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Monopole currents and Dirac sheets in U(1) lattice gauge theory
6 April 1995 PHYSICS
LETTERS
3
PhysicsLettersR 348 ( 19(iS) 565-570
ElSfNER
Monopole currents and Dirac sheetsin U( 1) lattice gaugetheory Werner Kerler a, Claudio Rebbi b, AntireasWebera a Fachbereich Physik, Universitiit Ma?bwR, D-3W.32 Mu’urbq, Germany b Department of Physics, Hoston Lhirwsr@, Boston, WA 02215 USA
Received24 January 1995 E&x 9V. landshoff
Abstract We show that the phases of the rtdimensional compact U( 1) lattice gauge theorj are unambiguous& characterized by the topological properties of minimal Dirac sheets as well as 0: monopole currents lines. ?\e obtain the minimal sheets by a simulated-annealing procedure. Our results indicate that the equivalence classes of sheet structures are the physical relevant quantities and that intersections are not important. In conclusion we get a percolation-type view of the phases which holds beyond the particular boundary conditions ‘used.
1. Introduction
We investigatetheCdimensiona1compactU( 1) lattice gaugetheory with Wilson action sdppler?ented by a .,ionopoleterm [ 1] : S=pC(l-cosB,,x)+ICIMp.xl, /OVA
PJ
where M,,X = epoP,,(6Pu,X+o- 6,,,)/4a and the physical flux 6)clV,xE [-7~, fl) is given by OpV,X= 6 pv,n+ 29r+,,, [ 21. We considerperiodic boundary conditionsexceptfor the discussionin Section5 where we extend our study to a systemwith open boundary conditions. As is well known, the systemhas two phases,separatedfor A = 0 by a first order phasetransition.The strengthof this transition decws with increasingA until the transition ultimately becomesof secondorder [ 3). We have used this property to set up a very efficient algorithm [ 3.41 for Monte Carlo simulations in which A becomesa dynamical variable.The A de0370-2693/95/$09.50
pcndcncehas also allowed us to study the dynamics of the transition in detail [ 3). We arguethat the analysisof the topologicalstrut-, ture of the configurationsis a valuabletool for studying the phasestructure,a tool which can provide much more detailedinformation thanjust the measu:ement of global observables.In the presentwork we use it by analyzingmonopolecurrentsand Dirac sheets. For individud loopsthe topologicalcharacterization is straightforward,but this becomesless !Fjvial when loops areentangledin netwoe of monopolecurrents. Recently we have beenable to producea matbsmatitally soundcharacterizationof the latter [ 31. This is not only necessaryfor the unartbiguousidentification of their physical featuresbut also for the analysisof huge networksby computer. Here the analysisof configurationsis extendedto Dirac sheetsand the fact that apprapriatefyspecilied topological structuressignal the phasesis confirmed in more detail. Enconclusionwe poirit out a general principle for characterizingthe phases,’ which doesnot rely on a particularchoiceof boundaryconditions.
@ 1995 Elsevii Science B.V. All rights reserved
SsDlO370-2693(95)00188-3
566
W. Kerler et al. /Physics Letters B 348 (I 995) 565-570
b
Fig. I. Example of current network with two possibilities for a Dirac sheet on a 2-dimensional lattice with periodic boundary conditions. The shaded (unshaded) area represents a topologically nontrivial (trivial) sheet with given boundary. (b) shifted against (a) by L/2 in both directions.
2. Dual-lattice structures
On the dua>lattice the current Jp,x = II~~,~+~, defined over the links, obeys the conservationlaw C,,( Jp,* - Jp& = 0. Current lines are defined in terms of the current as follows: for JP,X= 0 there is no line on the link, for Jp,x = fl there is one line, and for Jo,x - -+2 there arc two lines, in positive or negativedirection, respectively.Networks of cunents are connectedsetsof current lines. For a’network N disconnectedfrom the rest one can define a net current flow f, with componentsfPj = CXIytXlrlXp2 JIL3,x,
Jp.xE N. The Dirac string content of the plaquetteson the dual lattice is describedin terms of a variableppu,x = - $rpVPVnflV,x+p+fl, which satisfiesthe field equation = Jp,x. To every plaquettein CAPm - p,,,,-,) the dual lattice we will associateno Dirac plaquette if pPcr,X= 0, one Dirac plaquetteif ppo,x = fl, with similar or oppositeorientationaccording to the sign of p, and two Dirac plaquettesif P,,~,~= f2. Dirac sheetsare formed by connectingDirac plaquetteswith a commonedge and appropriateorientation, so that Jp,x = 0 except a’ the boundariesof the sheets.Dirac sheet structuresare not gauge invariant. However,they belong to equivalenceclasses which &mot be deformedinto each other by gauge transformations.For given boundariesthere are topologically distinct possibilitiesfor the sheetstructures. This is illustratedin Fig. 1 by a simple examplein 2 dimensions,where a topologically nontrivial current network may be accompaniedby a trivial or a non-
C
d
Fig. 2. Typical possibilities for intersections of Dirac sheets.
trivial Dirac sheet.The equivalenceclassesof Dirac sheetsthus carry more information than the related current networks. We representthe equivalenceclassesof the,sheet structuresby their memberswith minimal area,which are obtainedby minimizing the numberof Dirac plaquettes by a gauge transformation.This permits a unique specificationand, in view of the large iaitial number of Dirac plaquettca,is necessaryfor a computationalanalysis. To addressthe connectednessof sheet structures, we constructminimal Dirac sheetsby making first the connectionswhere only two Dirac plaquettesmeet at an edge with appropriateorientation. Then we considerthe placeswheremore than two Dirac plaquettes meet at an edge. If the sheetscan be deformed (by a gaugetransformation)in such a way that no more than two plaquettesmeet we connect the respective pairs. Otherwiseall must to be connected. In Fig. 2 we use somesimple 3-dimensionalexamples to illustrate typical possibilities for the intersection of Dime sheetsor for the meeting of more than two plaquettes.In Fig. 2a, deformingone of the sheets by a gauge transformation,the sheetscan be separate& Consequentlythere is no connection.The same holds for Fig. 2b, where one must realize that common sites arenot consideredas a connectionof sheets. In Fig. 2c, althoughone can identify two sheets,the structurecannotbe separatedby a gaugetransforma-
W. Kerier et al. /Physics Letters B 348 (1995) 565-.570
tion and, therefore,is to be consideredas connected. In Fig. 2d separationis obviously not possible.
567
I_I.-.. I__ -7-a 1.0 =I I I 3 Pnet j i * 1 * 0.8 Ei
3. Topolo&al
analysis
The elementsof the fundamentalhomoto, -I group it (X, b) of a spaceX with basepoint b are ,rivalence classesof paths starting and ending at b which can be deformed continuously into each other. Its generatorsmay be obtainzl embeddinga sufficiently dense network N into X and performing suitable transformationswhich preserve homotopy. We observethat, if a give? network N doesnot wrap around in all dixecrions,then only the generatorscf a subgroup are produced. This provides an unambiguous characterizationof networks. Choosingone vertex point of N to be the basepoint b and consideringall paths which start and end ~7:b, we note that a mappingwhich shrinksone edgeto zero length preservesthe homotopy of all of thesepaths. By a sequenceof such mappingswe can then shift all other verticesto b witbout changingthe groupcontent until we finally obtain a bouquet of paths which all start and end at b. Describing a path by a vector which is the sum of oriented steps along the path, for N with KO vertices and KI edgesthe bouquethas K = KI - Ko + 1 loops, representedby vectors Sir with i ‘= 1, . . . , K and jth componentsSij = WijLj, where Lj is the lattice size. The bouquetmatrix Wij must then be analyzedfor the content of generatorsof TT~<@,b) = Z4. Current networks ha*re the additional properties of path orientation and current conservat.ion.The maps reducing the bouquetmatrix Wij, therefore,in addition to homotopy have to respectcurrentconservation. This leads to a modified Gauss elimination procedure,whereby the addition of a row to another requires its simultaneoussubtractionfrom a further row. In this way one arrives at L&eminimal form, with rows ut,. . . , Q~,~,O,.. . ,0, where r < 4. The significance of this minimal matrix becomes obvious if one switches to the pair form with rows 01, -at,. . . , a,, -a,, f, 0,. . . ,O, which exhibits the relation to the net current flow f explicitly. The number of independentpairs determinesthe number of nontrivial directions. For the topologicalanalysisof minimal Dirac sheets
0.8 -
f
i 4 & _ i I
i j
0.4 0.2 -
0.0 L
= * - - x ” : x ?‘~~ -0.3
0.0
A
0.3
fig. 3. F*obability Pact for the occurrenceof a nontivial network in the hst and cold plme at the transition point as function of A for lattice sizes S4 (crosses) and 164 (dots).
we usethe networkof plaquettesas a sufficiently dense auxiliary network,The bouquetmatrix thenis obtained as describedbefore.The reductionof the bouquetmatrix is simpler here becausethe usual Gausselimination applies,which gives the minimal form with rows at,. . . , &, 0, . . . , 0. Now the numberof independent vectorscorrespondsto the numberof nontrivial directions. 4. Numerical resuits Fig. 3 shows our results for the probability P,,t to find a current network which is nontrivial in four directionsas function of A, with p chosenat the corrcspondingphasetransition points [ 3,4). For larger L or negativeA, where the peaksof the energydistribution relatedto the phasesarewell separated[ 3,4], P,,t is seento bc very close to P for the hot (confining) phase,and very close to 0 for the cold (Coulomb) phase.We thus seethat the two phaseshavean unambiguoustopologicalcharacterization,provided by the existenceof a nontrivial network in the hot phaseand its absencein the cold phase. For A = 0 the events with f # 0 are rare; for lattice sizes L = 8 and L = 16 their percentageis less than 2% and OS%, respectively.For psitive I they get slightly more frequent[ 31. As is obviousfrom the pair representationof the bouquetmatrix in Section3, the net currentflow f is not centralfor the topoiogical ,characterization.In this context we should note that the quantityintroducedin [ 51 to classify the networks, bkcauseof current conservation,is equal to f.
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W. Kerler et al. / Phy&s Letters B 348 (199.5) 56.5-570
For minimizing the number of Dirac plaquettes via a sequenceof gauge transformationswe have usedan annealingtechnique[ 61. Our procedureuses Metropolis sweepsbased on a probability distribution P(q)
N exp(-au?)), where q = C,,,,,
l~~,~l,
implementedthrough random local gauge transformations with anglesmaintainedwithin C-P, r). In subsequentsimulationswe increasethe parameterCY in appropriatelychosen steps.First we determine~7 as function of LY(which changeslittle with L and A) using tiny step sizes and large sweep numbers. Then we continuethe procedurewith 4.5valuesof (Y (chosenin the interval [0.03,100] in such a way as to get approximatelyequal changesin r]), perfotming up to 100 sweepsfor each CY.To be sure about the results,for each configurationwe have generally performedthree such runs as wel! as someadditional runs with many more sweeps.R,,eated applications of the method, with different random numbers,lead to consistentvalues for minimal numbersof Dirac plaquettes,with deviations smaller than 2%. Thus our procedureappearsto produce a rather reliable determinationof the absoluteminimum. We found that thecarefulannealingdescribedabove is absolutelynecessary.A minimizationof the number of Dirac plaquettesbasedonly on iterativegaugetransformationsgivesstrongly fluctuatingresults,which on the averageare roughly factors 2 and 1.3 larger in the cold and hot phase,respectively,than the minimal numbersobtainedby our procedure.This is not only inappropriatein principle but also not sufficient for a propercomputationalidentificationof the topological structures. Before applying the annealingprocedurethe numbersof Dirac plaquettesare similar in both phases,but the numbersbecomemarkedly different after annealing. For example,for L = 8 and A = 0, in typical configurations with 8703 and 8793 plaquettesin the hot and cold phase,respectively,our procedurereduced the numbersto 306 and 846. The strongerreduction in the cold phaseis mainly due to the disappearance of closedtrivial structures.We also note that the procedurenot only affects sheetsseparately,but involves the full set of them. For example,evenpairs of oppositely orientednontriviaI sheets,first observedin [ 81, disappearwith our annealingprocedure. Our analysisof minimal Dirac sheetstructuresleads to’results which agree with those derived from the
study of currentnetworks.The hot phaseis characterized by the existenceof a Dirac sheet which is nontrivial in all directions and the cold phaseby its absence.To understandthe significanceof this result it is important to rememberthat, given a nontrivial current network, the sheetscould be trivial or nontrivial (as illustrated by the example of Fig. 1). Thus our results show that in the hot phasethe placeme:,tof the largestsheetis one which makes it topologically nontrivial. This points to the remarkablefact that the (equivalenceclassesof) Dirac sheetstructuresarethe physically relevantobjects. In this context it should be rememberedthat some time ago Griisch et al. [7] were able to identify nontrivial Dirac sheets without boundaries as the structuresresponsiblefor the metastablestatesoccurring undercertain conditionsin the cold phase.Here we have shown that nontrivial sheet structureswith boundariescharacterizethe hot phase. To study the role of connectedness,we construct minimal Dirac sheetsby ..n&ing first the connections where only two Dirac plaquettes meet at an edge with appropriateorientation.We note then that making either all or none of the connectionswhere more than two plaquettesmeet (with correct orientation) changesthe numbersof sheetsand networks, and the sizes of the largest sheets,only by 10% or less and doesnot affect the topology.Looking into more detail we find that, for the most part, thesechangesare due to intersectionsof the type of Figs. 2a and 2b, where the sheetsmust actually be kept separated.Taking this into accountthe changereducesto 3% or less.We thus obtain the further remarkableresult that intersections of minimal sheetsare not important. In Fig. 4 we presentthe sizesof the largestcurrent network and of the largestminimal Dirac sheet.Obviously the sheetsgive a more sensitivesignal for the different phases.This may be explained by the fact that areainsteadof length enters.It is to be noted that the definition we have used for the connectednessof sheetsplays an important role in this result. If one considersas connectedsheetswhich havejust sites in common,then the signal provided by the size of the largestminimal sheetsbecomescomparableto the one given by the networks. Fig. 5 indicatesthat the probability Pnetof the occurrenceof a nontrivial network, taking values very close 1 or 0 for hot and cold’phase,respectively,may
W. Kerler et ul. /Physics
?-~~-----c-/
links
._--
4
-0.3
0.0
- --I”-----:
( plaq.
l 1 600
hot !sheet)
h
Latrers B 348 (1995) 565-570
0.3
-
Fig. 4. Size of largest current network (squares, links)andlargest minimal Dirac sheet (dots, plaquettes)in thehot and cold phase at thetransition point,asfunctionof A on 84 lattice.
.‘tot 1 t--*-----+-s
0.4 0.2
0.0
1
- n.
I
0.6
\r
c
1.000
1.010
i
p
1.020
Fig. 5. Order parametersPnet andnmar;/nt,,t forA= 0 asfunctions of /3 on 84 lattice.
be a moresensiti+eorderparameterthan nmax/ntot, the relativesize of the largestnetwork,advocatedin [ 91. 5. Discussion
The successof our topologicalorderparameterconfirms that the phase transition of the 4-dimensional compact U( 1) gauge field theory coincideswith a percolation-typetransition of the topological structures. Examplesof theorieswith different behaviors haverecentlybe discussedin [ 9 3. In order to see the generalmeaningof this results we note that a topologically nontrivial structureon a finite lattice with periodicboundaryconditions,i.e. on the torus p, &responds to an infinite structureon an infinite lattice. Thus the hot phaseis more gener-
569
ally characterizedby the existenceof an infinite current networkor Dirac sheetand the cold phaseby its absence,whereon finite lattices“infinite” is to be defined in accordancewith the particularboundaryconditions which are used. To test this c;laraclrization we alsoperformedsimulations with open boundaryconditions [lo]. In the correspondinganalysisthe prescription“nontrivial in all directions”is replacedby “touching the boundary in all directions”.In this way we again get an unambiguoussignal for the phases.The comparisonof the order parameterslooks very similar to Fig. 5, the advantageof our order parameterbeing even more pronounced.Becauseof finite-sizeeffects, however,the width of transitionregionbecomeslarger,for L = 16, by a factor of approximately100 . In order to obtain information about the order of the phasetransitionthe most immediatething to do is to look whetherthereis an energygap for sufficiently large L. With periodic boundariesL = 8 is sufficient for seeinga gap whereasL = 4 is not. With open boundariesa gapis not yet seenfor L = 16. The width of the peak is relatively narrow (comparableto the width of the peakswith L = 8 in the periodic-boundary case) [ 101. It appearsto us that lattices well larger than L = 16 will be neededin order to seea possible gap in systemswith open boundaryconditions. In [ 111 the simulationshave been performedon the surfaceof a 5-dimensionalcube which is homeomorphicto the sphereS4.Consideringlatticesup to L = 10 the authorsobserveonly one peak. Its width for L = 10 is comparableto those of the peakson @ for L = 8. The extensionof the transitionregion, which could alsoprovideinformationaboutfinite-size effects,hasnot beenreported. In [ 121 simulationsfor latticesup to L 7 16 have beendonewith fixed boundaries(i.e. settingthe group elementsto 1 at the boundaries),whichmakesthe lattitie againhomeomorphicto S4. With theseboundary conditionsthe transition region was observedto be wider by a factor.sf about 103,which reflectsthe huge’ finite-sizeeffectscausedby the stronginhomogeneity of the system.Becauseof low statisticsthe resultsin this work are of semiqualitativenatureand the energy distributionis not given. It seemsto us that in the two geometrieshomeomorphic to S4 mentionedabovelargerlatticeswould also be necessary for decidingaboutthe existenceof a gap.
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W. Kerler et al. /Physics
Further,in thesegeometriesthe analysisof configurations of monopoIecrlrrentsand Dirac sheetswould be of great interest, too. Appropriatedefinitions of “infinite” on finite lattices should of coursebe give?. It should be realized that thesegeometriesare different from a physical point of view. Fixed boundaryconditions can be expectedto lead to the same(flat-space) limit as periodic or open boundaryconditions,while for the surface of a S-dimensionalhypercubesome curvature(dependingor. ihe d;tii; of the limit) could persist. A further type of boundarycondition,proposedin [ 131,consistsin the suppressionof (all) monopolesat the boundaries.This is doneby usingan action leading to the suppressionof monopoles[ 141 at the boundaries and the usual Wilson action elsewhere.Though the definition of “infinite” getssomewhatdilute in that case we expect it to fit into our picture as well. The existenceof a gap with this type of boundarycondition remainsto be confirmed. Acknowledgement
This researchwas supportedin part under DFG grantsKe 25017-2and25011l-l andunderDOE grant DE-FGO2-91ER40676. The computationswere done on the CM5 of the Centerfor ComputationalScience of Boston University and on the CM5 of the GMD at St. Augustin.
Letters B 348 (1995) 56.5-570
References [I ] J.S. Barber, R.E. Shrock and R. Schrader,Phys.Len. B 152 ( 1985) 221. [2] T. De&and and D, Toussaint,Phys. Rev. ~322 ; 1980) 2478. i3] W. Kerler, C. Rebbi and A. Weber,Phys. Rev. D SO (1994) 6984.
[4] W. Kerler, C. Rebbi and A. Webcr,Boston University preprint BUREP-94-36 ( 1994). [5] A. Bode, T. Lippert and K. Schilling, Nuci. Phys. B (Proc.
Suppl.)34 (1994)549. [61 S. Kirkpatrick, C.D. Gerlatt Jr. and M.P. Vecchi, Science 220(1983)671. I71 V. Grlisch, K. Hansen,J. retxik, C.B. Lang, T. Neuhausand C. Rebbi, Phys. Len. B 162 (1985) 171. [S] V.G. Bomyakov, V.K. Mitrujushkin and M. Miiller-Pteussker, Phys. Lett. B 317 (1943) 596. [9] M. Baig, H. Fort and J.B. Kogut, Phys. Rev. D 50 (1994) 5920.
[lo] W. Kerler, C. Rabbi and A. Weber, in preparation. 111 ] C.B. Lang and T. NeuhaLs,Nucl. Phys B (Proc. Suppl.) 34 (1994) 543; Nucl. Phys. B 431 (1994) 119; J. Jets& C.B. Lang and T. Neuhaus, preprint heplatI9412051 (1994). [ 121 M. Baig and H. Fott, Phys. Lea. I3 332 (1994) 428. [ 131 lb. Lippert, A. Bode, V. Bomyakov and K. Schilling, Wuppertal University preprint WUB 94-38 (1994). [ 141 V.G. Bomyakov,V.K. Mitruius~.~inand ‘1. Mtiller-Preussker. Nucl. Phys. B (Pox. Suppi.) 30 (1993) 587.
LETTERS
3
PhysicsLettersR 348 ( 19(iS) 565-570
ElSfNER
Monopole currents and Dirac sheetsin U( 1) lattice gaugetheory Werner Kerler a, Claudio Rebbi b, AntireasWebera a Fachbereich Physik, Universitiit Ma?bwR, D-3W.32 Mu’urbq, Germany b Department of Physics, Hoston Lhirwsr@, Boston, WA 02215 USA
Received24 January 1995 E&x 9V. landshoff
Abstract We show that the phases of the rtdimensional compact U( 1) lattice gauge theorj are unambiguous& characterized by the topological properties of minimal Dirac sheets as well as 0: monopole currents lines. ?\e obtain the minimal sheets by a simulated-annealing procedure. Our results indicate that the equivalence classes of sheet structures are the physical relevant quantities and that intersections are not important. In conclusion we get a percolation-type view of the phases which holds beyond the particular boundary conditions ‘used.
1. Introduction
We investigatetheCdimensiona1compactU( 1) lattice gaugetheory with Wilson action sdppler?ented by a .,ionopoleterm [ 1] : S=pC(l-cosB,,x)+ICIMp.xl, /OVA
PJ
where M,,X = epoP,,(6Pu,X+o- 6,,,)/4a and the physical flux 6)clV,xE [-7~, fl) is given by OpV,X= 6 pv,n+ 29r+,,, [ 21. We considerperiodic boundary conditionsexceptfor the discussionin Section5 where we extend our study to a systemwith open boundary conditions. As is well known, the systemhas two phases,separatedfor A = 0 by a first order phasetransition.The strengthof this transition decws with increasingA until the transition ultimately becomesof secondorder [ 3). We have used this property to set up a very efficient algorithm [ 3.41 for Monte Carlo simulations in which A becomesa dynamical variable.The A de0370-2693/95/$09.50
pcndcncehas also allowed us to study the dynamics of the transition in detail [ 3). We arguethat the analysisof the topologicalstrut-, ture of the configurationsis a valuabletool for studying the phasestructure,a tool which can provide much more detailedinformation thanjust the measu:ement of global observables.In the presentwork we use it by analyzingmonopolecurrentsand Dirac sheets. For individud loopsthe topologicalcharacterization is straightforward,but this becomesless !Fjvial when loops areentangledin netwoe of monopolecurrents. Recently we have beenable to producea matbsmatitally soundcharacterizationof the latter [ 31. This is not only necessaryfor the unartbiguousidentification of their physical featuresbut also for the analysisof huge networksby computer. Here the analysisof configurationsis extendedto Dirac sheetsand the fact that apprapriatefyspecilied topological structuressignal the phasesis confirmed in more detail. Enconclusionwe poirit out a general principle for characterizingthe phases,’ which doesnot rely on a particularchoiceof boundaryconditions.
@ 1995 Elsevii Science B.V. All rights reserved
SsDlO370-2693(95)00188-3
566
W. Kerler et al. /Physics Letters B 348 (I 995) 565-570
b
Fig. I. Example of current network with two possibilities for a Dirac sheet on a 2-dimensional lattice with periodic boundary conditions. The shaded (unshaded) area represents a topologically nontrivial (trivial) sheet with given boundary. (b) shifted against (a) by L/2 in both directions.
2. Dual-lattice structures
On the dua>lattice the current Jp,x = II~~,~+~, defined over the links, obeys the conservationlaw C,,( Jp,* - Jp& = 0. Current lines are defined in terms of the current as follows: for JP,X= 0 there is no line on the link, for Jp,x = fl there is one line, and for Jo,x - -+2 there arc two lines, in positive or negativedirection, respectively.Networks of cunents are connectedsetsof current lines. For a’network N disconnectedfrom the rest one can define a net current flow f, with componentsfPj = CXIytXlrlXp2 JIL3,x,
Jp.xE N. The Dirac string content of the plaquetteson the dual lattice is describedin terms of a variableppu,x = - $rpVPVnflV,x+p+fl, which satisfiesthe field equation = Jp,x. To every plaquettein CAPm - p,,,,-,) the dual lattice we will associateno Dirac plaquette if pPcr,X= 0, one Dirac plaquetteif ppo,x = fl, with similar or oppositeorientationaccording to the sign of p, and two Dirac plaquettesif P,,~,~= f2. Dirac sheetsare formed by connectingDirac plaquetteswith a commonedge and appropriateorientation, so that Jp,x = 0 except a’ the boundariesof the sheets.Dirac sheet structuresare not gauge invariant. However,they belong to equivalenceclasses which &mot be deformedinto each other by gauge transformations.For given boundariesthere are topologically distinct possibilitiesfor the sheetstructures. This is illustratedin Fig. 1 by a simple examplein 2 dimensions,where a topologically nontrivial current network may be accompaniedby a trivial or a non-
C
d
Fig. 2. Typical possibilities for intersections of Dirac sheets.
trivial Dirac sheet.The equivalenceclassesof Dirac sheetsthus carry more information than the related current networks. We representthe equivalenceclassesof the,sheet structuresby their memberswith minimal area,which are obtainedby minimizing the numberof Dirac plaquettes by a gauge transformation.This permits a unique specificationand, in view of the large iaitial number of Dirac plaquettca,is necessaryfor a computationalanalysis. To addressthe connectednessof sheet structures, we constructminimal Dirac sheetsby making first the connectionswhere only two Dirac plaquettesmeet at an edge with appropriateorientation. Then we considerthe placeswheremore than two Dirac plaquettes meet at an edge. If the sheetscan be deformed (by a gaugetransformation)in such a way that no more than two plaquettesmeet we connect the respective pairs. Otherwiseall must to be connected. In Fig. 2 we use somesimple 3-dimensionalexamples to illustrate typical possibilities for the intersection of Dime sheetsor for the meeting of more than two plaquettes.In Fig. 2a, deformingone of the sheets by a gauge transformation,the sheetscan be separate& Consequentlythere is no connection.The same holds for Fig. 2b, where one must realize that common sites arenot consideredas a connectionof sheets. In Fig. 2c, althoughone can identify two sheets,the structurecannotbe separatedby a gaugetransforma-
W. Kerier et al. /Physics Letters B 348 (1995) 565-.570
tion and, therefore,is to be consideredas connected. In Fig. 2d separationis obviously not possible.
567
I_I.-.. I__ -7-a 1.0 =I I I 3 Pnet j i * 1 * 0.8 Ei
3. Topolo&al
analysis
The elementsof the fundamentalhomoto, -I group it (X, b) of a spaceX with basepoint b are ,rivalence classesof paths starting and ending at b which can be deformed continuously into each other. Its generatorsmay be obtainzl embeddinga sufficiently dense network N into X and performing suitable transformationswhich preserve homotopy. We observethat, if a give? network N doesnot wrap around in all dixecrions,then only the generatorscf a subgroup are produced. This provides an unambiguous characterizationof networks. Choosingone vertex point of N to be the basepoint b and consideringall paths which start and end ~7:b, we note that a mappingwhich shrinksone edgeto zero length preservesthe homotopy of all of thesepaths. By a sequenceof such mappingswe can then shift all other verticesto b witbout changingthe groupcontent until we finally obtain a bouquet of paths which all start and end at b. Describing a path by a vector which is the sum of oriented steps along the path, for N with KO vertices and KI edgesthe bouquethas K = KI - Ko + 1 loops, representedby vectors Sir with i ‘= 1, . . . , K and jth componentsSij = WijLj, where Lj is the lattice size. The bouquetmatrix Wij must then be analyzedfor the content of generatorsof TT~<@,b) = Z4. Current networks ha*re the additional properties of path orientation and current conservat.ion.The maps reducing the bouquetmatrix Wij, therefore,in addition to homotopy have to respectcurrentconservation. This leads to a modified Gauss elimination procedure,whereby the addition of a row to another requires its simultaneoussubtractionfrom a further row. In this way one arrives at L&eminimal form, with rows ut,. . . , Q~,~,O,.. . ,0, where r < 4. The significance of this minimal matrix becomes obvious if one switches to the pair form with rows 01, -at,. . . , a,, -a,, f, 0,. . . ,O, which exhibits the relation to the net current flow f explicitly. The number of independentpairs determinesthe number of nontrivial directions. For the topologicalanalysisof minimal Dirac sheets
0.8 -
f
i 4 & _ i I
i j
0.4 0.2 -
0.0 L
= * - - x ” : x ?‘~~ -0.3
0.0
A
0.3
fig. 3. F*obability Pact for the occurrenceof a nontivial network in the hst and cold plme at the transition point as function of A for lattice sizes S4 (crosses) and 164 (dots).
we usethe networkof plaquettesas a sufficiently dense auxiliary network,The bouquetmatrix thenis obtained as describedbefore.The reductionof the bouquetmatrix is simpler here becausethe usual Gausselimination applies,which gives the minimal form with rows at,. . . , &, 0, . . . , 0. Now the numberof independent vectorscorrespondsto the numberof nontrivial directions. 4. Numerical resuits Fig. 3 shows our results for the probability P,,t to find a current network which is nontrivial in four directionsas function of A, with p chosenat the corrcspondingphasetransition points [ 3,4). For larger L or negativeA, where the peaksof the energydistribution relatedto the phasesarewell separated[ 3,4], P,,t is seento bc very close to P for the hot (confining) phase,and very close to 0 for the cold (Coulomb) phase.We thus seethat the two phaseshavean unambiguoustopologicalcharacterization,provided by the existenceof a nontrivial network in the hot phaseand its absencein the cold phase. For A = 0 the events with f # 0 are rare; for lattice sizes L = 8 and L = 16 their percentageis less than 2% and OS%, respectively.For psitive I they get slightly more frequent[ 31. As is obviousfrom the pair representationof the bouquetmatrix in Section3, the net currentflow f is not centralfor the topoiogical ,characterization.In this context we should note that the quantityintroducedin [ 51 to classify the networks, bkcauseof current conservation,is equal to f.
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W. Kerler et al. / Phy&s Letters B 348 (199.5) 56.5-570
For minimizing the number of Dirac plaquettes via a sequenceof gauge transformationswe have usedan annealingtechnique[ 61. Our procedureuses Metropolis sweepsbased on a probability distribution P(q)
N exp(-au?)), where q = C,,,,,
l~~,~l,
implementedthrough random local gauge transformations with anglesmaintainedwithin C-P, r). In subsequentsimulationswe increasethe parameterCY in appropriatelychosen steps.First we determine~7 as function of LY(which changeslittle with L and A) using tiny step sizes and large sweep numbers. Then we continuethe procedurewith 4.5valuesof (Y (chosenin the interval [0.03,100] in such a way as to get approximatelyequal changesin r]), perfotming up to 100 sweepsfor each CY.To be sure about the results,for each configurationwe have generally performedthree such runs as wel! as someadditional runs with many more sweeps.R,,eated applications of the method, with different random numbers,lead to consistentvalues for minimal numbersof Dirac plaquettes,with deviations smaller than 2%. Thus our procedureappearsto produce a rather reliable determinationof the absoluteminimum. We found that thecarefulannealingdescribedabove is absolutelynecessary.A minimizationof the number of Dirac plaquettesbasedonly on iterativegaugetransformationsgivesstrongly fluctuatingresults,which on the averageare roughly factors 2 and 1.3 larger in the cold and hot phase,respectively,than the minimal numbersobtainedby our procedure.This is not only inappropriatein principle but also not sufficient for a propercomputationalidentificationof the topological structures. Before applying the annealingprocedurethe numbersof Dirac plaquettesare similar in both phases,but the numbersbecomemarkedly different after annealing. For example,for L = 8 and A = 0, in typical configurations with 8703 and 8793 plaquettesin the hot and cold phase,respectively,our procedurereduced the numbersto 306 and 846. The strongerreduction in the cold phaseis mainly due to the disappearance of closedtrivial structures.We also note that the procedurenot only affects sheetsseparately,but involves the full set of them. For example,evenpairs of oppositely orientednontriviaI sheets,first observedin [ 81, disappearwith our annealingprocedure. Our analysisof minimal Dirac sheetstructuresleads to’results which agree with those derived from the
study of currentnetworks.The hot phaseis characterized by the existenceof a Dirac sheet which is nontrivial in all directions and the cold phaseby its absence.To understandthe significanceof this result it is important to rememberthat, given a nontrivial current network, the sheetscould be trivial or nontrivial (as illustrated by the example of Fig. 1). Thus our results show that in the hot phasethe placeme:,tof the largestsheetis one which makes it topologically nontrivial. This points to the remarkablefact that the (equivalenceclassesof) Dirac sheetstructuresarethe physically relevantobjects. In this context it should be rememberedthat some time ago Griisch et al. [7] were able to identify nontrivial Dirac sheets without boundaries as the structuresresponsiblefor the metastablestatesoccurring undercertain conditionsin the cold phase.Here we have shown that nontrivial sheet structureswith boundariescharacterizethe hot phase. To study the role of connectedness,we construct minimal Dirac sheetsby ..n&ing first the connections where only two Dirac plaquettes meet at an edge with appropriateorientation.We note then that making either all or none of the connectionswhere more than two plaquettesmeet (with correct orientation) changesthe numbersof sheetsand networks, and the sizes of the largest sheets,only by 10% or less and doesnot affect the topology.Looking into more detail we find that, for the most part, thesechangesare due to intersectionsof the type of Figs. 2a and 2b, where the sheetsmust actually be kept separated.Taking this into accountthe changereducesto 3% or less.We thus obtain the further remarkableresult that intersections of minimal sheetsare not important. In Fig. 4 we presentthe sizesof the largestcurrent network and of the largestminimal Dirac sheet.Obviously the sheetsgive a more sensitivesignal for the different phases.This may be explained by the fact that areainsteadof length enters.It is to be noted that the definition we have used for the connectednessof sheetsplays an important role in this result. If one considersas connectedsheetswhich havejust sites in common,then the signal provided by the size of the largestminimal sheetsbecomescomparableto the one given by the networks. Fig. 5 indicatesthat the probability Pnetof the occurrenceof a nontrivial network, taking values very close 1 or 0 for hot and cold’phase,respectively,may
W. Kerler et ul. /Physics
?-~~-----c-/
links
._--
4
-0.3
0.0
- --I”-----:
( plaq.
l 1 600
hot !sheet)
h
Latrers B 348 (1995) 565-570
0.3
-
Fig. 4. Size of largest current network (squares, links)andlargest minimal Dirac sheet (dots, plaquettes)in thehot and cold phase at thetransition point,asfunctionof A on 84 lattice.
.‘tot 1 t--*-----+-s
0.4 0.2
0.0
1
- n.
I
0.6
\r
c
1.000
1.010
i
p
1.020
Fig. 5. Order parametersPnet andnmar;/nt,,t forA= 0 asfunctions of /3 on 84 lattice.
be a moresensiti+eorderparameterthan nmax/ntot, the relativesize of the largestnetwork,advocatedin [ 91. 5. Discussion
The successof our topologicalorderparameterconfirms that the phase transition of the 4-dimensional compact U( 1) gauge field theory coincideswith a percolation-typetransition of the topological structures. Examplesof theorieswith different behaviors haverecentlybe discussedin [ 9 3. In order to see the generalmeaningof this results we note that a topologically nontrivial structureon a finite lattice with periodicboundaryconditions,i.e. on the torus p, &responds to an infinite structureon an infinite lattice. Thus the hot phaseis more gener-
569
ally characterizedby the existenceof an infinite current networkor Dirac sheetand the cold phaseby its absence,whereon finite lattices“infinite” is to be defined in accordancewith the particularboundaryconditions which are used. To test this c;laraclrization we alsoperformedsimulations with open boundaryconditions [lo]. In the correspondinganalysisthe prescription“nontrivial in all directions”is replacedby “touching the boundary in all directions”.In this way we again get an unambiguoussignal for the phases.The comparisonof the order parameterslooks very similar to Fig. 5, the advantageof our order parameterbeing even more pronounced.Becauseof finite-sizeeffects, however,the width of transitionregionbecomeslarger,for L = 16, by a factor of approximately100 . In order to obtain information about the order of the phasetransitionthe most immediatething to do is to look whetherthereis an energygap for sufficiently large L. With periodic boundariesL = 8 is sufficient for seeinga gap whereasL = 4 is not. With open boundariesa gapis not yet seenfor L = 16. The width of the peak is relatively narrow (comparableto the width of the peakswith L = 8 in the periodic-boundary case) [ 101. It appearsto us that lattices well larger than L = 16 will be neededin order to seea possible gap in systemswith open boundaryconditions. In [ 111 the simulationshave been performedon the surfaceof a 5-dimensionalcube which is homeomorphicto the sphereS4.Consideringlatticesup to L = 10 the authorsobserveonly one peak. Its width for L = 10 is comparableto those of the peakson @ for L = 8. The extensionof the transitionregion, which could alsoprovideinformationaboutfinite-size effects,hasnot beenreported. In [ 121 simulationsfor latticesup to L 7 16 have beendonewith fixed boundaries(i.e. settingthe group elementsto 1 at the boundaries),whichmakesthe lattitie againhomeomorphicto S4. With theseboundary conditionsthe transition region was observedto be wider by a factor.sf about 103,which reflectsthe huge’ finite-sizeeffectscausedby the stronginhomogeneity of the system.Becauseof low statisticsthe resultsin this work are of semiqualitativenatureand the energy distributionis not given. It seemsto us that in the two geometrieshomeomorphic to S4 mentionedabovelargerlatticeswould also be necessary for decidingaboutthe existenceof a gap.
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Further,in thesegeometriesthe analysisof configurations of monopoIecrlrrentsand Dirac sheetswould be of great interest, too. Appropriatedefinitions of “infinite” on finite lattices should of coursebe give?. It should be realized that thesegeometriesare different from a physical point of view. Fixed boundaryconditions can be expectedto lead to the same(flat-space) limit as periodic or open boundaryconditions,while for the surface of a S-dimensionalhypercubesome curvature(dependingor. ihe d;tii; of the limit) could persist. A further type of boundarycondition,proposedin [ 131,consistsin the suppressionof (all) monopolesat the boundaries.This is doneby usingan action leading to the suppressionof monopoles[ 141 at the boundaries and the usual Wilson action elsewhere.Though the definition of “infinite” getssomewhatdilute in that case we expect it to fit into our picture as well. The existenceof a gap with this type of boundarycondition remainsto be confirmed. Acknowledgement
This researchwas supportedin part under DFG grantsKe 25017-2and25011l-l andunderDOE grant DE-FGO2-91ER40676. The computationswere done on the CM5 of the Centerfor ComputationalScience of Boston University and on the CM5 of the GMD at St. Augustin.
Letters B 348 (1995) 56.5-570
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