Universal low-temperature behavior of two-dimensional lattice scalar chromodynamics
aa r X i v : . [ h e p - l a t ] J a n Universal low-temperature behavior of two-dimensional lattice scalar chromodynamics
Claudio Bonati, Andrea Pelissetto, and Ettore Vicari Dipartimento di Fisica dell’Universit`a di Pisa and INFN Largo Pontecorvo 3, I-56127 Pisa, Italy Dipartimento di Fisica dell’Universit`a di Roma Sapienza and INFN Sezione di Roma I, I-00185 Roma, Italy (Dated: January 22, 2020)We study the role that global and local nonabelian symmetries play in two-dimensional latticegauge theories with multicomponent scalar fields. We start from a maximally O( M )-symmetricmulticomponent scalar model, Its symmetry is partially gauged to obtain an SU( N c ) gauge theory(scalar chromodynamics) with global U( N f ) (for N c ≥
3) or Sp( N f ) symmetry (for N c = 2), where N f > S M /SU( N c ) where S M is the M -dimensional sphere and M = 2 N f N c . In agreement with the Mermin-Wagner theorem,the system is always disordered at finite temperature and a critical behavior only develops in the zero-temperature limit. Its universal features are investigated by numerical finite-size scaling methods.The results show that the asymptotic low-temperature behavior belongs to the universality class ofthe 2D CP N f − field theory for N c >
2, and to that of the 2D Sp( N f ) field theory for N c = 2. Theseuniversality classes correspond to 2D statistical field theories associated with symmetric spaces thatare invariant under Sp( N f ) transformations for N c = 2 and under SU( N f ) for N c >
2. Thesesymmetry groups are the same invariance groups of scalar chromodynamics, apart from a U(1)flavor symmetry that is present for N f ≥ N c >
2, which does not play any role in determining theasymptotic behavior of the model.
I. INTRODUCTION
Nonabelian gauge symmetries are known since longtime to describe fundamental interactions [1]. More re-cently, it has been pointed out that they may also charac-terize emerging phenomena in condensed-matter physics,see, e.g., Refs. [2–6] and references therein. As a conse-quence, the large-scale properties of gauge models arealso of interest in two or three dimensions.We consider a lattice model of interacting scalar fieldsin the presence of nonabelian gauge symmetries, whichmay be named scalar chromodynamics or nonabelianHiggs model. In four space-time dimensions it repre-sents a paradigmatic example to discuss the nonabelianHiggs mechanism, which is at the basis of the Stan-dard Model of fundamental interactions. The three-dimensional model may also be relevant in condensed-matter physics, for systems with emerging nonabeliangauge symmetries. Its phase diagram and its behaviorat the finite-temperature phase transitions has been in-vestigated in Refs. [7, 8]. In this paper we extend such astudy to two-dimensional (2D) systems.We consider a 2D lattice nonabelian gauge theorywith multicomponent scalar fields. It is defined start-ing from a maximally O( M )-symmetric multicomponentscalar model. The global symmetry is partially gauged,obtaining a nonabelian gauge model, in which the fieldsbelong to the coset S M /SU( N c ), where M = 2 N f N c , N f is the number of flavors, and S M = SO( M ) / SO( M − M -dimensional sphere. According to the Mermin-Wagner theorem [9], the model is always disordered forfinite values of the temperature. However, a critical be-havior develops in the zero-temperature limit. We inves-tigate its universal features for generic values of N c and N f ≥
2, by means of finite-size scaling (FSS) analyses ofMonte Carlo (MC) simulations. The results provide numerical evidence that theasymptotic low-temperature behavior of these latticenonabelian gauge models belongs to the universality classof the 2D CP N f − field theory when N c ≥
3, and to thatof the 2D Sp( N f ) field theory for N c = 2. This sug-gests that the renormalization-group (RG) flow of the2D multiflavor lattice scalar chromodynamics associatedwith the coset S M /SU( N c ) is asymptotically controlledby the 2D statistical field theories associated with thesymmetric spaces [10, 11] that have the same global sym-metry, i.e., SU( N f ) for N c ≥ N f ) for N c = 2.The paper is organized as follows. In Sec. II we intro-duce the lattice nonabelian gauge models that we con-sider. In Sec. III we discuss the general strategy we useto investigate the nature of the low-temperature criticalbehavior. Then, in Secs. IV and V we report the numer-ical results for lattice models with N c ≥ N c = 2,respectively. Finally, in Sec. VI we summarize and drawour conclusions. In App. A we report some results onthe minimum-energy configurations of the models con-sidered. II. MULTIFLAVOR LATTICE SCALARCHROMODYNAMICS
We consider a 2D lattice scalar nonabelian gauge the-ory obtained by partially gauging a maximally symmetricmodel of complex matrix variables ϕ af x , where the indices a = 1 , .., N c and f = 1 , ..., N f are associated with thecolor and flavor degrees of freedom, respectively.We start from the maximally symmetric action S s = − t X x ,µ Re Tr ϕ † x ϕ x +ˆ µ , Tr ϕ † x ϕ x = 1 , (1)where the sum is over all sites and links of a squarelattice and ˆ µ = ˆ1 , ˆ2 denote the unit vectors along thelattice directions. Model (1) with the unit-length con-straint for the ϕ x variables is a particular limit of amodel with a quartic potential P x V (Tr ϕ † x ϕ x ) of theform V ( X ) = rX + u X . Indeed, it can be obtained bysimply setting r + u = 0 and taking the limit u → ∞ . Inthe following we set t = 1 for simplicity, which amountsto an appropriate choice of the temperature unit. It issimple to see that the action S s has a global O( M ) sym-metry, with M = 2 N f N c . Indeed, it can also be writtenin terms of M -component real vectors s x (which are thereal and imaginary parts of ϕ af x ) as S s = − X x ,µ s x · s x +ˆ µ , s x · s x = 1 . (2)This is the standard nearest-neighbor M -vector latticemodel.We proceed by gauging some of the degrees of freedomusing the Wilson approach [12]. We associate an SU( N c )matrix U x ,µ with each lattice link [( x , µ ) denotes the linkthat starts at site x in the ˆ µ direction] and add a Wilsonkinetic term for the gauge fields. We obtain the actionof the 2D lattice scalar chromodynamics defined by S g = − N f X x ,µ Re Tr ϕ † x U x ,µ ϕ x +ˆ µ − γN c X x Re Tr Π x , (3)where Π x is the plaquette operatorΠ x = U x , ˆ1 U x +ˆ1 , U † x +ˆ2 , U † x , . (4)The plaquette parameter γ plays the role of inverse gaugecoupling, and the N f and N c factors in Eq. (3) are con-ventional. The partition function reads Z = X { ϕ,U } e − β S g , β ≡ /T . (5)The lattice model (3) is invariant under SU( N c ) gaugetransformations: ϕ x → W x ϕ x , U x ,µ → W x U x ,µ W † x +ˆ µ , (6)with W x ∈ SU( N c ). For γ → ∞ , the link variables U x become equal to the identity (modulo gauge trans-formations), thus one recovers the ungauged model (1),or equivalently the O( M ) vector model (2).For N f = 1 the model is trivial. Because of the unit-length condition, using gauge transformations we can fix ϕ x to any given unit-length vector on the whole lattice:there is no dynamics associated with the scalar field. Aswe shall see, multiflavor models with N f ≥ N f and of colors N c . For N c ≥ ϕ x → ϕ x V , V ∈ U( N f ) , (7) thus it has a global U( N f ) / Z N c symmetry, Z N c being thecenter of SU( N c ). As discussed in Ref. [8], when N f < N c ϕ x → e iθ ϕ x (8)can be realized by an appropriate SU( N c ) local transfor-mation. Thus, the actual global symmetry group reducesto SU( N f ).For N c = 2, model (3) is invariant under the largergroup Sp( N f ) / Z , where Sp( N f ) ⊃ U( N f ) is the com-pact complex symplectic group, see also Refs. [2, 7, 8,13, 14]. Indeed, if one defines the 2 × N f matrix fieldΓ af x = ϕ af x , Γ a ( N f + f ) x = X b ǫ ab ¯ ϕ bf x , (9)where f = 1 , ..., N f , ǫ ab = − ǫ ba , ǫ = 1, the action (3)is invariant under the global transformationΓ al x → N f X m =1 Γ am x Y ml , Y ∈ Sp( N f ) . (10)We recall that the compact complex symplectic groupSp( N f ) is the group of the 2 N f × N f unitary matrices U sp satisfying the condition U sp J U T sp = J , J = (cid:18) − II (cid:19) , (11)where I is the N f × N f identity matrix. III. UNIVERSAL FINITE-SIZE SCALING
We exploit FSS techniques [15–18] to study the na-ture of the asymptotic critical behavior of the model for T →
0. For this purpose we consider models defined onsquare lattices of linear size L with periodic boundaryconditions.We mostly focus on the correlations of the gauge-invariant variable Q x defined by Q fg x = P fg x − N f δ fg , P fg x = X a ¯ ϕ af x ϕ ag x , (12)which is a hermitian and traceless N f × N f matrix. Thecorresponding two-point correlation function is definedas G ( x − y ) = h Tr Q x Q y i , (13)where the translation invariance of the system has beentaken into account. We define the susceptibility χ = P x G ( x ) and the correlation length ξ = 14 sin ( π/L ) e G ( ) − e G ( p m ) e G ( p m ) , (14)where e G ( p ) = P x e i p · x G ( x ) is the Fourier transform of G ( x ), and p m = (2 π/L, U = h µ ih µ i , µ = 1 V X x , y Tr Q x Q y , (15)where V = L .To identify the universality class of the asymptoticzero-temperature behavior, we consider the Binder pa-rameter U as a function of the ratio R ξ ≡ ξ/L . (16)Indeed, in the FSS limit we have (see, e.g., Ref. [19]) U ( β, L ) ≈ F ( R ξ ) , (17)where F ( x ) is a universal scaling function that com-pletely characterizes the universality class of the tran-sition. Eq. (17) is particularly convenient, as it al-lows us to check the universality of the asymptotic zero-temperature behavior without the need of tuning any pa-rameter. Corrections to Eq. (17) decay as a power of L .In the case of asymptotically free models, such as the 2DCP N − and O( N ) vector models, corrections decrease as L − , multiplied by powers of ln L [19, 20].Because of the universality of relation (17), we can usethe plots of U versus R ξ to identify the models that be-long to the same universality class. If the data of U fortwo different models follow the same curve when plottedversus R ξ , their critical behavior is described by the samecontinuum quantum field theory. This implies that anyother dimensionless RG invariant quantity has the samecritical behavior in the two models, both in the thermo-dynamic and in the FSS limit. An analogous strategy wasemployed in Ref. [19] to study the critical behavior of the2D Abelian-Higgs lattice model in the zero-temperaturelimit.The asymptotic values of F ( R ξ ) for R ξ → R ξ →∞ correspond to the values that U takes in the small- β and large- β limits. For R ξ → R ξ → U = N f + 1 N f − . (18)independently of the value of N c . The large- β limit isdiscussed in App. A. For N c ≥ U = 1.In the following we study the large- β critical behav-ior of lattice scalar chromodynamics for several valuesof N f and N c . We perform numerical simulations usingthe same upgrading algorithm employed in three dimen-sions [7, 8]. The analysis of the data of U versus R ξ outlined above allows us to conclude that the critical be-havior only depends on the global symmetry group ofthe model. For any N c ≥
3, the critical behavior be-longs to the universality class of the 2D CP N f − fieldtheory. Indeed, the FSS curves (17) for the model (3)agree with those computed in the CP N − model (we use the results reported in Ref. [19]). For N c = 2, instead,the critical behavior is associated with that of the 2DSp( N f ) field theory. Note that the parameter γ appearsto be irrelevant in the RG sense (at least for | γ | not toolarge). Indeed, for all positive and negative values of N c , N f , and γ we investigated, the universal critical behaviordoes not depend on γ . IV. SU( N c ) GAUGE MODELS WITH N c ≥ A. Numerical results
In this section we study the critical behavior of scalarchromodynamics for some values of N f and of N c ≥ N f − model.Such a comparison provides evidence that the asymptoticzero-temperature behavior for finite values of γ in a wideinterval around γ = 0 is described by the 2D CP N f − field theory.In Figs. 1 and 2 we show MC data for the two-flavormodel (3) with SU(3) gauge symmetry, i.e., for N f = 2and N c = 3, and γ = 0. In Fig. 1 the results for theBinder parameter are shown as a function of β for sev-eral lattice sizes. The curves corresponding to differ-ent lattice sizes do not intersect, confirming the absenceof a phase transition at finite β , as expected from theMermin-Wagner theorem. The ratio R ξ behaves analo-gously. For each lattice size R ξ is an increasing functionof β , seemingly divergent for β → ∞ , but no crossingis present between curves corresponding to different L values. In Fig. 2 the data of U appear to approach aFSS curve in the large- L limit when plotted versus R ξ ,in agreement with the FSS prediction (17). This asymp-totic FSS curve is consistent with that of the 2D CP universality class (equivalent to that of the O(3) vectormodel , see, e.g., Ref. [11]) determined in Ref. [19]. More-over, scaling corrections are consistent with the expected O ( L − ) behavior.The behavior of the data for different values of theinverse gauge coupling γ shows that the FSS curve isindependent of γ , at least in a wide interval around γ = 0,as can be seen from Fig. 3, where data for γ = ± N f = 2 and N c = 4, see Fig. 4.These results should be considered as a robust evi-dence that the asymptotic low-temperature behavior oftwo-flavor chromodynamics with SU(3) and SU(4) gaugesymmetry belongs to the universality class of the 2D CP [equivalently, O(3)] field theory.In Fig 5 we report results for the three-flavor latticetheory with SU(3) gauge symmetry. In this case, for both γ = 0 and γ = 2, data appear to approach the FSS curveof the 2D CP model, obtained in Ref. [19] by numericalsimulations. This excellent agreement provides a robustindication that the three-flavor lattice theory with SU(3)gauge theory has the same asymptotic critical behavior β U L=20 γ = 0
L=40L=60L=80L=100 N f = 2 N c = 3 FIG. 1: Plot of U versus β for N f = 2, N c = 3, and γ =0. The horizontal dashed line corresponds to U = 5 /
3, theasymptotic value for β → R ξ U L=20 γ = 0
L=40L=60L=80L=100 CP N f = 2 N c = 3 FIG. 2: Plot of U versus R ξ for N f = 2, N c = 3, and γ = 0.Data approach the universal FSS curve of the 2D CP or O(3)universality class (full line, taken from Ref. [19]). The hor-izontal dashed line corresponds to U = 5 /
3, the asymptoticvalue for R ξ → as the 2D CP model. Analogous results are obtained for N f = 4, see Fig. 6 for results for γ = 0. The FSS curveappears to approach that of the 2D CP model. We notethat for N f = 4 larger scaling corrections are present.However, they appear to be consistent with an O ( L − )behavior.Up to now we have discussed the critical behavior of Q correlations. However, note that the model has theadditional U(1) global invariance, Eq. (8). As we havealready discussed, for N f < N c such an invariance isonly apparent, but in principle it may be relevant for N f ≥ N c . To understand its role, we have studied thebehavior of an appropriate order parameter. As discussedin Ref. [8], for N f = N c an order parameter is providedby the composite operator Y x = det ϕ x , (19)which is invariant under both the SU( N c ) gauge transfor- R ξ U L=20 γ = −3
L=40L=60L=80L=100 CP N f = 2 N c = 3 R ξ U L=20 γ = 3
L=40L=60L=80L=120L=160 CP N f = 2 N c = 3 FIG. 3: Plot of U versus R ξ for N f = 2, N c = 3, and γ = − γ = 3 (lower panel). Data approach theuniversal FSS curve of the 2D CP or O(3) universality class(full line, taken from Ref. [19]). The horizontal dashed linecorresponds to U = 5 /
3, the asymptotic value for R ξ → R ξ U L=20 γ = 0
L=40L=60L=80L=100 CP N f = 2 N c = 4 FIG. 4: Plot of U versus R ξ for N f = 2, N c = 4, and γ = 0.Data approach the universal FSS curve of the 2D CP or O(3)universality class (full line, taken from Ref. [19]). The hor-izontal dashed line corresponds to U = 5 /
3, the asymptoticvalue for R ξ → R ξ U L=20 γ = 0
L=40L=60L=80L=100L=30 CP L=40L=50L=70L=100 N f = 3 N c = 3 R ξ U L=40 γ = 2
L=80L=120L=160L=30 CP L=40L=50L=70L=100 N f = 3 N c = 3 FIG. 5: Plot of U versus R ξ for N f = 3, N c = 3. Resultsfor γ = 0 (top) and γ = 2 (bottom). Data (empty symbols)approach the universal FSS curve of the 2D CP universalityclass (CP results, taken from Ref. [19], are reported with fullsymbols). The horizontal dashed line corresponds to U = 5 / R ξ → R ξ U L=20 γ = 0
L=40L=60L=80L=100L=120L=50 CP L=70L=96L=120 N f = 4 N c = 3 FIG. 6: Plot of U versus R ξ for N f = 4, N c = 3, and γ = 0.Data (empty symbols) Data (empty symbols) approach theuniversal FSS curve of the 2D CP universality class (CP results, taken from Ref. [19], are reported with full symbols).The horizontal dashed line corresponds to U = 17 /
15, theasymptotic value for R ξ → β ξ Y L=20L=40 N f = 3 N c = 3 γ = 0 FIG. 7: Plot of the correlation length ξ Y associated with thecorrelation function h ¯ Y x Y y i versus β . The quantity Y x is de-fined in Eq. (19). Results for N f = N c = 3 and γ = 0. mations (6) and the global transformations ϕ x → ϕ x V with V ∈ SU( N f ). Starting from Y x , one can define acorrelation function G Y ( x − y ) = h ¯ Y x Y y i , (20)and a correlation length ξ Y , using Eq. (14). Results for ξ Y for N f = N c = 3 and γ = 0 are presented in Fig. 7.Apparently, ξ Y remain finite and very small ( ξ Y ≈ . β increases. The U(1) flavor modes are clearly notrelevant for the critical behavior, which is completelycontrolled by the U(1)-invariant modes encoded in Q x .The possibility of a U(1) critical behavior, which wouldimply the presence of a finite-temperature Berezinskii-Kosterlitz-Thouless transition [21–24], is excluded by theMC data. The behavior we observe is completely anal-ogous to what occurs in three dimensions at finite tem-perature [8]. B. Universality class of the asymptoticlow-temperature behavior
The numerical FSS analyses reported above suggestthat, for N c ≥
3, the low-temperature asymptotic be-havior of scalar chromodynamics with N f flavors dependsonly on N f . Irrespective of the values of N c and of γ , thecritical behavior is the same as that of the 2D CP N f − model.Before presenting further arguments to support sucha conclusion, we recall some features of the 2D CP N − model [11, 25]. This is a 2D quantum field theory definedon a complex projective space, isomorphic to the sym-metric space U( N ) / [U(1) × U( N − L = 12 g D µ z · D µ z , ¯ z · z = 1 , (21) D µ = ∂ µ + iA µ , A µ = i ¯ z · ∂ µ z , where z is an N -component complex field, and A µ is acomposite gauge field. The Lagrangian in invariant underthe local U(1) gauge transformations z ( x ) → e iθ ( x ) z ( x ),and the global transformations z ( x ) → W z ( x ) with W ∈ SU( N ). The global invariance group is SU( N ) / Z N (againglobal transformations differing by a Z N factor are gaugeequivalent).For N = 2 the CP field theory is locally isomorphic tothe O(3) non-linear σ model with the identification of thethree-component real vector s a x = P ij ¯ z i x σ aij z j x , where a = 1 , , σ a are the Pauli matrices. Various latticeformulations of CP N − models have been considered, see,e.g., Refs. [26, 27]. The simplest formulation is S CP = − J X x µ | ¯ z x · z x +ˆ µ | = − J X x µ Tr P x P x +ˆ µ , (22)where P ab x = ¯ z a x z b x (23)is a projector, i.e., it satisfies P x = P x . This explicitlyshows that 2D CP N − theories describe the dynamicsof projectors on N -dimensional complex spaces. CP N − models can also be obtained by considering the action (3)with γ = 0, z replacing the field ϕ and using U(1) gaugefields U x ,µ .In 2D CP N − models correlations are always short-ranged at finite β [9]. A critical behavior is only observedfor T →
0. In this limit the correlation length increasesexponentially as [11, 26] ξ ∼ T p e c/T . (24)This behavior is related to the asymptotic-freedom prop-erty of these models, which is shared with quantum chro-modynamics, the four dimensional theory of strong inter-actions. An analogous exponential behavior is expectedto characterize all statistical lattice theories belongingto the same universality class, therefore also the 2D N -component Abelian-Higgs lattice model [19] (which is alattice version of scalar electrodynamics), and, as we shallargue, the 2D scalar chromodynamics with N f flavorsand N c ≥ N c ≥ N f − modelsor, equivalently, of the 2D N f -component Abelian-Higgsmodel with a U(1) gauge symmetry. This is certainlyquite surprising. We shall now argue that the corre-spondence is strictly related to the identical nature ofthe minimum-energy configurations, which represent thebackground for the spin waves that are responsible forthe zero-temperature critical behavior.The nature of the minimum-energy configurations isdiscussed in App. A. For γ ≥
0, such configurations arethose for which Re Tr ϕ † x U x ,µ ϕ x +ˆ µ = 1 (25)on each lattice link. In the appendix, by combining ex-act and numerical results, we show that, for β → ∞ and N c ≥
3, by appropriately fixing the gauge, the config-urations that dominate the statistical average have theform Π x = (cid:18) V
00 1 (cid:19) (26)where V is an SU( N c −
1) matrix, and ϕ af = 0 a < N c ,ϕ af = z f a = N c , (27)where z f is a unit-length N f -dimensional vector. In otherwords, the analysis shows that gauge and ϕ fields com-pletely decouple. Moreover, the ϕ field becomes equiva-lent to a single unit-length N f -dimensional vector, whichis the fundamental field of the CP N f − model. Stated dif-ferently, the operator P x becomes a projector, i.e., sat-isfies P x = P x , for T →
0. However, we cannot yet,at this point, argue that the large- β behavior of scalarchromodynamics and of the CP N f − model is the same,because in our factorization there is no U(1) gauge sym-metry. However, our numerical data also show that thecritical behavior is only associated with the order pa-rameter Q x : the U(1) modes do not order in the large- β limit. This is also confirmed by the detailed analysis ofthe low-temperature configurations presented in Ref. [8].Therefore, in the effective theory we can quotient out theU(1) degrees of freedom, which are irrelevant for the be-havior of the order parameter Q x , i.e., we can reintroducethe U(1) gauge symmetry. If this occurs, scalar chromo-dynamics and CP N f − model are expected to have thesame critical large- β behavior.It is interesting to observe that CP N f − behavior hasalso been observed for several negative values of γ . This isnot an obvious result, as the system is frustrated. Also inthis case, the result is explained by the nature of the low-temperature configurations. As discussed in App. A fora specific value of γ , γ = −
1, the relevant configurationscan again be parametrized as in Eq. (27), modulo gaugetransformations.This phenomenological argument explains the numer-ical evidence that the asymptotic zero-temperature be-haviors for N c ≥ N f − continuum theory. Note that this scenario does not applyto nonabelian gauge theories with N c = 2. As discussedin App. A, the typical low-temperature configurationscannot be parametrized as in Eq. (27), implying a differ-ent critical behavior. We shall argue that it correspondsto that of the 2D Sp( N f ) field theories. V. SU(2) GAUGE MODELS
We now discuss the behavior of models with SU(2)gauge symmetry. In this case the global symmetry group[7, 8] is Sp( N f )/ Z . In the two-flavor case, because ofthe isomorphism Sp(2) / Z =SO(5), an O(5) symmetryemerges. Because of the symmetry enlargement, theorder-parameter field is the 2 N f × N f matrix T lm x = X a Γ al x Γ am x − δ lm N f , (28)where the matrix Γ x is defined in Eq. (9). If f, g =1 , ..., N f , T lm x can be written in the block form T f,g x = Q fg x T f,g + N f x = ¯ D fg T f + N f ,g x = − D fg T f + N f ,g + N f x = Q gf x (29)where D fg x = X ab ǫ ab ϕ af x ϕ bg x . (30)The order parameter T x is hermitian and satisfies J T x J + T x = 0 , (31)where the matrix J is defined in Eq. (11).For N f = 2 the matrix T x can be parametrized by afive-dimensional real vector Φ x . The first three compo-nents are given byΦ k x ≡ X fg σ kfg Q fg x , k = 1 , , , (32)while the fourth and fifth component are the real andimaginary parts of12 X fg ǫ fg D fg x ≡ Φ x + i Φ x . (33)The parametrization of T x in terms of Φ x effectively im-plements the isomorphism between the Sp(2)/ Z and theSO(5) groups, since an Sp(2) transformation of T x mapsto an SO(5) rotation of Φ x . Moreover, the unit-lengthcondition for ϕ implies Φ x · Φ x = 1 . (34)The discussion of the previous section leads us to con-jecture that the global Sp( N f ) / Z symmetry uniquelydetermines the asymptotic zero-temperature critical be-havior. For N f = 2 this would imply that the SU(2)gauge theory has the same zero-temperature behavior ofthe O(5) vector model. An analogous conjecture provedto be true in the three-dimensional case [7, 8]. To per-form the correct universality check for N f = 2, as dis-cussed in detail in Ref. [8], it is important to consider aBinder parameter in the SU(2) gauge theory that mapsonto the usual vector O(5) Binder parameter under theisomorphism Sp( N ) / Z → SO(5). The Binder parame-ter U defined in Eq. (15) is not the appropriate one sinceit only involves three components of Φ k x , see Eq. (32).A straightforward group-theory computation shows that R ξ U r L=20 γ = 0
L=40L=60L=80
O(5) N f = 2 N c = 2 FIG. 8: Plot of U r versus R ξ for N f = 2, N c = 2, and γ = 0.The data approach a universal FSS curve, which correspondsto that of the standard O(5) nearest-neighbor vector model(full line, see Fig. 9). The horizontal dashed line correspondsto the asymptotic value U r = 7 / R ξ → R ξ U L=50 L=100
2D O(5) vector model
FIG. 9: Plot of U versus R ξ for the O(5) vector universalityclass, as obtained by MC simulations of the nearest-neighborO(5) vector lattice model. The full line [28], is an interpola-tion the MC data up to R ξ . .
8. It provides an approxi-mation of the universal FSS curve, with an accuracy smallerthan 0.5% (we include the uncertainty arising from scalingcorrections). The horizontal dashed line corresponds to theasymptotic value U = 7 / R ξ → U → R ξ → ∞ . the correct correspondence is achieved by defining therelated quantity [7, 8] U r = 2125 U . (35)As for R ξ , the quantity computed using Eq. (14) corre-sponds exactly to the analogous quantity computed inthe O(5) vector model.The results shown in Fig. 8 clearly support the con-jecture. Indeed, the MC data of U r collapse (withoutappreciable scaling violations) on a unique curve whenplotted versus R ξ , which is consistent with that of theBinder parameter U versus R ξ for the 2D O(5) vector R ξ U r L=20 γ = 2
L=40L=60L=80L=20 γ=−2
L=40L=60L=80
O(5) N f = 2 N c = 2 FIG. 10: Plot of U r versus R ξ for N f = 2, N c = 2, and γ = ±
2. The data approach the O(5) FSS curve in the large L limit (full line [28]). The horizontal dashed line correspondsto the asymptotic value U r = 7 / R ξ → model (with U and R ξ defined analogously in terms of Φcorrelations [19]). The O(5) FSS curve is obtained by MCsimulations (using the cluster algorithm) of the nearest-neighbor O(5) vector model (2), see Fig. 9. Again therole of the inverse gauge coupling is irrelevant. It doesnot change the universal features of the low-temperatureasymptotic behavior, as shown in Fig. 10, where we re-port results for γ = 2 and − N f flavors be-longs to the universality class associated with the 2DSp( N f ) field theory. The fundamental field is a complex2 N f × N f order-parameter field Ψ x , which formally rep-resents a coarse-grained versio of T x , defined in Eq. (28).It is hermitian, traceless, and satisfies Eq. (31). If wewrite Ψ = (cid:18) A A A A (cid:19) , (36)where A i are N f × N f matrix fields, the conditions re-quired are that A is hermitian and traceless, A is anti-symmetric, A = ¯ A , and A = − ¯ A . The corresponding2D field theory is defined by the Lagrangian L Sp = 1 g Tr (cid:2) ∂ µ Ψ † ∂ µ Ψ (cid:3) , Tr Ψ † Ψ = 1 . (37)For N f = 2, using the correspondence A = 12 (cid:18) Φ Φ − i Φ Φ + i Φ − Φ (cid:19) , (38) A = 12 (cid:18) + i Φ − Φ − i Φ (cid:19) , one can easily show that the Sp(2) field theory is equiv-alent to the O(5) σ -model with Lagrangian L O = 1 g ∂ µ Φ · ∂ µ Φ , Φ · Φ = 1 . (39) VI. CONCLUSIONS
We have studied a 2D lattice nonabelian gauge modelwith multicomponent scalar fields, focusing on the rolethat global and local nonabelian gauge symmetries playin determining the universal features of the asymptoticlow-temperature behavior. The lattice model we con-sider is obtained by partially gauging a maximally O( M )-symmetric multicomponent scalar model, using the Wil-son lattice approach. The resulting theory is locally in-variant under SU( N c ) gauge transformations ( N c is thenumber of colors) and globally invariant under SU( N f )transformations ( N f is the number of flavors). The fieldsbelong to the coset S M /SU( N c ), where M = 2 N f N c and S M is the M -dimensional sphere. The model is alwaysdisordered at finite temperature, in agreement with theMermin-Wagner theorem [9]. However, it develops a crit-ical behavior in the zero-temperature limit. The cor-responding universal features are determined by meansof numerical analyses of the FSS behavior in the zero-temperature limit.We observe universality with respect to the inversegauge coupling γ that parametrizes the strength of thegauge kinetic term, see Eq. (3). The RG flow is alwayscontrolled by the infinite gauge-coupling fixed point, cor-responding to γ = 0, as it also occurs in three dimen-sions [7, 8], and in 2D and 3D models characterized byan abelian U(1) gauge symmetry [19, 29]. Indeed, mod-els corresponding to different values of γ have the sameuniversal behavior for T →
0, at least in a large inter-val around γ = 0. We conjecture that the same criticalbehavior is obtained for all positive finite values of γ ,since, by increasing γ , we do not expect any qualitativechange in the structure of the minimum-energy configu-rations that control the statistical average. On the otherhand, the behavior for negative values of γ , i.e., whenthe system is frustrated, is not completely understood.Therefore, we cannot exclude that the behavior changesfor large negative values of γ . This issue remains anopen problem. It is important to note that by consider-ing a positive value of γ , we are effectively investigatingthe behavior close to the multicritical point β = ∞ and β g = βγ = ∞ . Our results show that approching thepoint along the lines β g /β = γ does not change the uni-versal features of the asymptotic behavior. However, weexpect that, by increasing β g faster than β (in a well-specified way), one can observe a radical change in thecritical behavior. For instance, if we take first the limit β g → ∞ at fixed finite β and then the limit β → ∞ , themodel becomes equivalent to the standard O( M ) vec-tor model, characterized by a different asymptotic low-temperature behavior.The numerical results and theoretical arguments pre-sented in this paper suggest the existence of a wide uni-versality class characterizing 2D lattice abelian and non-abelian gauge models, which only depends on the globalsymmetry of the model. The gauge group does not ap-parently play any particular role. Indeed, we report nu-merical evidence that, for any N c , the asymptotic low-temperature behavior of the multiflavor scalar gauge the-ory (3) belongs to the universality class of the 2D CP N f − model. This also implies that it has the same universalfeatures of the N f -component lattice scalar electrody-namics (abelian Higgs model) [19]. It is important to notethat the global symmetry group of model (3) is U( N f ),while the global symmetry group of the CP N f − model isSU( N f ) (we disregard here discrete subgroups), so thatthe global symmetry group of the two models differs bya U(1) flavor group. As we have discussed in Ref. [8], theU(1) symmetry is only apparent for N f < N c . and there-fore, the symmetry groups of scalar chromodynamics andof the CP N f − model are the same for N f < N c . ThisU(1) symmetry is instead present for N f ≥ N c . How-ever, our numerical results indicate that the U(1) flavorsymmetry does not play any role in model (3). The uni-versal critical behavior is only associated with the U(1)-invariant modes that are encoded in the local bilinearoperator Q x , so that the global symmetry group thatdetermines the asymptotic behavior is SU( N f ). Note,however, that the decoupling of the U(1) flavor modesmay not be true in other models with the same globaland local symmetries. If the U(1) modes become criti-cal, a different critical behavior might be observed. Thisissue deserves further investigations.For N c = 2, the global symmetry group changes: Theaction is invariant under Sp( N f ) transformations. Inthis case the asymptotic low-temperature behavior is ex-pected to be described by the Sp( N f ) continuum theory.We have numerically checked it for the two-flavor model,for which the global symmetry Sp(2) / Z ≃ SO(5).Our results lead us to conjecture that the RG flowof the 2D multiflavor lattice scalar chromodynamics inwhich the fields belong to the coset S M /SU( N c ), where M = 2 N c N f , is asymptotically controlled by the 2Dstatistical field theories associated with the symmetricspaces [10, 11] that are invariant under SU( N f ) (for N c ≥
3) or Sp( N f ) (for N c = 2) global transformations.These symmetry groups are the same invariance groupsof scalar chromodynamics, apart from a U(1) flavor sym-metry that is present for N f ≥ N c >
2, which does notplay any role in determining the asymptotic behavior ofthe model.This conjecture may be further extended to modelswith different global and local symmetry groups, for in-stance, to those considered in Refs. [10, 30]. It would beinteresting to verify whether generic nonabelian modelshave an asymptotic critical behavior which is the sameas that of the model defined on a symmetric space thathas the same global symmetry group. This issue deservesfurther investigations.
Acknowledgement . Numerical simulations have beenperformed on the CSN4 cluster of the Scientific Comput-ing Center at INFN-PISA.
Appendix A: Minimum-energy configurations
In this appendix we identify the minimum-energy con-figurations for the action (3), summarizing the main ar-guments reported in Ref. [8] and extending them to the γ = 0 case.
1. Behavior for γ = 0 We start by considering the simplest case γ = 0. Theglobal minimum is obtained by the configurations thatsatisfy the maximum conditionRe Tr ϕ † x U x ,µ ϕ x +ˆ µ = 1 (A1)on each link. This condition is trivially satisfied if ϕ x +ˆ µ = U † x ,µ ϕ x , (A2)which implies Q x = Q x +ˆ µ , and, thus, the breaking of theSU( N f ) / Z N c symmetry for β → ∞ .If we apply repeatedly the relation (A2) along a pla-quette, we obtain the consistency condition ϕ x = Π x ϕ x , (A3)where Π x is the plaquette operator (4).For N c = 2, Eq. (A3) implies that Π x is the identitymatrix. The same argument allows us to show that alsoall Polyakov loops can be reduced to the identity matrix,so that all gauge configurations are trivial. The energy-minimum configurations can therefore be written as ϕ af x = W ab x A bf , U x ,µ = W x W † x +ˆ µ , (A4)where A af is a generic space-independent 2 × N f complexmatrix satisfying Tr A † A = 1, and W x ∈ SU(2). To verifythese conclusions we have performed simulations on a 4 lattice for very large values of β ( β varies between 30 and100). The results are reported in Table I. As expectedRe Tr Π x = N c . (A5)We have also computed U and h Tr P x i , confirming thepredictions of Ref. [8] obtained only assuming that theconfigurations of minimal energy are those of the form(A4): h Tr P x i ≡ N f + N c N f N c , (A6)and U = (1 + N f N c )( N f N c + 4 N f + N f N c − N f − N f N c )(3 + N f N c ) , (A7)with N c = 2.0 ( N c , N f ) h Re Tr Π x i /N c S g / (2 N f ) U Eq. (A7) h Tr P x i Eq. (A6)(2, 2) 1.00004(2) − . . . − . . . . . . (3, 2) 0.3421(3) − − − β → ∞ on a 4 lattice for γ = 0. For N c ≥
3, the minimum-energy condition (A3)has several classes of different solutions. If Π x satisfiesEq (A3), we can always write it asΠ x = V ⊕ V
00 1 ! (A8)where V is an SU( N c −
1) matrix, modulo a gauge trans-formation. The corresponding configurations of the fields ϕ x depend on the structure of the matrix V . If V is ageneric unitary matrix which does not have unit eigen-values, Eq. (A3) implies that the field ϕ is necessarilygiven by ϕ af = 0 a < N c ,ϕ af = z f a = N c , (A9)where z f is a unit N f -dimensional vector. Different ϕ configurations are only possible if V has some unit eigen-values. For instance, if V = V ⊕
1, with V belonging tothe SU( N c −
2) subroup, then the ϕ field configurationsof the form ϕ af = 0 a < N c − ,ϕ af = w f a = N c − ,ϕ af = z f a = N c , (A10)( z f and w f are generic N f -dimensional vectors) satisfythe condition (A3). To understand which type of config-urations dominate, we have again resorted to numericalsimulations on small lattices. The results are reported inTable I. For Π x results are consistent with h Re Tr Π x i = 1 . (A11)This relation is consistent with Eq. (A8) only if we as-sume that the matrix V is a randomly chosen SU( N c − V = V ⊕ V ∈ SU( N c − h Re Tr Π x i =2. This result constraints the field ϕ to be of the form(A9). If this is the case, the operator P takes the form P fg = ¯ z f z g in the large- β regime. Therefore, P becomesa projector and Tr P is predicted to be one. Analo-gously, also the Binder parameter should converge to one.The numerical results reported in Table I are in perfectagreement.
2. Behavior for γ = 0 Let us now determine the minimum-energy configura-tions for γ >
0. For N c = 2, the introduction of a positive γ is of course irrelevant: the gauge part of the action isalready minimized for γ = 0. The analysis for N c ≥ β g = βγ , fixing the valueof γ corresponds to considering a particular way of ap-proaching the limiting point β = β g = ∞ . We will nowargue that the relevant ϕ configurations, that is thosethat dominate the ensemble average, strongly depend onhow the limit is taken. Imagine that one first takes thelimit β g → ∞ at fixed, finite β and then the limit β → ∞ .In this case, the limiting configurations would have theform (A4) and h Tr P x i and U would assume the values(A6) and (A7), respectively. On the other hand, considerthe opposite approach: first, we take the limit β → ∞ at fixed, finite β g , followed by β g → ∞ . For finite β g ,the operator Π x should have the form (A8). The matrix V would not be random (it should become closer to theidentity as β g increases). Nonetheless, it is not expectedto have an eigenvector with an eigenvalue exactly equalto one. There, the relevant ϕ configurations should al-ways be of the form (A9). It is not possible to predicta priori what are the relevant configurations if the limitis taken keeping the ratio γ = β g /β fixed and we havetherefore performed simulations on very small lattices.Results for γ = 1 are reported in Table II. For N f = 2results on a lattice with L = 4 are definitely consistentwith U = 1 and h Tr P x i = 1. For N f = 3 we observe sig-nificantly larger size corrections. We have performed adetailed study for N c = 3. We observe that the data con-verge to the expected results U = 1 and h Tr P x i = 1. U converges quite fast, while the second quantity convergeswith the expected behavior L − . For N f = 4, size cor-rections are even larger, but the extrapolations are againconsistent with the expected results. If we extrapolatethe estimates of h Tr P x i reported in Table II assumingcorrections that decay as 1 /L we obtain results that areconsistent with one.Let us finally discuss the case γ <
0, which is muchless obvious, as the system shows frustration. Indeed, ifwe minimize the contribution of the action that dependson the fields ϕ , we obtain the consistency condition (A3),which requires each plaquette to have at least one uniteigenvalue. On the other hand, minimizing the plaquetteterm we would expect the plaquette Π x to converge to1 ( N c , N f ) h Re Tr Π x i /N c S φ / (2 N f ) U Eq. (A7) h Tr P x i Eq. (A6) L (2, 2) 1.000018(8) − . . . − . . . . . . − − − − − − − − − − − − β → ∞ on a L lattice for γ = 1. Here S φ is the part of the action (3) that depends on the ϕ field (for γ = 0 we have S g = S φ ).( N c , N f ) h Re Tr Π x i /N c S φ / (2 N f ) U Eq. (A7) h Tr P x i Eq. (A6) L (2,2) − − . . . − − . . . − − . . . − − . . . . . . − − . . . . . . − − . . . . . . − − − − − − − − − − − − − − − − − − β → ∞ on a L lattice for γ = −
1. Here S φ is the part of the action (3) that depends onthe ϕ field (for γ = 0 we have S g = S φ ). − I or − e ± iα I , for even or odd N c , respectively, where I is the N c × N c identity matrix and α = π/N c [correspond-ingly, Tr Π x would converge to − N c or − N c cos( π/N c )].Therefore, one cannot simultaneously minimize all localcontributions: the system is frustrated.To understand the effective behavior along the lines β g = γβ , we have again studied numerically the systemfor a specific value of γ , γ = −
1. The results are reportedin Table III. Per N c = 2, they show the clear presenceof frustration. The plaquette is not identical to the ma-trix − I and the minimal condition (A1) is not satisfied.Nonetheless, the estimates of the Binder parameter andof the trace of P x are completely consistent with the theresults obtained for γ = 0. Even for γ = − ϕ are uniformly distributed on the N -dimensional sphere( N = 4 N f ). We conclude that γ plays a role only on thegauge properties, but not on the behavior of ϕ correla-tions, which dominate the large- β behavior.For N c = 3, the results for the trace of the plaque-tte are completely consistent with Π x = diag ( − , − , γ = 0, Eq. (A8). Theonly difference is that the matrix V is no longer a ran-dom SU(2) matrix, but is simply − I (where I is the two-dimensional identity). Finally, we have some results for N c = 4. In this case, it is difficult to take the limit β → ∞ , using data in the range 10 ≤ β ≤
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