We measure the free energies in SU(2) of static fundamental charges and
center monopoles. Dual to temporal center fluxes, the former provide a
well-defined (dis)order parameter for deconfinement. In contrast, the monopole
free energies vanish in the thermodynamic limit at all temperatures and are
thus irrelevant for the transition.Comment: 3 pages, LaTeX2e (espcrc2.sty), 4 figures (epsfig), for
  Lattice2002(topology

't Hooft

Davis

de Forcrand

Fingberg

Gonzalez-Arroyo

Kovacs

Lorenz von Smekal

Philippe de Forcrand

Nuclear Physics B - Proceedings Supplements

English

arXiv

Crossref

Electric and magnetic fluxes inSU(2) Yang-Mills theory

We measure the free energies in SU(2) of static fundamental charges and center monopoles. Dual to temporal center fluxes, the former provide a well-defined (dis)order parameter for deconfinement. In contrast, the monopole free energies vanish in the thermodynamic limit at all temperatures and are thus irrelevant for the transition.We measure the free energies in SU(2) of static fundamental charges and center monopoles. Dual to temporal center fluxes, the former provide a well-defined (dis)order parameter for deconfinement. In contrast, the monopole free energies vanish in the thermodynamic limit at all temperatures and are thus irrelevant for the transition

von Smekal, Lorenz

de Forcrand, Philippe

CERN Document Server

FAU-TP3-02/25, hep-lat/0209149 1Electric and Magnetic Fluxes in SU(2) Yang-Mills TheoryLorenz von Smekal,a∗ with Philippe de Forcrandba Institut fu¨r Theoretische Physik III, Universita¨t Erlangen-Nu¨rnberg, D-91058 Erlangen, Germanyb ETH-Zu¨rich, CH-8093 Zu¨rich and CERN Theory Division, CH-1211 Geneva 23, SwitzerlandWe measure the free energies in SU(2) of static fundamental charges and center monopoles. Dual to temporalcenter fluxes, the former provide a well-dened (dis)order parameter for deconnement. In contrast, the monopolefree energies vanish in the thermodynamic limit at all temperatures and are thus irrelevant for the transition.1. IntroductionFor pure SU(N) gauge theory without quarks’t Hooft’s gauge invariant electric and magneticfluxes [1] describe, respectively, the eect of astatic fundamental color charge and a centermonopole in a nite volume. The partition func-tion of a certain amount of electric/magnetic fluxyields the free energy of a static electric/magneticcharge with boundary conditions to imitate thepresence of its ’mirror’ (anti)charge in a neigh-boring box along the direction of the flux.Herein, we extend our previous study [2] of thepurely electric fluxes in SU(2) by measuring alsothe magnetic ones and combinations of the two.2. Confinement: Twist vs. Electric FluxImposing ’t Hooft’s twisted b.c.’s xes the totalnumbers modulo N of ZN -vortices through the 6planes of the 4-dimensional Euclidean 1/T L3box. In SU(2) for example, twist in one planecorresponds to an ensemble with an odd numberof Z 2-vortices through that plane. It diers by atleast one from the periodic ensemble with an evennumber; and their free-energy dierence is whatit costs to add one such vortex to the system.Intuitively it may help to assume that vorticescan lower their free energy by spreading. At nitetemperature T > 0 it is then clear that we needto distinguish between the twists of two types:Magnetic twist is dened in a purely spatialplane in which the vortex can spread independent∗Talk presented by L. von Smekalof T . It xes the conserved, ZN -valued and gauge-invariant magnetic flux ~m through the plane.Correspondingly, its free energy, or that of a staticcenter monopole, will vanish for L!1 at all Tas we demonstrate in the next section.Twist in a temporal plane is classied by a vec-tor ~k2Z 3N parallel to its spatial edge. The otheredge being of nite length 1/T , the vortices aresqueezed in such a plane more and more with in-creasing temperature. They can no-longer spreadarbitrarily and this is what drives the connementphase transition. In the thermodynamic limit,their free energy approaches zero (innity) for Tbelow (above) Tc [2{4]. This is the reversed be-havior of a static fundamental charge.As shown by ’t Hooft, the partition functionsof xed units of electric and magnetic fluxes,~e, ~m2Z 3N , which we denote by Ze(~e, ~m), are ob-tained as 3-dimensional ZN -Fourier transforms,w.r.t. the temporal ~k-twist, of those with twistedb.c.’s, Zk(~k, ~m). With the free energy of a purelyelectric flux one measures that of static funda-mental charge in a perfectly well-dened (UV-regular) way [2]. This follows from the gauge-invariant denition of the Polyakov loop P (~x) inpresence of temporal twist which entails,Ze(~e, 0) =〈P (~x)P †(~x + L~e)L,T, (1)relative to the no-flux ensemble (i.e., the expecta-tion value is taken therein and the l.h.s. is normal-ized such that Ze(0, 0) = 1). In Ref. [2] we mea-sured the ratios of partition functions Zk(~k, 0)with dierent ~k-twists and the Ze(~e, 0) for the20.30.40.50.60.70.80.911.13 4 5 6 7 8 9 10 11L/aT=Tcexp(-fc e- σs(Tc) L2 )Zk(0,1)Ze(0,1)Figure 1. Finite volume partition functions of onemagnetic flux at Tc, Zk(0, 1) and Ze(0, 1) relativeto the periodic and no-flux ensemble, respectively.various electric fluxes in SU(2). We explicitlydemonstrated the Kramers-Wannier duality, be-tween the temporal center fluxes and the staticcharges, of the general patternZe(~e, 0)L→∞−!(0 , Zk(~k, 0)!1, T < Tc1 , Zk(~k, 0)!0, T > Tc(2)reflecting the dierent realization of the 3-dim.(electric) center symmetry in both phases. As ex-pected from universality, this duality is the ana-logue in SU(2) of that between the Wilson loopsof the 3d Z 2-gauge theory and the 3d-Ising spins.3. Magnetic FluxThere is no analogue in the 3d-spin systems tothe magnetic fluxes, however. These are temporal’t Hooft loops of maximal size winding on the duallattice around the temporal 1/TL planes. Themagnetic current along the ’t Hooft loop in thiscase creates a pair of static center monopoles ata distance L. Magnetic flux ~m thus correspondsto one static center monopole inside the L3 boxwith mirror image in a neighboring box along thedirection of ~m { the magnetic analogue of thePolyakov loop correlator on the r.h.s. of Eq. (1).In agreement with this electric-magnetic dual-ity, the temporal ’t Hooft loops in SU(2) showa screening behavior in both phases, for T < Tcand for T >Tc [3,5], just as spatial Wilson loopsexhibit an area law in either case. One can thus00.20.40.60.810.6 0.8 1 1.2 1.4 1.6 1.8 2T / TcZk(0,1)4463x483x4Figure 2. Magnetic flux versus temperature.anticipate likewise that the monopole free energyin both phases must vanish in the thermodynamiclimit. This would not exclude that it approachesa nite value at T = Tc as in the case of electriccharges and temporal center fluxes. We obtainedin Ref. [2], e.g., with ~k = (0, 0, 1) for one unit ofthe latter, Zk(~k, 0) = 0.54(1) at Tc in agreementwith the universally related ratio in the Isingmodel. In particular, if the massless phase whichthe system passes through at Tc was selfdual, onewould also expect a nite monopole free energy atT =Tc. This is, however, not the case, as shownin Fig. 1. The free energies of the magnetic fluxesin SU(2) at Tc are well described by an exponen-tial decrease / expf−σs(Tc)L2g, where σs is thespatial string tension. As a check, our t thenyields for the spatial string tension at Tc,σs = (2.2 0.2)T 2c , or Tc/pσs = 0.675 0.03 ,consistent with published values for the zero tem-perature SU(2) string tension [6]. This screeningof the temporal ’t Hooft loops is similar to thatof spatial ones in the low temperature phase:In [2] we demonstrated that − ln Zk(~k, 0) /expf−σ(T )L/T g for temporal ~k-twist at T < Tcand suciently large L. The important dierenceis the temperature dependence of the standardstring tension σ(T ), as compared to the practi-cally constant spatial one, σs here. As a result,the dominant behavior of the monopole free en-ergies cannot be described by nite size scaling.That they indeed vanish for all other T also, isseen in Fig. 2 where we plot the partition function300.20.40.60.810.6 0.8 1 1.2 1.4 1.6 1.8 2T / Tc63x4Ze(0,1)Ze(1,0)Ze(1,1) par.Ze(1,1) orth.Figure 3. Electric and magnetic fluxes on the 634lattice versus temperature.Zk(0, ~m) of one magnetic twist (e.g., ~m=(0, 0, 1))versus temperature. While it is indistinguishablefrom a temporal twist on the symmetric 44 latticeas expected, with increasing spatial lattice size itrapidly and smoothly approaches unity over thewhole temperature range of our simulations.Our measurements of the flux partition func-tions Ze(~e, ~m) from two dierent lattice sizes areshown for various combinations of magnetic withelectric fluxes in Figs. 3 and 4. The dierencesbetween parallel (~e  ~m odd) and orthogonal (~e  ~meven) fluxes on the smaller volumes have previ-ously been observed at T =0 also [7]. They werethen found in good agreement with semiclassicalpredictions. As the purely magnetic free energyvanishes, or Ze(0, ~m)! 1 with increasing spatialsize, these dierences disappear and the partitionfunctions approach those of purely electric fluxesfor which we include the t from [2] in the gures.Similarly, our data for all other combinations oftwists and fluxes in SU(2) indicate, at all T ,Ze(~e, ~m)L→∞−! Ze(~e, 0) , Zk(~k, ~m) L→∞−! Zk(~k, 0).Magnetic fluxes are irrelevant for the phase tran-sition, and center monopoles always ’condense’.The corresponding 3-dim. magnetic center sym-metry remains unbroken at all temperatures.4. ConclusionsTo summarize, changing the spatial twist isa gauge-invariant way of introducing one more00.20.40.60.810.6 0.8 1 1.2 1.4 1.6 1.8 2T / Tc83x4 Ze(0,1)Ze(1,0)Ze(1,1) par.Ze(1,1) orth.Figure 4. Same as in Fig. 3 for the 834 lattice.static center monopole. The monopole free en-ergy in the 3d Georgi-Glashow model was recentlystudied in an analogous way [8]. As therein, weobserve for SU(2) that the monopole free energyis zero, in the thermodynamic limit, at all tem-peratures. In contrast to the crossover behaviorof the Georgi-Glashow model, however, in SU(2)there is no indication of a plateau with nitemonopole mass at intermediate volumes either.Simulations were performed on the SGI Originsystems at RRZE, Erlangen, and ZHR, Dresden.REFERENCES1. G. ’t Hooft, Nucl. Phys. B153 (1979) 141.2. Ph. de Forcrand and L. von Smekal, Phys.Rev. D 66 (RC) (2002) 011504; Nucl. Phys.B106 (PS) (2002) 619.3. T. G. Kovacs, E. T. Tomboulis, Phys. Rev.Lett. 85 (2000) 704; Nucl. Phys. B106 (PS)(2002) 670; Phys. Rev. D 65 (2002) 074501.4. L. von Smekal, with Ph. de Forcrand, to app.in Confinement, Topology and other Non-Per-turbative Aspects of QCD, Eds. J. Greensiteand S. Olejnik, [hep-ph/0205002].5. Ph. de Forcrand, M. D’Elia, M. Pepe, Phys.Rev. Lett. 86 (2001) 1438.6. J. Fingberg, U. Heller and F. Karsch, Nucl.Phys. B392 (1993) 493.7. A. Gonzalez-Arroyo and P. Martinez, Nucl.Phys. B459 (1996) 337.8. A. C. Davis et al., Phys. Rev. D 65 (2002)125008; JHEP 0011 (2000) 010.

Electric and Magnetic Fluxes in SU(2) Yang-Mills Theory

Copyright © 2003 Published by Elsevier B.V.We measure the free energies inSU(2) of static fundamental charges and center monopoles. Dual to temporal center fluxes, the former provide a well-defined (dis)order parameter for deconfinement. In contrast. the monopole free energies vanish in the thermodynamic limit at all temperatures and are thus irrelevant for the transition.Lorenz von Smekal and Philippe de Forcrandhttp://www.elsevier.com/wps/find/journaldescription.cws_home/505717/description#descriptio

von Smekal, Lorenz Johann Maria

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Electric and magnetic fluxes in(2) Yang-Mills theory

Electric and Magnetic Fluxes in SU(2) Yang-Mills Theory

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