CERN-PH-TH/2015-291 Mean distribution approach to spin and gauge theories Oscar Akerlund1 and Philippe de Forcrand1,2 1Institut fu¨r Theoretische Physik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland 2CERN, Physics Department, TH Unit, CH-1211 Geneva 23, Switzerland (Dated: January 7, 2016) We formulate self-consistency equations for the distribution of links in spin models and of pla- quettes in gauge theories. This improves upon known mean-field, mean-link, and mean-plaquette approximations in such that we self-consistently determine all moments of the considered variable instead of just the first. We give examples in both Abelian and non-Abelian cases. I. INTRODUCTION 6 1 It is always of interest to think about methods that allow easy extraction of approximate results, even though the 0 computerpoweravailableforexactsimulationsisgrowingataneverincreasingpace. Mean-fieldmethodsareoften 2 qualitatively reliable in their self-consistent determination of the long-distance physics, and have a wide range of applications, with spin models as typical examples. For a gauge theory, formulated in terms of the gauge links, n a however, it is questionable what a mean link would mean, because of the local nature of the symmetry. This J can be addressed by fixing the gauge, but the mean-field solution will then in general depend on the gauge-fixing 6 parameter. Nevertheless, Drouffe and Zuber developed techniques for a mean field treatment of general Lattice Gauge Theories in [1] and showed that for fixed βd, where β is the inverse gauge coupling and d the dimension, ] the mean-field approximation can be considered the first term in a 1/d expansion. They established that the mean t a field approximation can be thought of as a resummation of the weak coupling expansion in a particular gauge and l - that there is a first order transition to a strong coupling phase at a critical value of β. Since it becomes exact in p the d limit, this mean field approximation can be used with some confidence in high-dimensional models [2]. e →∞ The crucial problem of gauge invariance was tackled and solved by Batrouni in a series of papers [3, 4], where he h [ firstchangedvariablesfromgauge-variantlinkstogauge-invariantplaquettes. TheassociatedJacobianisaproduct of lattice Bianchi identities, which enforce that the product of the plaquette variables around an elementary cube 1 is the identity element. In the Abelian case this is easily understood, since each link occurs twice (in opposite v directions) and cancels in this product, leaving the identity element. In the non-Abelian case the plaquettes in 5 7 each cube have to be parallel transported to a common reference point in order for the cancellation to work. It is 1 worthnotingthatintwodimensionstherearenocubessotheJacobianofthetransformationistrivialandthenew 1 degrees of freedom completely decouple (up to global constraints). 0 This kind of change of variables can be performed for any gauge or spin model whose variables are elements . 1 of some group. Apart from gauge theories, examples include ZN-spin models, O(2)- and O(4)-spin models and 0 matrix-valuedspinmodels. Inspinmodels, thechangeofvariablesisfromspinstolinksandtheBianchiconstraint 6 dictates that the product of the links around an elementary plaquette is the identity element. A visualization of 1 the transformation and the Bianchi constraint for a 2d spin model is given in Fig. 1. : v i X r a FIG. 1. The change of variables from spins s (left panel) to links l (right panel) that leads to the Bianchi identity i ij l l l l (=s s†s s†s s†s s†)=1. 12 23 34 41 1 2 2 3 3 4 4 1 Let us review the change of variables for a gauge theory [4]. The original variables are links. The new ones are plaquettes. Undertheactionoftheoriginalsymmetryofthemodel,thenewvariablestransformwithinequivalence classes and it is possible to employ a mean field analysis to determine the “mean equivalence class”. As usual we 2 first choose a set of live variables, which keep their original dynamics and interact with an external bath of mean- valued fields. Interactions are generated through the Jacobian, which is a product of Bianchi identities represented by δ-functions δ U 1 , (1) P (cid:32) − (cid:33) P∈∂C (cid:89) where P denotes a plaquette and ∂C denotes the oriented boundary of the elementary cube C. The δ-functions can be represented by a character expansion in which we can replace the characters at the external sites by their expectation,ormean,values. Upontruncatingthenumberofrepresentations,thisyieldsaclosedsetofequationsin the expectation values which can be solved numerically. The method can be systematically improved by increasing the number of representations used and the size of the live domain. While this method works surprisingly well, even at low truncation, it determines the expectation value of the plaquette in only a few representations. Here, we propose a method that self-consistently determines the complete distributionoftheplaquettes(orlinks)andthustheexpectationvalueinallrepresentations. Thisisduetoanexact treatment of the lattice Bianchi identities which does not rely on a character expansion. The only approximation then lies in the size of the live domain which can be systematically enlarged, as in any mean field method. It is worth noting that our method works best for small β and low dimensions: it does not become exact in the infinite dimension limit. In this way it can be seen as complementary to the mean field approach of [1]. We will also see that the mean distribution approach proposed here actually works rather well for both small and large β. The paper is organized as follows. In section II we describe the method in general terms and compare it to the mean field, mean link and mean plaquette methods before describing more detailed treatments of spin models and gauge theories in sections III and IV respectively. Finally, we draw conclusions in section V. II. METHOD A. Mean Field Theory Let us for completeness give a very brief reminder of standard mean field theory. Consider for definiteness a lattice modelwithasingletypeofvariablesswhichliveonthelatticesites. Thelatticeactionisassumedtobetranslation invariant and of the form 1 S = J s†s + V(s ), (2) −2 |i−j| i j i i,j i (cid:88) (cid:88) where i,j labels the lattice sites and V(s) is some local potential. Let us now split the original lattice into a live domain D and an external bath Dc. The variables s i Dc all take a constant “mean” values. The mean field i { | ∈ } action then becomes (up to a constant) 1 S = J s†s + V(s ) J s†s , (3) MF −2 |i−j| i j  i − |i−j| i  i,j∈D i∈D j∈Dc (cid:88) (cid:88) (cid:88)   wheresis determined by the self-consistency condition that the average value of s in the domain D is equal to the average value in the external bath, ds s e−SMF i i (cid:90) i∈D ! s = (cid:89) =s. (4) (cid:104) (cid:105) ds e−SMF i (cid:90) i∈D (cid:89) Onceshasbeendeterminedthemeanfieldaction(3)canbeusedtomeasureotherobservableslocaltothedomain D. 3 B. Mean Distribution Theory To generalize the mean field approach we relax the condition that the fields at the live sites interact only with the mean value of the external bath. Instead, the fields in the external bath are allowed to vary and take different values distributed according to a mean distribution. The self-consistency condition is thus that the distribution of the variables in the live domain equals the distribution in the bath. Consider a real scalar theory for illustration purposes. Starting from the action S = 2κ φ φ + V (φ ), (5) i j i − (cid:104)i,j(cid:105) i (cid:88) (cid:88) withnearestneighborcouplingκandageneralon-sitepotentialV, weexpandthefieldφ δφ+φarounditsmean ≡ valueφand integrate out all the fields except the field at the origin φ =φ+δφ and its nearest neighbors, denoted 0 0 φ , i=1,...,z, where z is the coordination number of the lattice. The partition function can then be written i z Z = dφ0e−V(φ0)+2zκφ¯δφ0 dδφipJ(δφ1,...,δφz)e2κδφ0(cid:80)zi=1δφi, (6) (cid:90) (cid:90) i=1 (cid:89) where p (δφ ,...,δφ ) is a joint distribution function for the fields around the origin and absorbs everything not J 1 z explicitly depending of δφ into its normalization. So far everything is exact and, given a way to compute p , we 0 J could obtain all local observables, for example φn . Now, p is in general not known, so we will have to make (cid:104) 0(cid:105) J some ansatz and determine the best distribution compatible with this ansatz. In standard mean field theory the z ansatz is p (δφ ,...,δφ ) = δ(δφ ) and only φ is left to be determined as explained above. In the mean J 1 z i=1 i distributionapproach we willassume thatthe distributionis a product distribution p (δφ ,...,δφ )= z p(δφ ) J 1 z i=1 i (cid:81) and determine p self-consistently to be equal to the distribution of δφ , i.e. 0 (cid:81) p(δφ )= 1e−V(δφ0+φ¯)+2zκφ¯δφ0 e2κδφ0δφi z, (7) 0 Z p(δφi) (cid:16)(cid:10) (cid:11) (cid:17) where f(φ) = dφp(φ)f(φ). Themeanvalueφhastobeadjustedsuchthatthedistributionphaszeromean. (cid:104) (cid:105)p(φ) After p andφhave been determined any observable, even observables extending outside the live domain, can be (cid:82) extracted under the assumption that every plaquette is distributed according to p. Local observables are given by simple expectation values with respect to the distribution p. This strategy can also be applied to spin and gauge models, taking as variables the links and plaquettes re- spectively, as discussed in the introduction. For a gauge theory, the starting point is the partition function in the plaquette formulation Z = dU δ U 1 e−S[UP], (8) P P (cid:32) − (cid:33) (cid:90) P C P∈∂C (cid:89) (cid:89) (cid:89) where S[U ] is any action which is a sum over the individual plaquettes, for example the Wilson action S[U ] = p P β (1 ReTrU ), or a topological action [5, 6] where the action is constant but the traces of the plaquette P − P variables are limited to a compact region around the identity. (cid:80) The difference to the mean plaquette method is that it is not assumed that the external plaquettes take some average value, but rather that they are distributed according to a mean distribution. More specifically, we assume that there exists a mean distribution for the real part of the trace of the plaquettes and that the other degrees of freedom are uniformly distributed with respect to the Haar measure. Such a distribution must exist and it can be measured for example by Monte Carlo simulations. For definiteness let us consider compact U(1) gauge theory with a single plaquette P0 as the live domain. The plaquette variables UP = eiθP U(1) can be represented with ∈ a single real parameter θ [0,2π] and the real part of the trace is cosθ . Our goal is to obtain an approximation P P ∈ to the distribution p(cosθ ), or equivalently p(θ )=Z(θ )/Z, where P0 P0 P0 Z(θP0)=e−S[UP0] dUP e−S[UP] δ UP(cid:48) −1, (9) (cid:90)P(cid:89)(cid:54)=P0 (cid:89)C P(cid:48)(cid:89)∈∂C   Z = dU Z(θ ). (10) P0 P0 (cid:90) Toobtainafinitenumberofintegralswenowmaketheapproximationthatallplaquetteswhichdonotshareacube with P are independently distributed according to some distribution p(θ). Clearly this neglects some correlations 0 4 among the plaquettes but this can be improved by taking a larger live domain. Again, let C denote an elementary cube with boundary ∂C and P denote a plaquette. We define U U , (11) C P ≡ P∈∂C (cid:89) C P ∂C , (12) 0 0 C ≡{ | ∈ } P C : P ∂C, P =P , (13) C 0 0 P ≡{ |∃ ∈C ∈ (cid:54) } i.e. isthesetofallcubescontainingP ,and isthesetofplaquettes,excludingP ,makingup . Thesought 0 0 C 0 0 C P C distribution is then determined by the self-consistency equation e−S[UP0] dUP p(θP) δ(UC 1) − p(θ )= (cid:90)P(cid:89)∈PC C(cid:89)∈C0 . (14) P0 dUP0e−S[UP0] dUP p(θP) δ(UC −1) (cid:90) (cid:90)P(cid:89)∈PC C(cid:89)∈C0 Thisself-consistencyequationissolvedbyiterativesubstitution: givenaninitialguessforthedistributionp(0)(θ ), P0 itisastraightforwardtasktointegrateouttheexternalplaquettesandobtainthenextiteratep(1)(θ )fromeq.(14), P0 andtoiteratetheprocedureuntilafixedpointisreached,i.e. p(n+1)(θ )=p(n)(θ ). Thisisafunctionalequation, P0 P0 which is solved numerically by replacing the distribution p by a set of values on a fine grid in θ or by a truncated P expansioninafunctionalbasis. Inthispaperwehavechosentodiscretizethedistributiononagrid. Asmentioned above, this can be done in a completely analogous way also for spin models and for different types of actions. In Fig.2wecomparethedistributionsofplaquettesinthe4dU(1)latticegaugetheorywiththeWilsonactioncloseto the critical coupling (left panel) and with the topological action at the critical restriction δ (right panel), obtained c by Monte Carlo on an 84 lattice and by the mean distribution approach with the normalized action eβcosθP. Below we give more details for a selection of models along with numerical results. 2.5 1.4 MeanMdHoisantatrreibmCuteaiarolsnou,,rehhcc,ooessβθθcPPosiiθP==/00I0..54(β0875) 1.2 2 1 ) 1.5 ) 0.8 δ =δc≈0.6202 P P θ θ p( p( 0.6 1 0.4 0.5 β =0.9 0.2 MMHaoeaanrntemdCiesaatsrruliobr,eu,htci1oons,θhPcio=sθP0.i59=30.549 0 0 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 θP θP FIG. 2. The distribution of plaquettes angles p(θ ) in the 4d U(1) lattice gauge theory with the Wilson action close to the P critical coupling (left panel) and with the topological action at the critical restriction δ (right panel) obtained by Monte c Carlo on an 84 lattice and by the mean distribution approach, together with the Haar measure. III. SPIN MODELS We will start by applying the method to a few spin models, namely Z2, Z4 and the U(1) symmetric XY-model and we will explain the procedure as we go along. Afterwards, only minor adjustments are needed in order to treatgaugetheories. Wewillderivetheself-consistencyequationsinanunspecifiednumberofdimensionsalthough graphical illustrations will be given in two dimensions for obvious reasons. Let us start with an Abelian spin model with a global ZN symmetry. The partition function is given by Z = exp β Res s† , (15)  i j {s} (cid:104)i,j(cid:105) (cid:88) (cid:88)   5 where si =ei2Nπni, ni 1, ,N ( ZN). In the usual mean field approach we would self-consistently determine ∈{ ··· } ∈ the mean value of s by letting one or more live sites fluctuate in an external bath of mean valued spins. However, i Batrouni [3, 7] noticed that by self-consistently determining the mean value of the links, or internal energy, U ij ≡ s s†,muchbetterestimatesofforexamplethecriticaltemperaturecouldbeobtainedforagivenlivedomain. Thus, i j wefirstchangevariablesfromspinstolinks. TheJacobianofthischangeofvariablesisaproductoflatticeBianchi identities, δ(U 1), one for each plaquette [8]. This can be verified by introducing the link variables U via P ij − dU δ U s s† 1 andintegratingoutthespinsinapedestrianmanner. SincetheBoltzmannweightfactorizes ij ij j i − over the(cid:16)link variabl(cid:17)es, all link interactions are induced by the Bianchi identities and hence the transformation (cid:82) trivially solves the one dimensional spin chain where there are no plaquettes [9]. As mentioned above, each δ-function can be represented by a sum over the characters of all the irreducible representations of the group. For ZN this is merely a geometric series, δ(UP −1) = N1 Nn=−01UPn. Since only the realpart enters in theaction itis convenient toreshuffle thesum sothat wesum onlyoverreal combinationsof the (cid:80) variables, N−1 (cid:98) 2 (cid:99) δ(U 1) 1+UN/2δN + Un+U−n , (16) P − ∝ P even P P n=1 (cid:88)(cid:0) (cid:1) where δN is 1 if N is even and 0 otherwise. even The next step is to choose a domain of live links. In this step, imagination is the limiting factor; for a given number of live links there can be many different choices and it is not known to us if there is a way to decide which is the optimal one. The simplest choice is of course to keep only one link alive but in our 2d examples we will make use also of a nine-link domain [7] to see how the results improve with larger domains. These two domains are shownintheleft(onelink)andright(ninelinks)panelsofFig.3. Inthecaseofasinglelivelink, thereare2(d 1) − plaquettes and thus there are 2(d 1) δ-functions of the type in eq. (16). − FIG. 3. Two choices of domains of live links for 2d spin models. The live links are denoted by the solid lines, whereas the dashed lines denote links which are assumed to take mean values or to be distributed according to the mean distribution. The left panel shows the unique domain with one live link and the right panel shows one of many domains with nine live links. A. Mean link approach Let us for simplicity consider the case of one live link, denoted U . The external links, denoted U by some 0 k enumeration ij k, are fixed to the mean value by demanding that Un = U−n = U n, k = 0. Each plaquette → k k (cid:104) (cid:105) ∀ (cid:54) containing the live link also contains three external links, and the δ-function eq. (16) becomes N−1 (cid:98) 2 (cid:99) 2πn n δ(U 1) 1+ U 3N/2( 1)n0δN +2 U 3ncos 0 . (17) P − ∝ (cid:104) (cid:105) − even (cid:104) (cid:105) N n=1 (cid:88) 6 For large N it is best to perform the sum analytically to obtain (for N =2M) 1 ( 1)n0 U 3M δ(U 1) − − (cid:104) (cid:105) . (18) P − ∝ 1+ U 6 2 U 3cosπn0 (cid:104) (cid:105) − (cid:104) (cid:105) M For U(1) we define πMn0 =θ0 as M →∞ and since (cid:104)U(cid:105)<1 we get −1 6 3 δ(U 1) 1+ U 2 U cosθ , (19) P 0 − ∝ (cid:104) (cid:105) − (cid:104) (cid:105) (cid:16) (cid:17) which can efficiently be dealt with by numerical integration. The partition functions for the single live link for Z2, Z4 and U(1) [10] spin models then become 2(d−1) ZZ2 ∝ eβU0 1+(cid:104)U(cid:105)3U0 , (20) U0(cid:88)=±1 (cid:16) (cid:17) 3 ZZ eβcosπn20 1+ U 6( 1)n0 +2 U 3cosπn0 2(d−1), (21) 4 ∝ (cid:104) (cid:105) − (cid:104) (cid:105) 2 n(cid:88)0=0 (cid:16) (cid:17) δ −2(d−1) Z dθeβcosθ 1+ U 6 2 U 3cosθ . (22) U(1) ∝ (cid:104) (cid:105) − (cid:104) (cid:105) −(cid:90)δ (cid:16) (cid:17) In the U(1) case, eq. (22) applies both to the standard action (β 0,δ = π) and to the topological action ≥ (β =0,δ π). ≤ B. Mean distribution approach In the mean distribution approach we sum over the external links assuming they each obey a mean distribution p(U),forwhichaone-to-onemappingtothesetofmoments Un exists. Thedifferencebetweenthetwomethods {(cid:104) (cid:105)} becomes apparent when expressed in terms of the moments, which are obtained by integrating the distributions of the external links against the δ-function given by the Bianchi constraint in eq. (16) N−1 3 (cid:98) 2 (cid:99) 2πn n p(U )p(U )p(U )δ(U 1)=1+ UN/2 UN/2δN +2 Un 3cos 0 . (23) 1 2 3 P − 0 even (cid:104) (cid:105) N {U1(cid:88),U2,U3} (cid:68) (cid:69) n(cid:88)=1 Comparing to eq (17), we see that for N 3 there is only one moment and the two methods are thus equivalent, but for larger N the mean link approach≤makes the approximation Un = U n whereas the mean distribution (cid:104) (cid:105) (cid:104) (cid:105) approach treats all moments correctly. Thus, for small N we do not expect much difference between the two approaches, and this is indeed confirmed by explicit calculations. For U(1), however, there are infinitely many moments which are treated incorrectly by the mean link approach and this renders the mean distribution approach conceptually more appealing. By using the Bianchi identities, one link per plaquette can be integrated out, giving 2(d−1) δ δ 2 Z = dθeβcosθ dθ dθ p(θ )p(θ ) p(2πn θ θ θ ) . (24) U(1) 1 2 1 2 1 2  − − −  (cid:90) (cid:90) n=−2 −δ −δ (cid:88)   It is often convenient not to work solely with distributions of single links, but also of multiple links, which are defined in the obvious way, N N p (Θ) dθ p(θ )δ θ Θ , (25) N i i i ≡ (cid:32) − (cid:33) (cid:90) i=1 i=1 (cid:89) (cid:88) and can efficiently be calculated recursively. The above partition function then simplifies slightly to 2(d−1) δ 2δ 2 Z = dθeβcosθ dΘp (Θ) p(2πn θ Θ) . (26) U(1) 2  − −  (cid:90) (cid:90) n=−2 −δ −2δ (cid:88)   7 In Figs. 4 and 5 we show results for 2d Z2, Z4 and U(1) spin models, the latter for the Wilson action S = β Res s† and the topological action eS = Θ(δ θ θ ). Note the remarkable accuracy of the mean (cid:104)ij(cid:105) i j (cid:104)ij(cid:105) −| i− j| distribution approach in the latter case, even when there is only one live link. (cid:80) (cid:81) 1 1 MonteCarlo Meanlink,onelivelink Meandist.,onelivelink 0.8 0.8 Meanlink,ninelivelinks Meandist.,ninelivelinks 2dZ2 0.6 0.6 2dZ4 lhi lhi 0.4 0.4 Exact 0.2 Onelivelink 0.2 Ninelivelinks βc Twolivespins βc Eightlivespins 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 β β FIG. 4. (left) Mean-field and mean-link approximation in the 2d Ising model for two choices of live domains. (Right) Mean-link and mean-distribution in the 2d Z4 model. In the Ising case, mean-link and mean-distribution are equivalent. 1 1 0.8 0.8 2dXY 2dXY 0.6 0.6 lhi S =β Resis†j lhi eS = Θ(δ−|θi−θj|) 0.4 βc Xhiji 0.4 Yhiji 0.2 0.2 MonteCarlo MonteCarlo Meanlink,onelivelink Meanlink,onelivelink δc Meandist.,onelivelink Meandist.,onelivelink 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 β δ FIG. 5. The mean link in the 2d XY spin model as a function of the Wilson coupling β (left panel) and of the restriction δ (right panel) from Monte Carlo, from the mean link and from the mean distribution methods. IV. GAUGE THEORIES Toextendtheformalismfromspinmodelstogaugetheories, wemerelyhavetochangefromlinksandplaquettes to plaquettes and cubes. The partition function for a U(1) gauge theory analogous to eq.(22) becomes δ −2(d−2) Z = dθeβcosθ 1+ U 10 2 U 5cosθ (27) U(1) (cid:104) (cid:105) − (cid:104) (cid:105) −(cid:90)δ (cid:16) (cid:17) in the mean plaquette approach and 2(d−2) δ 4δ 3 Z = dθeβcosθ dΘp (Θ) p(2πn θ Θ) (28) U(1) 4  − −  (cid:90) (cid:90) n=−3 −δ −4δ (cid:88)   8 in the mean distribution approach. Results for d=4 are shown in Fig. 6 for the Wilson action (left panel) and for the topological action (right panel). 1 1 4dU(1) LGT Wilson action 0.8 0.8 4dU(1) LGT 0.6 0.6 Topological action i i P P U U h h 0.4 0.4 βc δc 0.2 0.2 MonteCarlo MonteCarlo Meanplaq,oneliveplaquette Meanplaq,oneliveplaquette Meandist.,oneliveplaquette Meandist.,oneliveplaquette 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 β δ FIG.6. Themeanplaquetteinthe4dU(1)gaugetheoryasafunctionoftheWilsoncouplingβ(leftpanel)andtherestriction δ (right panel) from Monte Carlo, and from the mean plaquette and the mean distribution methods. Anothernicefeatureofthemeandistributionapproachisthatotherobservablesbecomeavailable,likeforinstance the monopole density in the U(1) gauge theory, under the assumption that each plaquette is distributed according to the mean distribution p. A cube is said to contain q monopoles if the sum of its outward oriented plaquette angles sums up to 2πq. Given the distribution p(θ) of plaquette angles the (unnormalized) probability p of finding q q monopoles in a cube is given by 6 6 p = dθ p(θ )δ θ 2qπ , q 2, 1,0,1,2 (29) q i i i (cid:32) − (cid:33) ∈{− − } (cid:90) i=1 i=1 (cid:89) (cid:88) and the monopole density n is given by monop 2p +4p 1 2 n = . (30) monop p +2p +2p 0 1 2 In Fig. 7 we show the monopole densities for 4d U(1) gauge theory as obtained by Monte Carlo simulations and by the mean distribution approach. Note that the monopole extends outside of the domain of a single live plaquette, whichwasusedtodeterminethe meandistributionp. Theleftpanelshows resultsfortheWilsonaction andinthe right panel the topological action is used. 0.5 0.5 MonteCarlo MonteCarlo 0.45 Meandist.,oneliveplaquette 0.45 Meandist.,oneliveplaquette 0.4 0.4 0.35 4dU(1) LGT 0.35 4dU(1) LGT 0.3 Wilson action 0.3 Topological action p p mono 0.25 mono 0.25 n n 0.2 0.2 0.15 0.15 δc 0.1 βc 0.1 0.05 Con(cid:28)ning Coulomb 0.05 Coulomb Con(cid:28)ning 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 β δ FIG. 7. The monopole density in the 4d U(1) gauge theory as a function of the Wilson coupling β (left panel) and the restriction δ (right panel) from Monte Carlo and the mean distribution method. 9 We can also treat SU(2) Yang-Mills theory without much difficulty. For the mean plaquette approach we need the character expansion of the δ-function ∞ sin(n+1)θ C δ(U 1) (n+1) , (31) C − ∝ sinθ C n=0 (cid:88) where θ is related to the trace of the cube matrix U through TrU =2cosθ . C C C C 5 In the mean plaquette approach we again make the substitution U U U in the case of a single live C 0 → (cid:104) (cid:105) plaquette. The above delta function then becomes ∞ sin(n+1)θ 5 5n 0 δ U U 1 U (n+1) 0 (cid:104) (cid:105) − ∝ (cid:104) (cid:105) sinθ 0 (cid:16) (cid:17) n(cid:88)=0 −2 10 5 1+ U 2cosθ U . (32) 0 ∝ (cid:104) (cid:105) − (cid:104) (cid:105) (cid:16) (cid:17) ForSU(2),theanalogueofarestrictionδ ontheplaquetteangleisarestrictiononthetraceoftheplaquettematrix to the domain [2α,2], where 1 α < 1. If we define a 1TrU = cosθ the approximate SU(2) partition − ≤ 0 ≡ 2 0 0 function can be written [11] in a way very similar to the U(1) partition function (27) 1 −4(d−2) Z = da 1 a2eβa0 1+ U 10 2 U 5a , (33) SU(2) 0 − 0 (cid:104) (cid:105) − (cid:104) (cid:105) 0 (cid:90)α (cid:113) (cid:16) (cid:17) from which U can be easily obtained as a function of α and β. (cid:104) (cid:105) The mean distribution approach works in a completely analogous way as for U(1), but let us go through the details anyway, since there are now extra angular variables to be integrated out. The starting point is again an elementary cube on the lattice. Five of the cubes faces have their trace distributed according to the distribution p(a ) and we want to calculate the distribution of the sixth face compatible with the Bianchi identity U = 1. In 0 C other words, taking U as the live plaquette, we want to evaluate 6 5 6 p(a ) 0,i p˜(a ) dΩ dU δ U 1 , (34) 0,6 6 i i ∝ (cid:90) (cid:90) i(cid:89)=1 (cid:113)1−a20,i (cid:32)i(cid:89)=1 − (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)TrU6=2a0,6 where we have decomposed U =Ω Uˆ Ω† withUˆ a diagonal SU(2) matrix with(cid:12)(cid:12) trace 2a , i.e. Ω is the angular 6 6 6 6 6 0,6 6 part of U . The choice to include the measure factor 1 a2 in the distribution is arbitrary but convenient. To 6 − 0 facilitate the calculation we recursively combine the product of four of the plaquette matrices into one matrix, U U U U U˜, by pairwise convolution of distributio(cid:112)ns (with p (a ) p(a )) 1 2 3 4 1 0 0 → ≡ p (a ) p (a ) p (a˜ ) dΩ˜dU dU i 0,1 i 0,2 δ U U U˜† 1 2i 0 1 2 1 2 ∝ (cid:90) (cid:113)1−a20,1(cid:113)1−a20,2 (cid:16) − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)TrU˜=2a˜0 1 1 (cid:12) (cid:12) da da p (a )p (a ) dcosθ δ a˜ a a 1 a2 1 a2 cosθ (35) ∝ 0,1 0,2 i 0,1 i 0,2 12 0− 0,1 0,2− − 0,1 − 0,2 12 α(cid:90)i −(cid:90)1 (cid:16) (cid:113) (cid:113) (cid:17) 1 p (a )p (a ) i 0,1 i 0,2 = da da χ , α(cid:90)i 0,1 0,2 1−a20,1 1−a20,2 |a˜0−a0,1a0,2|≤√1−a20,1√1−a20,2 (cid:113) (cid:113) where α α, α = max(2α 1, 1) and χ is the characteristic function on the domain A. The domain 1 2i i A ≡ − − of integration in the (a ,a )-plane is simply connected with parametrizable boundaries and comes from the 0,1 0,2 condition that the argument of the delta function has a zero for some cosθ [ 1,1]. We then obtain for the 12 ∈ − sought distribution p(a ) p (a˜ ) p˜(a ) dΩ dU 0,5 dU˜ 4 0 δ U˜U U 1 , (36) 0,6 6 5 5 6 ∝ (cid:90) (cid:90) (cid:113)1−a20,5 (cid:90) (cid:112)1−a˜20 (cid:16) − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)TrU6=2a0,6 (cid:12) (cid:12) 10 where it is now easy to integrate out U˜ = U†U†. If we denote by θ the angle between U and U , the angular 6 5 56 5 6 integral over Ω contributes just a multiplicative constant and we obtain 6 p a a 1 a2 1 a2 cosθ 4 0,5 0,6− − 0,5 − 0,6 56 p˜(a ) da dcosθ p(a ) , (37) 0,6 ∝ 0,5 56 0,5 (cid:16) (cid:113) (cid:113) (cid:17) (cid:90) a a 1 a2 1 a2 cosθ 0,5 0,6− − 0,5 − 0,6 56 (cid:114) (cid:113) (cid:113) which can be evaluated numerically in a straightforward manner. In the end, since there are 2(d 2) cubes sharing − the plaquette P0, and since the a priori probability for P0 to have trace 2a0 is 1−a20eβa0, with respect to the uniform measure, we obtain for one live plaquette (cid:112) 1 1 Z = da p(a )= da 1 a2eβa0p˜(a )2(d−2) SU(2) 0 0 0 − 0 0 (cid:90)α (cid:90)α (cid:113) 2(d−2) 1 1 p a x 1 a2√1 x2cosθ = da 1 a2eβa0 dxp(x)dcosθ 4 0 − − 0 − , (38) (cid:90)α 0(cid:113) − 0 (cid:90)α (cid:16)a0x−(cid:112)1−a20√1−x2cosθ(cid:17)  (cid:113)  (cid:112) which also defines the functional self-consistency equation for p(a ). 0 Results for the Wilson and topological actions can be seen in Fig. 8 in the left and right panels, respectively [12]. 1 1 MonteCarlo MonteCarlo Meanplaquette[3,4] Meanplaquette 0.8 Meandistribution(thiswork) 0.8 Meandistribution Meanlink[1] 0.6 4dSU(2) LGT 0.6 4dSU(2) LGT 2 2 /i Wilson action /i Topological action U U Tr Tr h 0.4 h 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 − − − − − β α FIG. 8. The average plaquette for the SU(2) gauge theory as a function of the Wilson coupling β (left panel) and the restrictionα(rightpanel)fromMonteCarlosimulation,themeanplaquettemethodandthemeandistributionmethod. For comparison the mean link result obtained with the formalism in [1] is also shown in the left panel. For SU(3) one can proceed in an analogous manner, only the angular integrals are now more involved and the trace of the plaquette depends on two diagonal generators so the resulting distribution function needs to be two dimensional. V. CONCLUSIONS It has been shown before [7] that determining a self-consistent mean-link gives a much better approximation than the traditional mean-field. Furthermore, the symmetry-invariant mean link can be generalized to a mean plaquette in gauge theories [3]. Here, we have shown that the approximation can be further improved by determining the self-consistent mean distribution of links or plaquettes. The extension from a self-consistent determination of the symmetry invariant mean link or plaquette to a self-consistent determination of the entire distribution of links and plaquettesisshowntoimproveupontheresultsobtainedbyBatrouniinhisseminalwork[3,4]. Especiallyappealing is the fact that the mean distribution approach yields a non-trivial result for the whole range of couplings and not justinthestrongcouplingregime,whichissometimesthecaseforthemeanlink/plaquetteapproach,orjustinthe weakcouplingregimewhichisaccessibletothemeanfieldtreatmentof[1]. Indeed,themeandistributionapproach gives a nearly correct answer when the correlation length is not too large, and by enlarging the live domain the