MONOPOLES AND CHAOS Harald Markum, RainerPullirsch, and Wolfgang Sakuler Atominstitut, Technische Universit¨at Wien, A-1040 Vienna, Austria 2 0 WedecomposeU(1)gaugefieldsintoamonopoleandphotonpartacrossthephase 0 transition from the confinement to the Coulomb phase. We analyze the leading 2 Lyapunov exponents of such gauge field configurations on the lattice which are initializedbyquantumMonteCarlosimulations. Itturnsoutthatthereisastrong n relationbetweenthesizesofthemonopoledensityandtheLyapunovexponent. a J 3 1 Motivation 1 ThemainquestioninQCDisthemechanismofconfiningquarksintohadrons. v One seeks for special classes of gauge fields which are responsible for confine- 1 ment. Intheframeworkofthedualsuperconductor,colormagneticmonopoles 0 arise as the appropriate candidates. Their investigation necessitates a projec- 0 1 tion of the non-Abelian gauge theory on the compact U(1) theory. In lattice 0 computations it was demonstrated that monopoles account for more than 90 2 percent of the string tension in the confinement and they drop toward zero 0 across the phase transition. / t The study ofchaotic dynamics ofclassicalfield configurationsinfield the- a l ory finds its motivation in phenomenological applications as well as for the - p understanding of basic principles. The role of chaotic field dynamics for the e confinementofquarksisalongstandingquestion. Here,weanalyzetheleading h Lyapunov exponents of compact U(1) configurations on the lattice. The real- : v time evolution of the classical field equations was initialized from Euclidean i equilibriumconfigurationscreatedbyquantumMonteCarlosimulations. This X waywe expect to see a relationshipbetweenthe appearanceofmonopolesand r a the strength of chaotic behavior in lattice simulations. 2 Monopoles in compact quantum field theories We have investigated 4d U(1) gauge theory described by the action S{U}= (1−cosθ ), (1) l p Xp with U =U =exp(iθ ) and θ =θ +θ −θ −θ (ν 6=µ) . l x,µ x,µ p x,µ x+µˆ,ν x+νˆ,µ x,ν At critical coupling β ≈1.01 U(1) gauge theory undergoes a phase transition c between a confinement phase with mass gap and monopole excitations for β <β and the Coulomb phase with a massless photon for β >β , see Fig. 1. c c 1 0.012 0.009 > 2 W| 0.006 <| 0.003 0 0.97 0.99 1.01 1.03 1.05 β Figure1: AverageoftheabsolutePolyakovloopsquaredh|W|2iasafunctionofβ ona124 lattice. We are interested in the relationship between monopoles of U(1) gauge theory and classical chaos across this phase transition. Following Refs. 1, 2, 3, 4, we have factorized our gauge configurations into monopole and photon fields. The U(1) plaquette angles θ are decomposed into the “physical” x,µν electromagneticflux throughthe plaquetteθ¯ andanumberm ofDirac x,µν x,µν strings passing through the plaquette θ =θ¯ +2πm , (2) x,µν x,µν x,µν where θ¯ ∈ (−π,+π] and m 6= 0 is called a Dirac plaquette. Monopole x,µν x,µν and photon fields are then defined in the following way θxm,oµn =−2π Gx,x′∂ν′ mx′,νµ (3) Xx′ θxp,hµot =− Gx,x′∂ν′ θ¯x′,νµ . (4) Xx′ Here ∂′ acts on x′, the quantities m and θ¯ are defined in Eq. (2) and x,µν x,µν Gx,x′ isthelatticeCoulombpropagator. Onecanshowthatθ˜x,µ ≡θxm,oµn+θxp,hµot is up to a gauge transformation identical with the original θ defined by x,µ U =exp(iθ ). x,µ x,µ 3 Classical chaotic dynamics from quantum Monte Carlo initial states Chaotic dynamics in general is characterized by the spectrum of Lyapunov exponents. These exponents, if they are positive, reflect an exponential diver- 2 genceofinitiallyadjacentconfigurations. Incaseofsymmetriesinherentinthe Hamiltonian of the system there are corresponding zero values of these expo- nents. Finally negative exponents belong to irrelevant directions in the phase space: perturbationcomponentsinthesedirectionsdieoutexponentially. Pure gaugefields onthe lattice showacharacteristicLyapunovspectrumconsisting ofonethirdofeachkindofexponents.5 Assumingthisgeneralstructureofthe Lyapunov spectrum we investigate presently its magnitude only, namely the maximal value of the Lyapunov exponent, L . max The general definition of the Lyapunov exponent is based on a distance measure d(t) in phase space, 1 d(t) L:= lim lim ln . (5) t→∞d(0)→0 t d(0) In case of conservative dynamics the sum of all Lyapunov exponents is zero according to Liouville’s theorem, L = 0. We utilize the gauge invariant i distancemeasureconsistingofthelPocaldifferencesofenergydensitiesbetween two 3d field configurations on the lattice: 1 ′ d:= |trU −trU |. (6) NP XP P P Herethesymbol standsforthesumoverallN plaquettes,sothisdistance P P ′ is bound in thePinterval (0,2N) for the group SU(N). UP and UP are the familiar plaquette variables, constructed from the basic link variables U , x,i U =exp aAc Tc , (7) x,i x,i (cid:0) (cid:1) located on lattice links pointing from the position x = (x ,x ,x ) to x+ae . 1 2 3 i ThegeneratorsofthegroupareTc =−igτc/2withτc beingthePaulimatrices in case of SU(2) and Ac is the vector potential. The elementary plaquette x,i variable is constructed for a plaquette with a corner at x and lying in the † † ij-plane as U = U U U U . It is related to the magnetic field x,ij x,i x+i,j x+j,i x,j strength Bc : x,k U =exp ε aBc Tc . (8) x,ij ijk x,k The electric field strengthEc is relat(cid:0)edto the can(cid:1)onicallyconjugate momen- x,i tum P =U˙ via x,i x,i Ec = 2atr TcU˙ U† . (9) x,i g3 x,i x,i (cid:16) (cid:17) The Hamiltonian of the lattice gauge field system can be casted into the form 1 1 H = hP,Pi + 1− hU,Vi . (10) (cid:20)2 4 (cid:21) X 3 Here the scalar product stands for hA,Bi = 1tr(AB†). The staple variable 2 V is a sum of triple products of elementary link variables closing a plaquette with the chosen link U. This way the Hamiltonian is formally written as a sum over link contributions and V plays the role of the classical force acting on the link variable U. Initial conditions chosen randomly with a given average magnetic energy per plaquette have been investigated in past years.6 In the present study we prepare the initial field configurations from a standard four dimensional Eu- clidean Monte Carlo program on a 123×4 lattice varying the inverse gauge couplingβ.7 We relatesuchfourdimensionalEuclideanlatticefieldconfigura- tions to Minkowskian momenta and fields for the three dimensional Hamilto- niansimulationbyidentifyingafixedtimesliceofthefourdimensionallattice. 4 Chaos and confinement We start the presentation of our results with a characteristic example of the time evolution of the distance between initially adjacent configurations. An initial state prepared by a standard four dimensional Monte Carlo simulation isevolvedaccordingtothe classicalHamiltoniandynamicsinrealtime. After- wards this initial state is rotated locally by group elements which are chosen randomly near to the unity. The time evolutionof this slightly rotatedconfig- uration is then pursued and finally the distance between these two evolutions is calculated at the corresponding times. A typical exponential rise of this distance followed by a saturation can be inspected in Fig. 2 from an example of U(1) gauge theory in the confinement phase and in the Coulomb phase. While the saturationis anartifactofthe compactdistance measureofthe lat- tice, the exponential rise (the linear rise of the logarithm) can be used for the determination of the leading Lyapunov exponent. The left plot exhibits that in the confinement phase the original field and its monopole part have similar LyapunovexponentswhilethephotonparthasasmallerL . Therightplot max in the Coulomb phase suggests that allslopes andconsequently the Lyapunov exponents of all fields decrease substantially. WenowturntoacomparisonwiththemonopoledensityinU(1)quantum field configurations, 1 Q = |m |. (11) mon x,µν 4V 4 xX,µν TheleftplotofFig.3exhibitsforastatisticsof100independentconfigurations thatQ decreasessharplybetweenthestrongandtheweakcouplingregime. mon It can be seen that the photon fields carry only a few monopoles. 4 4 4 ’full’ ’full’ ’monopole’ ’monopole’ 2 ’photon’ 2 ’photon’ 0 0 d(t) -2 d(t) -2 ln -4 ln -4 -6 -6 -8 -8 -10 -10 0 10 20 30 40 50 0 10 20 30 40 50 t t Figure2: Exponentially divergingdistanceofinitiallyadjacent U(1)fieldconfigurations on a 123 lattice prepared at β = 0.9 in the confinement phase (left) and at β = 1.1 in the Coulombphase(right). 0.3 1 ’full’ ’full’ ’monopole’ ’monopole’ 0.25 ’photon’ ’photon’ 0.75 0.2 > mon 0.15 max 0.5 Q L < 0.1 0.25 0.05 0 0 0.7 0.8 0.9 1 1.1 1.2 1.3 0.7 0.8 0.9 1 1.1 1.2 1.3 β β Figure 3: Monopole density (left) and Lyapunov exponent (right) of the decomposed U(1) fieldsasafunctionofcoupling. The main result of the present study is the dependence of the leading Lyapunov exponent L on the inverse coupling strength β, displayed in the max rightplotofFig.3. Asexpectedthestrongcouplingphase,whereconfinement of static sources has been established many years ago by proving the area law behavior for large Wilson loops, is more chaotic. The transition reflects the critical coupling to the Coulomb phase. The right plot of Fig. 3 shows that the monopole fields carry Lyapunov exponents of nearly the same size as the full U(1) fields. The photon fields yield a non-vanishing value in the confinementascendingtowardβ =0forrandomizedfieldswhichindicatesthat the decomposition (1) works perfectly for ideal configurations only. 5 5 Conclusion We investigated the classical chaotic dynamics of U(1) lattice gauge field con- figurations prepared by quantum Monte Carlo simulation. The fields were decomposed into a photon and monopole part. The maximal Lyapunov ex- ponent shows a pronounced transition as a function of the coupling strength indicating that on a finite lattice configurations in the strong coupling phase are substantially more chaotic than in the weak coupling regime. The compu- tations give evidence that the Lyapunov exponents in the original U(1) field and in its monopole part are very similar. The situation for the monopole density is analogous and serves as a consistency check of the decomposition. We conclude that classical chaos in field configurations and the existence of monopoles are intrinsically connected to the confinement of a theory. Acknowledgments This work has been supportedby the AustrianNationalScientific Fund under the project FWF P14435-TPH. We thank Bernd A. Berg and Urs M. Heller as well as Tama´s S. Bir´o and Natascha Ho¨rmann for previous cooperation concerningtopologicalobjects andclassicalchaosin U(1) theory, respectively. [1] J.D. Stack and R.J. Wensley, Nucl. Phys. 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