Topological order and the vacuum of Yang-Mills theories
TTopological order and the vacuum of Yang-Mills theories.
G. Burgio and H. Reinhardt Institut f¨ur Theoretische Physik, Auf der Morgenstelle 14, 72076 T¨ubingen, Germany
We study, for SU (2) Yang-Mills theories discretized on a lattice, a non-local topo-logical order parameter, the center flux z . We show that: i) well defined topologicalsectors classified by π ( SO (3)) = Z can only exist in the ordered phase of z ; ii)depending on the dimension 2 ≤ d ≤ T . Yang-Mills theories can thus existin two different continuum phases, characterized by an either topologically orderedor disordered vacuum; this reminds of a quantum phase transition, albeit controlledby the choice of symmetries and not by a physical parameter. a r X i v : . [ h e p - l a t ] J a n I. INTRODUCTION
Of all ideas applied to the confinement problem in non-abelian Yang-Mills theories [1, 2]the most popular still involve topological degrees of freedom of some sort [3–11]. Amongthese center vortices [12–14] have enjoyed broad attention, in particular in the lattice lit-erature. Although most of the effort was put in dealing with gauge fixed schemes, someinvestigations actually attempted to tackle the problem in a gauge invariant way [17–23]and are therefore directly related to ’t Hooft’s original idea.Non-abelian gauge fields transform under the group’s adjoint representation, SU ( N ) / Z N .Such group is not simply connected, with a non trivial first homotopy class: π ( SU ( N ) Z N ) = Z N , (1)the center of SU ( N ). Following Ref. [13], let us consider the Euclidean Yang-Mills theoryon a d -dimensional torus, i.e. with all directions compactified, and choose one of the d Euclidean directions as time. If large gauge transformations classified by Eq. (1) inducea super-selection rule, we can decompose the physical Hilbert space H of gauge invariantstates [25] in sub-spaces H (cid:126)k, (cid:126)m labeled by topological indices, the Z N electric and magneticfluxes (vortices) (cid:126)k = ( k , . . . , k n t ) and (cid:126)m = ( m . . . , m n s ) [13]: H = N − (cid:77) k i ,m j =0 H (cid:126)k, (cid:126)m . (2)Here n t = d − n s = ( d − d − the space-space planes and k i , m i ∈ Z N . As ’t Hooft pointed out, a sufficient condition for confinement is realized ifthe low-temperature phase of pure Yang-Mills theories corresponds to a superposition ofall (electric) sectors, while above the deconfinement transition such Z N symmetry must getbroken to the trivial one [12, 13]. One can check such scenario by calculating, e.g. in lattice simulations, how the freeenergy for Z N flux creation: F ( (cid:126)k ) = ∆ U (cid:126)k − T ∆ S (cid:126)k = − log Z ( (cid:126)k ) Z ( (cid:126)
0) (3) See e.g. Refs. [15, 16] for early results and Ref. [2], Chapts. 6, 7 for a comprehensive review. The goalhere is to isolate some relevant degrees of freedom, usually called P-vortices, assumed to be related to ’tHooft’s topological excitations; we will comment on this in Sec. IV. See e.g. Ref. [24] for an extensive introduction to the subject. Magnetic sectors, on the other hand, can remain unbroken and be responsible for screening effects. changes with the temperature T across the deconfinement transition. Here ∆ U (cid:126)k is the energy(action) cost to generate the (cid:126)k th (electric) vortex from the vacuum, ∆ S (cid:126)k the correspondingentropy change and Z ( (cid:126)k ) the partition function restricted to the topological sector labeledby (cid:126)k . For the one-vortex sector, F is nothing but the free energy of a maximal ’t Hooftloop, giving a confinement criterion dual to Wilson’s: in the thermodynamic limit F shouldvanish in the confined phase while it should diverge as ˜ σ ( T ) L above the deconfinementtemperature T c , where ˜ σ ( T ) is the dual string tension [2, 13, 17–21, 27]. In other words, aperimeter law for the Wilson loop implies an area law for the ’t Hooft loop and vice-versa[12, 13].Of course, when considering the theory at T = 0, the distinction between electric andmagnetic fluxes is artificial. In this case all the N d ( d − topological sectors must be takeninto account when establishing whether Z N -symmetry is unbroken, i.e. whether the vacuum | Ψ (cid:105) is indeed a symmetric superposition of states belonging to H (cid:126)k, (cid:126)m : | Ψ (cid:105) = (cid:88) (cid:126)k, (cid:126)m | Ψ (cid:126)k, (cid:126)m (cid:105) . (4)In the following we will use either definition, depending on whether we are considering the T = 0 or the T > SU ( N ) / Z N . One possibility amongmany (see e.g. Ref. [28]) is given by the adjoint Wilson action with periodic boundaryconditions [27]: S A = β A (cid:88) P (cid:18) − N − A U P (cid:19) , (5)where U P is the standard plaquette. For N = 2 it was indeed shown in Refs. [21, 22, 29]that for simulations based on Eq. (5):i) Z topological sectors are well defined in the continuum limit, both below and above T c ,i.e. the decomposition in Eq. (2) holds; In the deconfined phase all electric sectors must be suppressed relatively to the trivial one, while in theconfined phase all (cid:126)k and (cid:126)m should be equally probable. For a representation of the ’t Hooft loop in the continuum see Ref. [26] Actually, in virtue of cubic symmetry, one can regard the sub-spaces H (cid:126)k,(cid:126)m with indices equal up toa permutation as equivalent and recombine them in Eq. (4) into weights given by their combinatorialmultiplicity [17–19]. ii) the partition function Z A = (cid:82) exp ( − S A ) dynamically includes all sectors;iii) in the deconfined phase all non trivial sectors are suppressed, while as T → T = 0 all the way up to T c is difficult; theevidence given in Refs. [21, 22] seems to point to a more complicated picture.Alternatively, universality [30] should allow the use of the fundamental Wilson action: S F = β F (cid:88) P (cid:18) − N (cid:60) [Tr F U P ] (cid:19) , (6)which is the quenched (mass → ∞ ) limit of the physical action coupling Yang-Mills theoriesto fundamental fermions, e.g. full QCD. In this case, however, some care must be taken indefining a SU ( N ) / Z N invariant theory. Indeed, in the presence of fundamental fermions thetopological classification of Eq. (1) breaks down. The extension to full QCD has been indeedone of the main obstacles in establishing the ’t Hooft vortex picture as a viable model forconfinement. We will comment on this in Sec. IV; for the moment, let us note that one canstill introduce vortex topological sectors “statically” by simply imposing twisted boundaryconditions [19, 20, 27]. We should then be able to reconstruct the “full” partition function Z F by taking the weighted sum of all partition functions Z F ( (cid:126)k, (cid:126)m ) = (cid:82) exp ( − S F ( (cid:126)k, (cid:126)m ))with boundary conditions corresponding to the sector labeled by (cid:126)k and (cid:126)m [19, 20]. Sinceeach Z F ( (cid:126)k, (cid:126)m ) must be determined via independent simulations, their relative weights canonly be calculated through indirect means. Still, such simulations are computationallymore efficient than in the Z A case and have therefore been the method of choice in mostinvestigations of Eq. (3) [17–20, 23].Investigations using Eq. (6) rely on the assumption that fixing the boundary conditions isenough to ensure that the Hilbert-space decomposition defined in Eq. (2) works. However,it is well known that upon discretization of Yang-Mills theories Z N magnetic monopoles aregenerated at strong coupling [28, 35–39], causing bulk phenomena in the β F − β A phase See e.g. Ref. [31] for a recent discussion. Such topological boundary conditions, relevant e.g. in investigations of large N reduction [24, 32, 33],allow adjoint fermions but no fundamental ones. Flavour twisted boundary conditions, on the other hand,are well established in full QCD [34]. See e.g. Ref. [23], Chapt. 3 for a detailed review of the methods involved. diagram. Now, since the Z N fluxes defining our topological sectors live on the co-set of atwo dimensional plane, they have a simple geometrical interpretation: they are describedin d = 4 by a closed world-sheet, i.e. they are string-like objects, and in d = 3 by a closed world-line, i.e. particle-like. On the other hand, topological lattice artifacts as the abovementioned Z N monopoles are themselves sources of Z N flux: in d = 4 they will be particle-like objects, their closed world-lines bounding open Z N flux world-sheets, while in d = 3 theywill be instanton-like objects and will be end-points of open Z N flux lines [27, 35–37, 39]. Z N monopoles are therefore in one to one correspondence with open center vortices; in otherwords, universality between the fundamental and adjoint actions can only be invoked whenjust closed , i.e. truly topological Z N vortices winding around the compactified directionscan form. Notice how in d = 2, where no Z N monopoles can exist, Z N fluxes are instantontype objects. The distinction between open and closed vortices is in this case blurred, butin a non-ergodic setup it can eventually be made through the flux allowed by the boundaryconditions chosen.The above discussion has a straightforward consequence. If one could “measure” whetheropen Z N vortices are absent in a given discretization, i.e. whether only topological vorticescan be generated from the vacuum, there would be no need to monitor Z N monopoles toestablish universality between S A and S F in the first place, since these must be absentanyway. This would have two advantages: first, such criterion could be generalized to d = 2. Second, absence of lattice artifacts, whether for S F , for S A or for both, would get“promoted” to a necessary condition for the super-selection rule of Eq. (2), and hence for theconjectured vacuum symmetry of Eq. (4), to be realized. Indeed, consider states belongingto distinct topological sectors labeled by the indices k , k (cid:48) ∈ Z N . The presence of openvortices immediately blurs the distinction among them: does the state pictured at the topof Fig. 1 belong to the k th sector, resulting from the superposition of k closed vortices withmod( k (cid:48) − k ) N open ones (middle picture), or does it belong to the k (cid:48) th sector, coming fromthe superposition of k (cid:48) closed vortices with mod( k − k (cid:48) ) N open ones winding in the otherdirection (bottom picture)? Clearly, there is no way to distinguish between them and assignthe configuration to Z ( k ) rather than Z ( k (cid:48) ) in Eq. (3). In other words, a Wilson loop willnever know if the vortex piercing it to generate the area law for its expectation value [12, 13]is open or closed: a confinement criterion based on the vortex free energy F and hence onthe ’t Hooft loop can only make sense if open vortices are absent at any temperature. mod(k’−k) N k kk k’k’mod(k−k’) N mod(k−k’) N FIG. 1. Illustration of the ambiguity in labeling the Z N topological sectors in presence of openvortices. Should the given configuration (top) be counted to the k th (middle) or to the k (cid:48) th (bottom)sector? In this paper we will investigate a topological order parameter, the center flux z , for thetransition between phases characterized by the presence of open or closed Z vortices in SU (2) Yang-Mills theories at T = 0, discretized through standard plaquette actions. Wewill show that, depending on the action, the dimensions and the volume, the theory can beeither in a topologically ordered or disordered phase; such distinction will persist at finite T .In the disordered phase open vortices dominate the vacuum and Z topological sectors are illdefined; the Hilbert space of Yang-Mills theories cannot be classified by a super-selection asin Eq. (2). Such disordered phase is compatible with the presence of fundamental fermions;the ordered phase, on the other hand, should be the correct one when coupling SU (2) withadjoint fermions, a popular candidate for infrared conformal gauge theories.Besides this (perhaps lengthy) introduction, the rest of the paper is organized as follows:Sec. II contains details on the lattice setup, observables and simulation techniques; in Sec. IIIthe main results will be presented; Sec. IV contains the conclusions and outlook. Preliminaryresults of this investigations have been presented in Refs. [40, 41]. II. SETUPA. Action and Observables
We will consider the SU (2) mixed fundamental-adjoint Wilson action with periodicboundary conditions in 2 ≤ d ≤ S = β A (cid:88) P (cid:18) −
13 Tr A U P (cid:19) + β F (cid:88) P (cid:18) −
12 Tr F U P (cid:19) a − d g = 14 β F + 23 β A , (7)where a is the (dimensionful) lattice spacing and U P denotes the 1 × SU ( N ) thegeneral picture should not change dramatically [29, 42–46]. However, other representationsthan just the fundamental and adjoint are allowed. Many details might therefore depend on N ; direct investigations of at least the SU (3) case would be welcome.In d ≥ Z monopoles can be defined for each elementary cube c through the product: σ c = (cid:89) P ∈ ∂c sign(Tr F U P ) , (8)over all plaquettes U P belonging to its surface ∂c [27–29, 38, 47]. Notice how rescaling anylink by a Z factor will leave σ c unchanged.The Z monopole density should vanish in the continuum limit g →
0. This happens,however, in different ways, depending on the dimensions d or the direction along which suchlimit is taken in the β F − β A plane, and has been the subject of intense investigations inthe pioneering years of lattice gauge theories [38, 42–46, 48–54]. For the SU (2) − SO (3)case considered here the resulting phase diagrams in d = 3 and 4 are sketched in Figs. 2,3; similar ones have been established for N ≥
3, see e.g. Refs. [42–46]. Continuous linesindicate bulk transitions [38, 49, 50, 52], dashed lines the roughening transition [48], dottedlines the crossover regions associated with Z monopoles [38, 52]. The vertical bulk transitionline coming down form β A = ∞ corresponds to the underlying Z gauge theory: in d = 3 itends at a finite point [52–54], while in d = 4 it joins the bulk transition line associated with Z monopoles [38, 49, 50]; from the endpoint of the latter a crossover region starts, extendingbeyond the β F axis. In d = 2, Z monopoles are of course absent. Furthermore, the Z gauge F ββ A
3d SU(2)
Z gauge theory FIG. 2. Phase diagram of the fundamental-adjoint plane for d = 3. Continuous lines indicate bulktransitions, dashed lines the roughening transition, dotted lines the crossover regions associatedwith Z monopoles. Similar diagrams hold for higher N .FIG. 3. Phase diagram of the fundamental-adjoint plane for d = 4. Continuous lines indicate bulktransitions, dashed lines the roughening transition, dotted lines the crossover regions associatedwith Z monopoles. Similar diagrams hold for higher N . theory has no phase transition; apart from the roughening transition [48], the correspondingphase diagram should therefore be free of any bulk effects, including crossovers.From Fig. 3 it is obvious that two distinct continuum limits in d = 4 exist, depending if g → What is thus the difference, if any, between them?A first hint towards an explanation to this (long neglected) puzzle is given by the resultsof Refs. [21, 22, 29, 47, 57–62]: in the continuum limit the d = 4 adjoint theory ( β F ≡ Z topological sectors, i.e. no openvortices: the Hilbert-space decomposition defined in Eq. (2) works! On the other hand, onecan easily check that in phase I, across all crossovers, the Z monopole density vanishes quiteslowly as g →
0: their persistence in the weak coupling phase should reflect itself in thepresence of open Z vortices, possibly spoiling Eq. (2). Could the difference between phaseI and II lie in whether such super-selection rule is indeed realized for the Hilbert space ofYang-Mills theories? To find out, we can start from the twist operator, which “counts” the Z vortices piercing all parallel planes for a fixed choice of µ - ν [13, 27]: z µν = 1 L d − (cid:88) ˆ y ⊥ µν − plane (cid:89) ˆ x ∈ µν − plane sign(Tr F U µν (ˆ x, ˆ y )) . (9) U µν and ˆ x , respectively, denote a 1 × µ - ν plane, while ˆ y denotes a point on its co-set, which is obviously empty in d = 2; only a single plane con-tributes to the sum in this case. Notice how z µν , like σ c , is unaffected by any multiplicationof links by a center element, i.e. it is insensitive to the spurious Z gauge degrees of freedom.If topological sectors are well defined, all parallel planes will contribute with the samesign to the sum in Eq. (9). For any fixed µ and ν , z µν can thus only take the values ± E.g., for the periodic boundary conditionsconsidered in this paper, the topological sector must always be trivial: z µν ≡ ∀ µ, ν .When, however, topological sectors are ill defined the contributions to the sum in Eq. (9) The authors of Ref. [27] also proved that the 1 st order line separating the two phases is just a finite volumeeffect: at high enough volume Phase I and II will be always separated by a 2 nd order line. Only for β F = 0, i.e. along the β A axis, the z µν are allowed to tunnel among different topological sectors,provided that an ergodic algorithm capable of overcoming the large barriers among them is used. In thiscase the z µν can take both values ± Z vortices pierce the planes randomly,all z µν will average to zero. To make such statement quantitative and characterize how thetransition from the disordered to the ordered regime takes place we define a (non-local!)order parameter, the center flux z , such that its expectation value (cid:104) z (cid:105) ≡ (cid:104) z (cid:105) ≡ Z fluxes are maximally randomized. For d ≥ z = 2 d ( d − d (cid:88) µ>ν =1 | z µν | , (10)while for d = 2, since | z | ≡
1, we will define: z = 1 − | z − (cid:104) z (cid:105)| . (11)Notice that the latter definition will only work as long as β F (cid:54) = 0, i.e. when the d = 2theory cannot tunnel among topological sectors. In the following we will investigate,either analytically (in d = 2) or via Monte-Carlo simulations (for d ≥ χ z = L d ( z − (cid:104) z (cid:105) ) . (12) B. Algorithm
Simulations for β A = 0, i.e. along the β F axis, have been performed using a standardheat-bath algorithm followed by micro-canonical steps. Although this cannot be extended to β A (cid:54) = 0, as long as also β F (cid:54) = 0 one can use the biased Metropolis + micro-canonical algorithmintroduced in Refs. [63, 64]. The lookup tables for the pseudo-heat-bath probability need tobe fixed beforehand: sizes between 32 ×
32 and 64 ×
64 were found to be sufficient [63, 64]. Aslong as β F (cid:29) β A , the algorithm is for all practical purposes just as efficient as an heat-bath, The definition of the center flux in d = 2 might also be adjusted to the pure adjoint theory as long as noergodic algorithm is available in the ordered phase. The issue is similar to that encountered for e.g. anIsing model when simulating the low-temperature phase with a cluster algorithm. Since z is non-local, one could argue that the volume factor should be substituted by the number of planes d ( d − L d − . This would however just change the critical exponent for χ z from L γ → L γ − , which couldbe re-absorbed in the definition of the hyper-scaling relations. Moreover for each plane up to L vorticescan form, summing up again to L d . To underline the analogies of our results with the Kosterlitz-Thoulessliterature we will thus stick to the standard definition. Anyhow, critical behaviours are controlled by adiverging correlation length ξ , which remains unaffected by any re-scaling of χ z . See e.g. Ref. [65] for a recent application. A similar algorithm had been proposed in Ref. [66] for SU (3). β F (cid:28) β A the rejected pseudo-heat-bath and micro-canonical updates increase considerably.This becomes a real issue when simulating around the peaks of the susceptibility Eq. (12),where auto-correlations for z and χ z become quite large. One can try to combat suchcritical slowing down, unavoidable when dealing with any phase transition, by increasingthe number of micro-canonical steps per biased Metropolis update. Unfortunately this turnsout to be less efficient than for the β F (cid:38) β A case or for the heat-bath algorithm; only withruns of order ∼ sweeps one eventually reaches a good signal-to-noise ratio for χ z . Sincethe d = 3 case will anyway turn out to be the most interesting from the point of view ofthe critical behaviour, while in d = 2 analytic results allow to gain otherwise control of theproblem, we have limited a precise finite size scaling (FSS) [67] analysis to determine theproperties of the transition to the β A = 0, d = 3 case. Still, we have performed simulationsfor a whole range of parameters and lattice sizes L in 2 ≤ d ≤
4, trying to explore thewhole β F − β A plane. We have nevertheless avoided phase II of the d = 4 phase diagramin Fig. 3, since it would have called for completely different simulation techniques; seeRefs. [21, 22, 29, 47, 57–62] for results in this parameter region. III. RESULTSA. d = 2 The SU (2) theory in d = 2 offers the chance to tackle our problem analytically [68, 69].The probability distribution for the mixed action in Eq. (7) reads: d ρ ( θ, β F , β A ) ∝ d θ sin θ e β F cos θ + β A cos θ , (13)so that the probability for a plaquette to have negative trace is simply given by: p ( β F , β A ) = (cid:82) π π d θ sin θ e β F cos θ + β A cos θ (cid:82) π d θ sin θ e β F cos θ + β A cos θ . (14) Other observables remain, on the other hand, mostly unaffected. The critical slowing down appears of course also in the limit g →
0, i.e. for large β F and/or β A .
10 12 14 16 18 20 22 24 26 28 3000.10.20.30.40.50.60.70.80.91 β F < z > L=512L=1024L=2048L=4096
FIG. 4. Order parameter (cid:104) z (cid:105) in d = 2 along β F for L = 512, 1024, 2048 and 4096. The limiting cases β F,A → , ∞ can be carried out explicitly, giving: p ( β F ≡ , β A ) = 12 p ( β F ≡ ∞ , β A ) = 0 (15) p ( β F , β A ≡
0) = 12 (cid:20) − L ( β F ) I ( β F ) (cid:21) p ( β F , β A ≡ ∞ ) = 11 + e β F , (16)where L and I denote the modified Struve and Bessel functions, respectively [70].For fixed volume L the order parameter z and its susceptibility χ z are given by [71]: (cid:104) z (cid:105) = e − L p ( β F ,β A ) (cid:104) χ z (cid:105) = L (cid:104) e − L p ( β F ,β A ) − e − L p ( β F ,β A ) (cid:105) . (17)The above expressions are plotted, for β A = 0, in Fig. 4 and 5; a similar behaviour extends tothe whole ( β F , β A ) plane, see Fig. 6, where the center flux is plotted for fixed L = 128. Wecan clearly distinguish a low β F , “strong” coupling regime, where (cid:104) z (cid:105) = 0 and the topology We are indebted to F. Bursa for precious correspondence on the derivation of the above expressions for z and χ z . Just to be on the safe side, we have also cross-checked all analytic results with Monte-Carlosimulations up to L = 1024; these become of course inefficient as β A gets large...
10 12 14 16 18 20 22 24 26 28 3000.511.522.533.54 x 10 β F < χ > L=512L=1024L=2048L=4096
FIG. 5. Susceptibility χ z in d = 2 along β F for L = 512, 1024, 2048 and 4096. is ill defined, from a high β F , “weak” coupling one, where (cid:104) z (cid:105) = 1, the correct value it shouldhave if the vacuum satisfies Eq. (4). For higher L the transition “front” simply moves to theright, i.e. higher β F ; see Fig. 7, where the curves along which the susceptibility χ z peaksare plotted for L = 64, 256, 1024 and 4096.As usual in a FSS analysis, we can determine the properties of the transition by definingthe pseudo-critical couplings ( β cF ( L ) , β cA ( L )) at finite L as those for which the correlationlength ξ (cid:39) L [67]. These can be identified through the peaks of the susceptibility χ z (seeFig. 7); since p ( β F , β A ) has no stationary points, from Eq. (17) one simply needs to solve: p ( β cF , β cA ) = log 24 L . (18)Substituting the above value into Eq. (17) we get for the scaling of the center flux and itssusceptibility with L : z ( β cF , β cA ) = 12 χ z ( β cF , β cA ) = L . (19)As for the scaling of the pseudo-critical points with L , from Eqs. (15) we have that along4 FIG. 6. Order parameter (cid:104) z (cid:105) in d = 2 for fixed L = 128. β F β A L=64L=256L=1024L=4096
FIG. 7. Peak curves of the susceptibility χ z in the β F − β A plane for L = 64, 256, 1024 and 4096. β A axis (cid:104) z (cid:105) = 0. Otherwise, we can fix a line β A = f ( β F ) and solve: p ( β F , f ( β F )) = log 24 L (20)for β F . In particular, from Eq. (16) and using the asymptotic expansion [70]: p ( β F , β A ≡ ∼ (cid:114) β F π e − β F (cid:20) O ( 1 β F ) (cid:21) (21)we get for the two limiting cases β A → , ∞ : β cF ( L ) | β A =0 ∼ log L + 12 log log L + O (1) (22) β cF ( L ) | β A = ∞ ∼
12 log L + O (1) : (23)shifting Eqs. (17) by Eqs. (22, 23) (cid:104) z (cid:105) and (cid:104) χ z (cid:105) will fall on top of each other.Inverting Eqs. (22, 23), one can extract the critical behaviour of the correlation length ξ ∼ L for β A → , ∞ , where the pre-factors come from the O (1) terms: ξ | β A =0 ∼ (cid:115) π log β F · e β F (24) ξ | β A = ∞ ∼ (cid:114) log 24 · e β F . (25)Similar expressions will hold for any direction β A = f ( β F ) along which the continuum limit β F → ∞ is taken. At T > L → L s · L t in Eqs. (18, 19,22, 23). The scaling behaviour remains thus, up to a factor, unchanged when taking thethermodynamic limit L s → ∞ : the critical behaviour will persist for any fixed L t , i.e. atany temperature.Compare now the above scaling with the critical behaviour of the Kosterlitz-Thoulessuniversality class [72–74] as a function of the reduced coupling β red : ξ KT ∼ K e A β ν red (26) β − = (cid:12)(cid:12) β − − β − c (cid:12)(cid:12) ∝ | T − T c | . (27)Albeit with a different critical exponent, ν = 1 and ν = 1 / β cF = ∞ . Mimicking now a well-known argument Of course, this only holds as long as f ( β F ) ∼ β F for large β F . f ∼ β F ; the density ofnegative plaquettes will thus be controlled by a Boltzmann factor ρ ∼ exp ( − f ). On theother hand the possible positions for this sign flip will scale like L and the balance betweenfree energy and entropy gives L ∼ ρ − / = exp ( β f ) (cid:39) ξ . Up to the power correction for β A = 0 case, Eq. (24), this simple argument works quite well, contrary to the XY -model,where it cannot explain renormalization effects leading to the non trivial critical exponent ν = 1 /
2. Moreover, since the minimal distance among vortices can be reliably estimatedwith that along a plane intersecting them, such picture should (roughly) hold in higherdimensions as well.We could in principle explore the similarities with the Kosterlitz-Thouless transitions fur-ther. Although, as far as we know, for the XY -model no local order parameter is available,one can couple the theory to an external magnetic field h and study the analytical continu-ation of the partition function Z ( β, h ) to the complex plane. (Hyper-) scaling relations willthen hold among the critical exponents of ξ , of the magnetic susceptibility χ h ∼ ξ − η log − r ξ ,of the specific heat C s and of the edge of the Lee-Yang zeroes [72]. We will avoid such athroughout analysis in our case, for which a dedicated paper would be needed. Let us how-ever just briefly comment on two points. First, from Eq. (13) we can explicitly calculate thereduced partition function and the specific heat in our usual limiting cases: Z ∝ β A → β F I ( β F ) (28) Z ∝ β A →∞ e β A (cid:112) β A cosh β F (29) C s = β A → β F (30) C s = β A →∞ − tanh β F . (31)Inserting Eqs. (22, 23) into Eqs. (30, 31) and assuming that no other contribution besides thesingular one exist [72], we see that the critical behaviour for C s should change (continuously?)from log − L to L − . Second, in our case we have direct access to a non local, topologicalorder parameter, for which we can determine a critical exponent, z ∼ M L − β ; from Eq. (19)we have β = 0. If we would like to study the extended partition function Z ( β F , β A , h ) we We wish to thank P. de Forcrand for useful comments about this point. S h = h z to the action. Although a direct calculation would gobeyond the scope of this paper, it is obvious that for fixed L a sufficient condition to alignthe center flux z is realized if β F → ∞ : the fundamental coupling plays the role of a ”mock” Z magnetic field. Indeed, from Eqs. (28, 29) the zeros of Z in the complex β F plane all lieon the imaginary axis, in agreement with the Lee-Yang theorem [75].Let us finally turn to the continuum limit. From the above discussion it is clear thattaking the thermodynamic limit L → ∞ before the weak coupling limit β F → ∞ , as oneshould, i.e. taking the Euclidean volume V = ( a L ) → ∞ (or, at finite temperature, V s = ( a L s ) → ∞ ), the theory remains stuck in the disordered phase (cid:104) z (cid:105) = 0: no vortextopological sector can be defined and the super-selection rule of Eq. (2) is not realized.On the other hand, assuming that the scaling of the string tension σ with the latticespacing a , known analytically for β A = 0: β F = 4 a g σ = 38 g , (32)will hold up to a different prefactor along any line f ( β F ) ∝ β F , we get: V = ( a L ) = 32 L σ β F . (33)Keeping now the volume V fixed as the continuum limit is approached, the values of thecoupling at which one needs to simulate for fixed L will scale as β F ∼ L , i.e. much higherthan the pseudo-critical coupling β cF ∼ log L . The theory will thus be in a pseudo-orderedphase with (cid:104) z (cid:105) = 1: on a finite Euclidean d = 2 torus the Wilson action can admit welldefined Z topological sectors. B. d = 3 Increasing the dimensions to d = 3 we expect interactions to arise among parallel planes,since vortices are now extended, one-dimensional objects. The simple picture we have found Some interpretation issues of course arise in this case. E.g., speaking of zero temperature for a compacti-fied, periodic time is at best misleading. Of course, one could also consider the case L s ∼ β F , but to fixthe temperature independently one must resort to an anisotropic (Hamiltonian) setup [76]. Transitionson finite toruses in the large N limit of the d = 2 Yang-Mills theories have been the subject of intenseinvestigations; see Refs. [77, 78] and references therein. L = 24 L = 32 L = 40 L = 48 L = 64 β cF ( L ) 5 . . . . . χ z ( β cF ( L )) 152 . . . . . z ( β cF ( L )) 0 . . . . . C s ( β cF ( L )) 0 . . . . . δβ F .
025 0 . . . . β F in d = 3. The third and fourthline give the values of the order parameter and of the specific heat at the pseudo-critical point β cF ( L ); the last line gives the coupling steps for the simulations used in the re-weighting. in d = 2 won’t probably work anymore and less trivial critical exponents might arise. Still,fluxes are inherently two-dimensional objects and most of the dynamics should thus takeplace on planes: many features of the d = 2 case should therefore survive. To check this, wehave performed sets of Monte-Carlo simulations along different lines in the β F − β A plane.Results are reported for β A = 0, β F = 0 .
5, 0 .
75 and lattice sizes between L = 24 and L = 80;other parameters have been checked and give a consistent picture.In the β A = 0 case approximately 20 to 50 simulations at coupling steps δβ F , each with10 independent configurations, were performed for each volume L . The data have beenre-weighted [79, 80] to determine the peak values β cF ( L ) and χ z ( β cF ( L )); this was viableonly up to L ∼
64. Indeed, as we shall see below, the d = 3 case shows a similar scalingbehaviour as Eq. (22), i.e. a logarithmic scaling of β cF ( L ) to a critical coupling β cF = ∞ .This has a practical drawback: the absolute width of the transition, i.e. the overall interval∆ β F one needs to simulate, varies very slowly, while the step-width δβ F one must scan inorder to keep the density of states computationally feasible decreases dramatically with L :the computational cost becomes eventually unmanageable.Results for all volumes considered are resumed in Tab. I, where the steps δβ F are alsolisted, along with the value of the center flux and, for sake of completeness, of the specificheat at the pseudo-critical point. To cross-check scaling results, similar simulation steps andstatistics have also been used for the other volumes not included in the re-weighting. The9data can be well fitted with the Ansatz: χ z ( β cF ( L )) ∼ A L − η log − r L (cid:0) O ( L − ) (cid:1) (34) z ( β cF ( L )) ∼ M L − β (cid:0) O ( L − ) (cid:1) (35) β cF ( L ) ∼ C log L + D log log L + O (1) , (36)For χ z we get A = 0 . r = − . η = 0 . χ /d.o.f.= 5 .
7; constraining η = 0 gives again A = 0 . r = − . χ /d.o.f.= 2 .
8. For β cF we get C =0 . D = − . χ /d.o.f.= 0 .
7; on the other hand, constraining D = 0 weget C = 0 . χ /d.o.f.= 0 .
6. Finally, for the order parameter, we get M = 1 . β = 0 . χ /d.o.f.= 1 .
4. Overall, the biggest source of systematic error is givenby the parameterization of the sub-leading corrections: leaving them out or parameterizingthem differently leads to changes of up to 10% for some of the critical exponents, not includedin our error estimates; obviously, more data at higher volumes are needed to pin the numbersdown. The data for the susceptibility χ z , rescaled by Eqs. (34, 36), are plotted in Fig. 8,showing very good agreement also for the volumes which have not been included in there-weighting analysis. In Fig. 9 we show the scaling of the order parameter z according toEqs. (35, 36); the agreement for L (cid:38)
40 is again very good. As for the specific heat, afit of the data in Tab. I with a logarithmic Ansatz C s ∼ log − α L gives α = 1 . χ /d.o.f.= 0 .
5. The signal-to-noise ratio for the MC is however not so good in this case,reflecting itself in the quality of the re-weighted data: more statistics would be definitelyneeded; anyway, checking any (hyper-) scaling relation is beyond our goals.The above result is quite surprising. Indeed, in contrast to d = 2, one could have ex-pected the Z monopole to control the open center vortices, since the density of the latter isproportional to that of the former. However, although monopoles per unit volume steadilydecrease beyond the cross-over, open vortices “connecting” them still cause a critical be-haviour cumulating to g → A possible explanation could be that their length increasesmore than linearly with the lattice size; multiple bendings in orthogonal directions wouldbe enough to randomize the fluxes. A direct investigation of any geometrical properties of Another possible issue could be the non-ergodicity of our set-up in the ordered phase. Indeed, a “good”algorithm would need to change boundary conditions to enable tunneling among different topologicalsectors around the transition, just like a cluster algorithm in an Ising model allows tunneling amongdifferent orientations of the spins in the spontaneously magnetized phase. See e.g. Ref. [23] for possiblesolutions to the problem. Implementing such algorithm is obviously beyond the scope of this paper. A somewhat cryptic comment regarding a possible critical behaviour, going as far as taking the XY -modelas a paradigm, can be found in Ref. [37]. FIG. 8. Data for the susceptibility of the order parameter in d = 3, including the re-weightedcurves, rescaled with the FSS Ansatz in Eqs. (34, 36). open vortices is however beyond the scope of this paper, since Eq. (9) is non local and gaugeinvariant and does not allow to isolate the topological defects on the planes.Going now to the β A (cid:54) = 0 case, since the Z monopoles undergo a cross-over also inthe low β F region of the phase diagram of Fig. 2, one would expect the center flux tobehave as in the β F case: one should find along β A a similar scaling as in Eqs. (34, 36).Also, the transition lines should not be effected by the bulk transition associated with theunphysical Z gauge degrees of freedom. However, such strong transition unavoidably makesany simulation near it quite noisy; on top of that the biased Metropolis algorithm, with e.g.3 micro-canonical steps, gets inefficient as β F gets small and β A large, reaching for z and χ z , around the peaks of the latter, integrated autocorrelation times of the order 10 − for 24 ≤ L ≤
40. Passable data were therefore only accessible for three volumes, whilegathering enough statistics to re-weight the susceptibility was out of the question. We have1 −0.06 −0.04 −0.02 0 0.02 0.04 0.060.90.920.940.960.9811.021.041.061.081.1 β F − β Fc (L) L β < z > L=24L=32L=40L=48L=56L=64L=72L=80
FIG. 9. FSS for the order parameter z as a function of the rescaled coupling as in Eq. (35). thus limited ourself to a consistency check near the bulk transition with a scaling Ansatzsimilar to Eqs. (34-36): χ z ( β cA ( L )) ∼ A L log − r L + O ( L ) (37) z ( β cA ( L )) ∼ M L − β (cid:0) O ( L − ) (cid:1) (38) β cA ( L ) ∼ C log L + O (1) ; (39)no fit has been attempted. The scaling of the pseudo critical point and of the order parameterare consistent with the d = 2, β A = ∞ case, C = 1 / β = 0, as can be seen from Fig. 10.On the other hand, the peaks of χ z are quite noisy and even a consistency check for thelogarithmic exponent is hopeless. In Fig. 11, 12 we show the results for the simulations along β F = 0 . β F = 0 .
75, lying respectively left and right of the bulk transition, rescaled byEq. (37, 39) with a “guessed” value for r = − /
2; of course, further work would be neededto determine the critical exponents reliably.We have also checked via Monte-Carlo simulations that all of the above results generalizeto
T > L → L s · L t in all the scaling relations for temporal fluxes,while the behaviour of all spacial fluxes remains unchanged. Again, as in the d = 2 case,this implies that, along any line in the β F − β A plane, when taking the thermodynamic limit2 −5 −4 −3 −2 −1 0 1 20.10.20.30.40.50.60.70.80.91 β A − β Ac (L) L β < z > L=24L=32L=40
FIG. 10. FSS for the order parameter z along the β F = 0 . −5 −4 −3 −2 −1 0 1 200.10.2 β A − β Ac (L) χ / χ (( β A c ( L )) L=24L=32L=40
FIG. 11. FSS for the susceptibility χ z along the β F = 0 . −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.10.20.30.40.50.60.70.80.9 β A − β Ac (L) χ / χ (( β A c ( L )) L=24L=32L=40
FIG. 12. FSS for the susceptibility χ z along the β F = 0 .
75 line with the Ansatz Eqs. (37, 39). before the weak coupling limit, i.e. sending the volume to infinity, the d = 3 theory remainsstuck in the disordered phase (cid:104) z (cid:105) = 0; again, Eq. (2) is not realized.What about the fixed volume limit? Taking as a blueprint for the continuum limit alongany direction the scaling of the string tension along β F [81]: β F = 4 a g a √ σ = c β F + c β F + O ( 1 β F ) . (40)we get immediately: V = ( a L ) ∝ L β F √ σ . (41)Keeping again V fixed as the continuum limit is approached, the values of the couplingcorresponding to a given L will now scale as β F ∼ L ; again, as in d = 2, they will alwaysbe much higher than the pseudo-critical coupling β CF ∼ log L and the Wilson action couldadmit well defined Z topological sectors on a finite d = 3 torus.4 L = 12 L = 16 L = 20 L = 24 β cF ( L ) 3 . . . . χ z ( β cF ( L )) 24(1) 42(1) 66(1) 95(1)TABLE II. Position and height of the susceptibility peaks along β F in d = 4. L = 16 L = 20 L = 24 β F = 1 . . . . β F = 1 . . . . β F = 1 . . . . β A in d = 4 for β F = 1 .
0, 1.2 and 1.3; theheights are all compatible with the results in Tab. II C. d = 4 The positions of the peaks of χ z , as obtained in the simulations along the β A = 0, β F = 1 . β F = 1 . β F = 1 . χ z ( β cF ( L )) ∼ A L + O ( L ) (42) z ( β cF ( L )) ∼ M L − β (cid:0) O ( L − ) (cid:1) (43) β cF ( L ) ∼ C log L + O (1) . (44)Results are shown in Figs. 13, 14 for the order parameter and its susceptibility along the β F axis; up to the values of C the behaviour along the lines parallel to the β A axis is basicallythe same, see e.g. Fig. 15. From the data in Tab. II we can estimate C = 0 . /
2, while for those in Tab. III we get C = 0 . β is compatible with 0.Direct simulations at T > L → L s · L t for thetemporal fluxes, while spacial fluxes remain unchanged. As in the d = 2 and d = 3 cases,in the thermodynamic limit the theory remains therefore stuck in the disordered phase.Moreover, starting from the 2-loop beta-function, the running of the physical scale with5 −1 −0.5 0 0.5 1.0 1.5 2 2.5 3 3.500.10.20.30.40.50.60.70.80.91 β F − β Fc (L) L β < z > L=12L=16L=20L=24
FIG. 13. FSS as in Eqs. (43, 44) for z in d = 4 along the β F axis. α lat = g / (4 π ) is given by:log (cid:0) a σ (cid:1) = − πβ α − + 2 β β log (cid:18) πβ α − (cid:19) + c + O ( α lat ) , (45)where c = log σ Λ and β = , β = , i.e. from Eq. (7): V = ( a L ) ∝ L (cid:18) π β β F (cid:19) β β e − π β β F . (46)When trying to keep the volume V fixed as β F →
0, up to log log corrections the couplingshould scale as: β F ∼ β π log L ; (47)the coefficient C in Eq. (44) is however larger than that coming from the beta-function andthe simulation parameters will lie in the disordered phase: topological sectors will alwaysbe ill-defined also on a finite torus T . The same holds along the lines parallel to the β A axis (see Tab. III); in this case, from Eq. (7), the coefficient in Eq. (47) coming from thebeta-function Eq. (45) is 3 β / (32 π ), again smaller than the corresponding value of C .As discussed in Sec. II B, we have excluded phase II (see Fig. 3) from the simulations.As mentioned above, vortex topology is however well understood in this case: the results of6 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.500.020.040.060.080.10.120.140.160.18 β F − β Fc (L) χ / χ (( β F c ( L )) L=12L=16L=20L=24
FIG. 14. FSS as in Eqs. (42, 44) for χ z in d = 4 along the β F axis. Refs. [21, 22, 29, 47, 60] show that, in contrast to phase I in d = 4 and to the d = 2 and 3cases, the theory possesses well defined Z topological sectors in the continuum limit. IV. CONCLUSIONS
We have studied a topological order parameter, the center flux z defined in Eqs. (10,11), for the SU (2) mixed action in 2 ≤ d ≤
4. Its ordered phase, (cid:104) z (cid:105) = 1, correspondsto well defined π ( SO (3)) = Z topological sectors, i.e. to a vacuum satisfying the super-selection rule of Eqs. (2, 4), while for (cid:104) z (cid:105) = 0 the vacuum state is disordered and no centertopology can be defined. This reminds of a quantum phase transition; however, one doesnot switch between vacua by tuning a physical parameter. Rather, the choice of dimensionsand the symmetry of the discretized action control in which phase the theory will be in thecontinuum limit.More specifically, discretized actions transforming in the fundamental representation pos-sess a disordered vacuum, with z showing an essential scaling to the critical coupling β c = ∞ .The critical exponent for the correlation length ξ is ν = 1, i.e. β c ( ξ ) ∝ log ξ ; explicit log log ξ corrections to scaling can be shown to exist for some choice of parameters. The susceptibil-7 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.020.040.060.080.10.120.140.160.18 β A − β Ac (L) χ / χ (( β A c ( L )) L=16L=20L=24
FIG. 15. FSS as in Eqs. (42, 44) for χ z in d = 4 along the β F = 1 . ity of the center flux scales as χ z ( ξ ) ∝ ξ in d = 2 and d = 4, while the order parameteritself scales trivially in these cases. On the other hand in d = 3, at least along the β F axis, the center flux has a non trivial critical exponent, z ( ξ ) ∝ ξ − β , with β = 0 . χ z ( ξ ) ∼ ξ log − r ξ , with r = − . β F − β A diagram, but more statistics would be needed to reach a conclusiveresult. A tentative critical exponent for the specific heat, C s ( ξ ) ∼ log − α ξ , gives α = 1 . T > d = 4 (see Fig. 3) is such an example [21, 22, 29].Large scale simulations with the adjoint action are hampered by strong finite-volume effects[27, 28, 38]. Therefore, although the techniques used in Refs [21, 22, 29] to tame them couldalso work in d = 3, a more viable alternative, applicable also in d = 2, would be to resort topositive plaquette models [14, 55, 56], where topological sectors are always well defined since8the operator given in Eq. (9) takes “by construction” the values dictated by the assignedboundary conditions. Indeed, a one-to-one mapping between configurations in such latticediscretization and those of the adjoint Wilson action with well defined vortex sectors wasconjectured in Ref. [27] and explicitly constructed in Ref. [29]. Finally, an ordered vacuumcould also be realized for a finite torus in d = 2 ,
3; here one could exploit the power-lawscaling of the physical mass with the coupling to define topological sectors when L → ∞ and a → V = ( a L ) d kept fixed.The above findings do not contradict universality, since non perturbatively the equivalencebetween fundamental and adjoint actions can only hold as long as no lattice artifacts arepresent [27–29, 36–39], while as we have seen for some discretizations the density of Z monopoles can not vanish at any finite coupling [37]. Does however such result have anyphysical consequences? The vacua of the two different phases can be essentially characterizedby the type of Z vortices they can carry:i) The ordered phase allows topological center vortices “`a la ’t Hooft” [12, 13]: a confinementmechanism based on the super-selection rule of Eqs. (2, 4) can be realized; at finite temper-ature the change in the vortex free energy as measured via Eq. (3) is thus a valid test toestablish how the symmetry is broken in the transition to the deconfined phase [21, 22, 29].No fundamental fields are allowed in this case [12, 13, 31]; however, adjoint fermions can beeasily incorporated in such scenario. It might therefore be interesting to investigate the vac-uum properties of the SU (2) gauge theory coupled to adjoint fermions, a popular candidatefor infrared conformality [65]. Numerical tests with the adjoint Wilson action or positiveplaquette model should be viable.ii) The disordered phase is dominated by (one huge, percolating?) open vortices, remindingof the Nielsen-Olesen “spaghetti vacuum” [4]. Such open vortices are not topological ac-cording to Eq. (1): Eqs. (2, 4) cannot be applied. One might conjecture some relationshipwith P-vortices [2, 15, 16], although it is still unclear how to test such hypothesis, since thecenter flux z is gauge invariant and constructed out of pure SO (3) variables while P-vorticesare gauge dependent and built out of the Z gauge degrees of freedom. Moreover, such openvortices persist at any temperature, not disappearing above T C . This disordered vacuum isdetached from the boundary conditions chosen and is therefore compatible with the pres-ence of fundamental matter fields. Of course, Eq. (3) is ill defined in this case; whether anyvortex related order parameter for the confinement-deconfinement phase transition could be9defined remains an open question. AKNOWLEDGEMENTS
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