---

The Savvidy “ferromagnetic vacuum”

arXiv:hep-lat/9210028v1 23 Oct 1992TRIUMF TRI-PP-92-98October 1992The Savvidy “ferromagnetic vacuum”in three-dimensional lattice gauge theoryHoward D. Trottier∗and R. M. WoloshynTRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3The vacuum effective potential of three-dimensional SU(2) lattice gauge theoryin an applied color-magnetic field is computed over a wide range of field strengths.The background field is induced by an external current, as in continuum field theory.Scaling and finite volume effects are analyzed systematically. The first evidence fromlattice simulations is obtained of the existence of a nontrivial minimum in the effectivepotential.

This supports a “ferromagnetic” picture of gluon condensation, proposedby Savvidy on the basis of a one-loop calculation in (3+1)-dimensional QCD.1990 PACS number(s):Typeset Using REVTEX1

The vacuum structure of quantum chromodynamics is known to play a fundamentalrole in strong interaction dynamics [1], yet a clear understanding of the physics underlyingsuch basic properties of the vacuum as the gluon condensate is lacking.An attractivephysical picture of gluon condensation was proposed more than ten years ago by Savvidy,who calculated the one-loop effective potential of the QCD vacuum in response to an appliedcolor-magnetic field [2]. The absolute minimum of the effective potential was found to lieat a nonzero value of the applied field, suggesting that the QCD vacuum behaves like aferromagnet, with a condensate in the color-field arising spontaneously.This result wasfurther analyzed by Matinyan and Savvidy [3], and was obtained independently by Pagelsand Tomboulis [4].However, the one-loop calculation is not adequate a priori to support the ferromagneticpicture of gluon condensation, since perturbation theory becomes untrustworthy near theminimum of the effective potential.Moreover it was discovered by Nielsen and Olesenthat a constant color-magnetic field generates unstable modes at the one-loop level [5].Nevertheless, further analysis by the Copenhagen group [6] suggested that the instability isremoved by a nonperturbative structure that leaves the nontrivial minimum of the one-loopeffective potential intact.

[This approach was challenged more recently in Ref. [7] however,where it was argued that the external field problem is fully nonperturbative, even in theregion of large fields.] From another line of argument, Adler reached the conclusion that theexistence of a truly nonperturbative minimum in the effective potential can be establishedfrom the leading-order calculation [8].

An attractive model of hadronic structure includingthe automatic formation of flux-tubes and bag-like excitations has also been developed basedon the occurrence of a nontrivial minimum in the effective potential [8,9].In order to truly substantiate the ferromagnetic picture of the QCD vacuum, a genuinenonperturbative calculation is required. In principle, the effective potential is well suitedto analysis in lattice gauge theory.In fact, two groups have recently attempted such alattice calculation in pure gauge SU(2) [10,11].

Unfortunately, the qualitative features of theeffective potential obtained by the two groups are in sharp disagreement. They acknowledgesignificant nonscaling and finite volume effects in their calculations.

Moreover, they wereunable to draw definitive conclusions about the behavior of the effective potential at smallfields, the region of greatest interest.We also note that very different algorithms for the introduction of an external field onthe lattice were used in Refs. [10] and [11], and we think that there are significant drawbacksto both methods.

In Ref. [10] a link variable is introduced to account for the external field;the external link is aperiodic (corresponding to the usual external vector potential in thecontinuum theory), while periodic boundary conditions for the “dynamical” links are used.We think that this procedure leads to undesired interaction terms in the action on thelattice boundary.

In Ref. [11] on the other hand an Abelian projection technique is used toconstruct the lattice action, and the results for the effective potential turn out to dependstrongly on the choice of operator that is diagonalized.In this Letter we present a calculation of the effective potential in three-dimensionalpure-gauge lattice SU(2).

Our rationale for doing the lattice calculation in three dimensionsis two-fold. First, one of us has recently demonstrated that the one-loop effective potential in2+1 dimensions has the same qualitative features as in 3+1 dimensions [12].

In particular,the effective potential has its absolute minimum at a nonzero value of the applied color-2

magnetic field in both theories. Thus the Savvidy “ferromagnetic vacuum” is found to occurat the one-loop level in both (2+1)- and (3+1)-dimensional QCD (the limitations of theone-loop calculation are of course the same in both cases).

Secondly, one can more readilyperform a thorough check of scaling and finite volume effects in a three dimensional latticecalculation that can be achieved (with the same computing power) for the correspondingfour-dimensional system.We have also developed an algorithm for the introduction of a background field in non-Abelian lattice gauge theory (in an arbitrary number of dimensions) that is free of thedrawbacks in the methods used in Refs. [10,11], outlined above.

Our algorithm is a straight-forward transcription of the procedure that is used in continuum field theory to induce abackground field by a coupling to an external current. This approach has previously beenconsidered in the context of the lattice Abelian Higgs model by Damgaard and Heller [13].In the continuum U(1) theory, the (Euclidean) action for a coupling of the gauge field toan external current isSU(1) =Zdxh14F 2µν + jextν Aνi,(1)where for our purposes the external current has the formjextν (x) = ∂µF extµν (x).

(2)If F extµν (x) is constant within a finite region, then jextµ (x) describes a solenoidal current whichcirculates around the boundary. An integration by parts in Eq.

(1) leads toSU(1) =Zdxh14F 2µν −12FµνF extµνi,(3)where now the region of support of the external field is unrestricted. The path integral inthis trivial theory leads to a “classical” behavior for expectation values such as ⟨Fµν⟩= F extµν .An extension of the above procedure to nonAbelian theories was proposed in Ref.

[11].The action isSSU(N) =Zdxh14F aµν2 + jext,aνAaνi,(4)where the current is a covariant generalization of Eq. (2)jext,aν(x) = Dabµ (Aext)F ext,bµν(x),(5)with Dabµ (A) ≡∂µδab −gf abcAcµ the usual covariant derivative.

F ext,aµνis the nonAbelianfield strength constructed out of Aext,aµ. Following the one-loop calculations of the effectivepotential [2,5,6,12], we consider an external field with a fixed direction in gauge spaceAext,aµ= Aextµ δa3⇒F ext,aµν= F extµν δa3.

(6)An integration by parts in Eq. (4) then yieldsSSU(N) =Zdxh14F aµν2 −12∂F a=3µνF extµνi,(7)3

where∂F a=3µν≡∂µAa=3ν−∂νAa=3µ. (8)Notice that the interaction term in Eq.

(7) only contains the derivative part ∂F of the fieldstrength for the “dynamical” gauge fields Aaµ (compare ∂F with the full field strength tensorF aµν = ∂µAaν −∂νAaµ −gf abcAbµAcν). If the full field strength appeared in the interactionterm, then Eq.

(7) would describe the same “trivial” physics as the U(1) action of Eq.(3). This difference between the two actions expresses the fact that the “charge” carried bynonAbelian gauge fields drives the different physics.Equations (4) and (7) represent a sensible way to introduce an external field in thenonAbelian theory since they lead, at the classical level, to an “induced” gauge field that isequal to the applied field; moreover, at the one-loop level, this action leads to the Savvidyeffective potential of interest.We now consider the lattice transcriptions of the above continuum theories.

We specializeto three dimensions. The lattice equivalent of the U(1) action Eq.

(3) is [13]SU(1) = SW −12XsitesbFµν bF extµν ,(9a)where SW is Wilson’s plaquette action, the external field in lattice units bF ext is given bybF extµν = a3/2F extµν ,(9b)and the “dynamical” field strength bFµν is given as usual in terms of the imaginary part ofthe plaquette Uµν:bFµν =qβ Im Uµν,β ≡1e2a(9c)(a the lattice spacing). In the SU(2) lattice theory:SSU(2) = SW −12Xsites∂bF a=3µνbF extµν ,(10a)where SW is Wilson’s plaquette action for SU(2), andbF extµν = 4β!3/2 1g3 F extµν ,β ≡4g2a.

(10b)To extract the derivative part ∂bF of the “dynamical” field strength on the lattice, we explicitlycompute the commutator of any two links Uµ, Uν which span the plaquette:∂bF a=3µν= −iqβ Trn(Uµν −[Uµ, Uν]) 12σ3o,(10c)where σ3 is the third Pauli matrix. The connection between the right hand side of Eq.

(10c)and ∂F in the continuum limit follows from the identification of the link variables in thislimit as Uµ ≡exp(iagAaµσa/2); the above lattice action reduces to the continuum action Eq. (7), up to corrections of O(a), which is the same order of accuracy as the continuum limit of4

the Wilson action. [The coupling to the external current which leads to Eq.

(10a) breaks thelocal SU(2) gauge invariance of the Wilson action to a local U(1) symmetry. This implies,for example, that Tr(Uµνσ3) is an invariant quantity in the theory defined by Eq.

(10a).] Ourconstruction of the lattice action using Eq.

(10c) differs from the procedure followed in Ref. [11], where ∂bF was defined by an Abelian projection (the results for the effective potentialobtained in Ref.

[11] depend strongly on the choice of operator that is diagonalized).In our calculations we take the field strength tensor for the external magnetic field (ascalar in three dimensions) to have nonvanishing components only in the (1,2) plane:F extµν = H (δµ1δν2 −δν1δµ2) . (11)An important aspect of our lattice simulation is our use of free boundary conditions forthe “dynamical” gauge links.

That is, we integrate over all links on the boundaries of thelattice [14]. This is motivated by the fact that the gauge field for a uniform magnetic fieldin the usual continuum theory is not periodic [cf.

Aextµ= xHδµ2, which generates Eq. (11)].In a perturbative calculation moreover, the continuum external potential induces quantumfluctuations which are also aperiodic.

Free boundary conditions seem most appropriate todescribe the corresponding physics on the lattice. Free boundary conditions also serve toeliminate long-lived metastable states that occur in the U(1) theory with periodic boundaryconditions, corresponding to closed Dirac strings winding through the (dual) lattice [15].

In asimulation of the U(1) external field problem on a lattice with periodic boundary conditions,it is necessary to use a global updating procedure to eliminate these strings [13].We compute the energy bE(H) in the “induced” gauge fields in lattice units according to(Refs. [16,10,11])bE(H) ≡β [⟨✷M⟩(H) −⟨✷E⟩(H)] ,(12a)where⟨✷M⟩≡1L3Xsitesh1 −12Tr U12i,⟨✷E⟩≡1L3Xsitesh2 −12Tr (U13 + U23)i,(12b)and L is the length of a side of the lattice.

To get the physical energy Ephys we performa vacuum subtraction, and convert to units of the coupling constant g (which in threedimensions has units of (mass)1/2). For SU(2):Ephys = g6 14β3 h bE(H) −bE(0)i.

(13)We now present results of our simulations of the U(1) and SU(2) theories. We haveperformed an extensive set of calculations over a wide range of values of H, β, and latticesizes.

We find that relatively modest statistics (similar to those employed in the previousfour-dimensional studies of Refs. [10,11]) are sufficient to identify the main features of theeffective potential.

We use a 10-hit Metropolis algorithm. Our main results in the SU(2) casewere obtained on a 323 lattice, using 1000 sweeps to thermalize at each value of H and β,followed by 5000 sweeps, keeping only every fourth configuration for data (this results in an5

integrated autocorrelation time in the energy data which in most cases satisfies τint <∼0.6).To check finite volume effects, we have also done some calculations in the SU(2) case on a163 lattice with an eightfold increase in statistics.Our U(1) calculations provide a useful check of our SU(2) code — we simply identifythe link variables Ua=1,2µ≡0 to do the U(1) simulation using our SU(2) code. Figure 1shows results obtained on a 163 lattice (using 1000 sweeps to thermalize at each value of H,followed by 10000 sweeps, keeping only every fourth configuration for data).

The agreementwith the expected classical behavior in the energy and in the induced field is very good, towithin about 5% at β = 10 (the difference from the classical values is due mainly to finitesize effects, and decreases with increasing β).Figures 2 and 3 show the energy in the SU(2) theory on a 323 lattice as a function of H,for various values of β. We note that at large H one must run at comparatively large valuesof β in order to approach the scaling limit, as evident in Fig.

2. [Roughly speaking, we mayexpect some components of the “induced” field to be of O(H), corresponding to a plaquetteangle of O(a2gH), which should remain ≪1 in order to approach the continuum limit (cf.Ref.

[10]).] Hence there may be appreciable finite volume effects at the largest fields in ourdata.In the region H/g3 <∼1 on the other hand, our data shows both good scaling behavior (cf.Fig.

3), and negligible finite volume effects, as shown in Fig. 4, where we compare our resultsfrom 163 and 323 lattices.

Our data at β = 7 and 10 show clear evidence of a minimum inthe energy. The data at β = 7 lie about eight standard deviations below zero in a region ofH/g3 near 1.

[The existence of a minimum is somewhat ambiguous in the case of our dataat β = 12, since the errors in Ephys with the present level of statistics are large comparedwith its magnitude for β = 12 and H/g3 <∼2 (finite volume effects in these data are alsosignificant, cf. Fig.

4). However, the data at β = 12 are generally consistent within errorswith the energy obtained at β = 7 and 10.] The energy is generally more than an order ofmagnitude smaller than the classical result 12H2 over the whole range of H that we havestudied.In summary, we have developed a method for the introduction of an external field innonAbelian lattice gauge theories that follows from a coupling to an external current, asused in continuum field theories.We have performed a systematic study of scaling andfinite volume effects, and we have obtained results for the effective potential over a widerange of field strengths.

We have obtained the first evidence from lattice simulations of theexistence of a nontrivial minimum in the effective potential for a background color-magneticfield in nonAbelian gauge theory. Our results support the Savvidy “ferromagnetic” picture ofgluon condensation in the QCD vacuum.

In future work, we will use the methods developedhere to compute the effective potential in four-dimensional lattice SU(2). It should alsobe possible to look for the nonperturbative vortex structure associated with the “would-be” Nielsen-Olesen unstable modes [5,6].

The three-dimensional gauge theory studied hereshould provide a very convenient system for such a study.HDT thanks Jan Ambjørn and the Niels Bohr Institute for their hospitality during astay which provided for fruitful discussion of this work. This work was supported in part bythe Natural Sciences and Engineering Research Council of Canada.6

REFERENCES∗Address after January 1, 1993: Physics Department, Simon Fraser University, Burnaby,B.C., Canada V5A 1S6. [1] See, e.g., E. V. Shuryak, The QCD vacuum, hadrons and the superdense matter (WorldScientific, Singapore, 1988).

[2] G. K. Savvidy, Phys. Lett.

71B, 133 (1977). [3] G. Matinyan and G. K. Savvidy, Nucl.

Phys. B134, 539 (1978).

[4] H. Pagels and E. Tomboulis, Nucl. Phys.

B143, 485 (1978). [5] N. K. Nielsen and P. Olesen, Nucl.

Phys. B144, 376 (1978).

[6] For a review see P. Olesen, Phys. Scripta 23, 1000 (1981).

[7] L. Maiani et al., Nucl. Phys.

B271, 275 (1986). [8] See S. L. Adler and T. Piran, Rev.

Mod. Phys.

56, 1 (1984), and references therein. [9] See R. R. Mendel and H. D. Trottier, Phys.

Rev. D 40, 3708 (1989); 42, 911 (1990);and references therein.

[10] J. Ambjørn et al., Phys. Lett.

225B, 153 (1989); 245B, 575 (1990). [11] P. Cea and L. Cosmai, Phys.

Rev. D 43, 620 (1991); Phys.

Lett. 264B, 415 (1991).

[12] H. D. Trottier, Phys. Rev.

D 44, 464 (1991). [13] P. H. Damgaard and U. M. Heller, Phys.

Rev. Lett.

60, 1246 (1988); Nucl. Phys.

B309,625 (1988). [14] K. H. M¨utter and K. Schilling, Nucl.

Phys. B200, 362 (1982); Phys.

Lett. 117B, 75(1982).

[15] V. Grosch et al., Phys. Lett.

162B, 171 (1985). [16] A. Hasenfratz and P. Hasenfratz, Nucl.

Phys. B193, 210 (1981); F. Karsch, ibid.

B205,285 (1982); J. Engels et al., ibid. B205, 545 (1982).7

FIGURESFIG. 1.Effectivepotentialand expectationvalue of the induced field strengthforthree-dimensional U(1) lattice gauge theory, as functions of the applied magnetic field strengthH.

These results are from a 163 lattice at β = 10.FIG. 2.

Effective potential for three-dimensional SU(2) lattice gauge theory, as a function ofthe applied color-magnetic field strength H, for various values of β. These results were obtainedon a 323 lattice.FIG.

3. Effective potential for three-dimensional lattice SU(2), as in Fig.

2, here shown on anexpanded scale for “small” field strengths H/g3 ≤2.FIG. 4.

Effective potential for three-dimensional lattice SU(2) on two different lattice sizes.Data points for H/g2 ≤1 were calculated at β = 7, while the points at larger H were taken atβ = 12.8

---

출처: arXiv:9210.028 • 원문 보기