also for the characterization of the magnetictransitions in spin systems a similar reduction step is necessary in the construction of the clusters.This may be a reduced linking probability between neighbors with equal spins [4], but also moregeneral approaches similar to the one used here were considered [8]. As a consistency check ofour cluster construction one may show [13] that the points that survive the cut carry most ofthe signal of a rising Polyakov loop above the transition/crossover temperature.In order to quantify the dependence of the cluster size on the temperature, in Fig. 4 we showthe expectation value of the weight W (i.e., the number of lattice points) of the largest cluster 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30Figure 3. Largest clusters for two quenched configurations below Tc (lhs.) and above (rhs.).normalized by the total number V of sites as a function of temperature. Again the lhs. is forthe quenched case (at a cut of 39% for all volumes and temperatures), while the rhs. is for fullQCD (19 % cut), and we compare three (two) different spatial volumes. Both in the quenchedand the dynamical case we observe that the clusters start out small below Tc and grow quicklyin size at the transition temperature. If one analyzes the percolation probability one finds thatfor the quenched case, where we have a true phase transition, this probability indeed approachesa step function as the volume is increased. For the dynamical case, where one observes only acrossover [14], the situation concerning percolation is not yet clear and further studies on severalvolumes will be necessary to settle this issue.4. Towards a continuum limit for the center clustersWe have shown that the phases ϕ(~x) of the local Polyakov loops L(~x) have preferred values nearthe center angles, and that neighboring sites have a tendency towards aligning these phases. Thecorresponding clusters were found to grow quickly near the transition/crossover temperature,and at least for the quenched case the deconfinement transition may be characterized by theonset of percolation for suitably defined clusters. So far this picture is established only for afixed lattice spacing a and some arbitrarily chosen value of the cutoff parameter δ which entersour cluster definition. If one wants to assign a physical significance to the center clusters, a wayof constructing them such that a continuum limit can be taken is necessary. In particular this0.6 0.8 1.0 1.2 1.4 T/Tc0.00.10.20.3< W/V >203 x 6303 x 6403 x 6Quenched SU(3) gauge theory100 150 200 250 300 T [MeV]0.00.10.20.3< W/V >183 x 6363 x 6Full QCDFigure 4. Weight W of the largest cluster normalized with the volume V as function of T .0.6 0.7 0.8 0.90.00.51.00.6 0.7 0.8 0.9 1.0 1.1 1.2 T/Tc0.02.04.06.08.010.0dphys [fm]303 x 6403 x 8Figure 5. Average cluster diameter in fm versus T for two resolution scales (quenched case).involves a prescription for connecting the scale a in physical units to the clusters.We begin with defining the linear extension (radius) of a cluster by considering the expectationvalue of two-point correlators C(|~x − ~y|) for sites ~x, ~y within the same cluster. These 2-pointfunctions decay exponentially, C(|~x − ~y|) ∝ exp(−|~x − ~y|/r), and we use the parameter r todefine the radius of the cluster in units of the lattice spacing a. Using the value of the latticeconstant a in fm, the diameter of the clusters in physical units (fm) is then given by dphys = ra.The result will depend on both, the lattice spacing a and the cutoff δ. In order to compare thephysical size of the clusters for different lattice resolutions a, we always adjust δ such that ata low temperature (T = 0.63Tc for the quenched case where this analysis is done) we fix thephysical diameter to a typical hadronic size, e.g., dphys = 0.5 fm. The corresponding value of δ isthen kept fixed for all other temperatures, and is used to study dphys as a function of T . In Fig. 5we show the result for the cluster diameter dphys in physical units as a function of temperaturefor the quenched case, comparing two different resolution scales a (i.e., Nt = 6 and Nt = 8).It is obvious that also in physical units the phase transition is signaled by a sudden increase ofthe cluster size. Furthermore, the results for the two different scales fall onto a universal curve,which suggests that a continuum limit for the cluster size might exist. This question is analyzedin detail in a future publication [13], where we study the flow of the cutoff δ as a function of theresolution scale a, arguing that the center clusters indeed have a continuum limit.Acknowledgments: We thank Christian Lang and Axel Maas for interesting discussions. Thiswork was partly supported by the DFG SFB TR 55 and the FWF DK 1203.References[1] Yaffe L.G., and Svetitsky B., 1982, Phys. Rev. D 26 963; Nucl. Phys. B 210 423.[2] Fortuin C.M., and Kasteleyn P.W., 1972, Physica 57 536.[3] Fortuin C.M., 1972, Physica 58 393; Physica 59 545.[4] Coniglio A., Klein W., 1980, J. Phys. A 13 2775.[5] Fortunato S., and Satz H., 2000, Phys. Lett. B 475 311.[6] Fortunato S., and Satz H., 2001, Nucl. Phys. A 681 466.[7] Fortunato S. et al., 2001, Phys. Lett. B 502 321.[8] Fortunato S., 2003, J. Phys. A 36 4269.[9] Schmidt A., Master Thesis, Graz 2010, http://physik.uni-graz.at/itp/files/schmidt/masterthesis.pdf[10] Gattringer C., 2010, Phys.Lett.B 690 179.[11] Aoki Y., et al., 2006, JHEP 0601:089; Phys. Lett. B 643 46.[12] Aoki Y., et al., 2009, JHEP 0906:088.[13] Work in preparation.[14] Aoki Y., et al., 2006, Nature 443 675.reserved for figure

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Coherent center domains from local Polyakov loops

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