Asymptotic low-temperature critical behavior of two-dimensional multiflavor lattice SO(Nc) gauge theories
Claudio Bonati, Alessio Franchi, Andrea Pelissetto, Ettore Vicari
aa r X i v : . [ h e p - l a t ] J un Asymptotic low-temperature critical behavior oftwo-dimensional multiflavor lattice SO( N c ) gauge theories Claudio Bonati, Alessio Franchi, Andrea Pelissetto, and Ettore Vicari Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Pisa, Italy Dipartimento di Fisica dell’Universit`a di Pisa, Pisa, Italy Dipartimento di Fisica dell’Universit`a di Roma Sapienza and INFN, Roma, Italy (Dated: June 30, 2020)We address the interplay between global and local gauge nonabelian symmetries in lattice gaugetheories with multicomponent scalar fields. We consider two-dimensional lattice scalar nonabeliangauge theories with a local SO( N c ) ( N c ≥
3) and a global O( N f ) invariance, obtained by par-tially gauging a maximally O( N f N c )-symmetric multicomponent scalar model. Correspondingly,the scalar fields belong to the coset S N f N c − /SO( N c ), where S N is the N -dimensional sphere. Inagreement with the Mermin-Wagner theorem, these lattice SO( N c ) gauge models with N f ≥ N f )symmetry. However, in the zero-temperature limit they show a critical behavior characterized by acorrelation length that increases exponentially with the inverse temperature, similarly to nonlinearO( N ) σ models. Their 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 two-dimensional RP N f − model. I. INTRODUCTION
Lattice gauge models provide effective theories in vari-ous physical contexts, ranging from fundamental interac-tions [1, 2] to emerging phenomena in condensed matterphysics [3, 4]. They provide mechanisms for fundamentalphenomena, such as confinement and the Higgs mecha-nism, which explain the spectrum of subnuclear systemsinteracting via strong and electroweak forces, supercon-ductivity, etc... The interplay between global and localgauge symmetries is crucial to determine the main fea-tures of the theory, such as the nature of the spectrum,the degeneracy of the energy levels, the phase diagram,the nature and universality classes of their thermal andquantum transitions.In the case of two-dimensional (2D) lattice gauge mod-els, the interplay of non-Abelian global symmetries andlocal gauge symmetries determines the large-scale prop-erties of the system in the zero-temperature limit, andtherefore, the statistical field theory realized in the corre-sponding continuum limit [5]. These issues have been ad-dressed in the multicomponent Abelian-Higgs model [6],characterized by a global U( N f ) symmetry ( N f ≥ N f ) symmetry and a local SU( N c ) gaugesymmetry. The results of Refs. [6, 7] provide numer-ical evidence that the asymptotic low-temperature be-havior of these 2D lattice gauge models always belongsto the universality class of the 2D CP N f − field theory [5].The universality class is only determined by the globalU( N f ) symmetry of the model. The local gauge symme-try apparently does not play any role: models with dif-ferent gauge symmetry but with the same global invari-ance have the same large-scale low-temperature behavior.These results may be interpreted as a numerical evidenceof a more general conjecture [7]: the renormalization- group flow determining the asymptotic low-temperaturebehavior is generally controlled by the 2D statistical fieldtheories associated with the symmetric spaces [5, 8] thathave the same global symmetry. This is indeed the caseof the Abelian-Higgs model and of scalar chromodynam-ics, whose low-temperature behavior is always controlledby the 2D CP N f − field theory.To achieve additional evidence of the above conjecture,we extend the analysis to other 2D lattice models, char-acterized by different global and local gauge symmetries.For this purpose, we consider 2D lattice models withreal scalar fields, which are invariant under global andlocal gauge transformations that belong to orthogonalgroups. In particular, we consider lattice gauge modelsthat are invariant under SO( N c ) local transformationsand under O( N f ) global transformations ( N c will be re-ferred to as the number of colors and N f as the num-ber of flavors), focusing on the case N f ≥
3, so thatthe symmetry group is nonabelian. According to theMermin-Wagner theorem [9], these lattice gauge mod-els do not present a finite-temperature transition associ-ated with the breaking of the global O( N f ) symmetry.However, they are expected to develop a critical behav-ior in the zero-temperature limit (for N f = 2 the globalsymmetry group is the abelian group O(2), so that afinite-temperature Berezinskii-Kosterlitz-Thouless tran-sition is possible). We study the universal features of theasymptotic zero-temperature behavior for N f = 3 , N c = 3 , N f − model, in which the fields effectively belongto the real projective space in N f dimension, a symmet-ric space which is invariant under global O( N f ) trans-formations. Note that the associated symmetric space isnot the N f -dimensional sphere, that has the same globalsymmetry group. This is due to the fact that the low-energy behavior is essentially characterized by a bilinearoperator (a projector) that is invariant under local Z transformations. We anticipate that the numerical re-sults confirm the conjecture.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 use toinvestigate the nature of the low-temperature critical be-havior. Then, in Sec. IV we report the numerical resultsfor lattice models with N f = 3 , N c = 3 ,
4. Finally,in Sec. V we summarize and draw our conclusions. InApp. A we report some results on the minimum-energyconfigurations of the models considered.
II. THE MULTIFLAVOR LATTICE SO( N c )GAUGE MODEL We define a 2D lattice scalar gauge theory by partiallygauging a maximally symmetric model of real matrixvariables φ af x , with a = 1 , .., N c and f = 1 , ..., N f (wewill refer to these two indices as color and flavor indices,respectively). We start from the action S sym = − t X x ,µ Tr φ t x φ x +ˆ µ , Tr φ t x φ x = 1 , (1)where the sum is over all links of a square lattice andˆ µ = ˆ1 , ˆ2 denote the unit vectors along the lattice direc-tions. Without loss of generality, we can set t = 1.The action S sym is invariant under global O( M ) trans-formations with M = N f N c . Indeed, it can be writtenin terms of M -component unit-length real vectors s x , as S sym = − P x ,µ s x · s x +ˆ µ , which is the standard nearest-neighbor M -vector lattice model.We proceed by gauging some of the degrees of freedomusing the Wilson approach [1]. We associate an SO( N c )matrix V 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 [1] for the gauge fields. We thus obtain themodel with action S g = − N f X x ,µ Tr φ t x V x ,µ φ x +ˆ µ − γN c X x Tr Π x , (2)where Π x is the plaquette operatorΠ x = V x , V x +ˆ1 , V t x +ˆ2 , V t x , . (3) Model (1) with the unit-length constraint for the φ x vari-ables is a particular limit of a model with a quartic potential P x V (Tr φ † x φ x ) of the form V ( X ) = rX + u X . Formally itcan be obtained by setting r + u = 0 and taking the limit u → ∞ . The plaquette parameter γ plays the role of inverse gaugecoupling. The partition function reads Z = X { φ,V } e − β S g , β ≡ /T . (4)On can easily check that the lattice model (2) is invariantunder SO( N c ) gauge transformations: φ x → W x φ x , V x ,µ → W x V x ,µ W t x +ˆ µ , (5)with W x ∈ SO( N c ). For γ → ∞ , the link variables V x become equal to the identity (modulo gauge trans-formations), thus one recovers the ungauged model (1),or equivalently the nearest-neighbor M -vector model.For N c = 2 the global symmetry group of model (2)is actually larger than O( N f ). Indeed, one can showthat [15] the model can be exactly mapped onto the lat-tice Abelian-Higgs model S AH = − N f X x ,µ Re [ ¯ z x · λ x ,µ z x +ˆ µ ] (6) − γ X x ,µ>ν Re [ λ x ,µ λ x +ˆ µ,ν ¯ λ x +ˆ ν,µ ¯ λ x ,ν ] , where z x is a unit-length N f -component complex vec-tor, and λ x ,ν a U(1) link variable. The Abelian-Higgsmodel is invariant under local U(1) and global U( N f )transformations. There is therefore an enlargement ofthe global symmetry of the model: the global symmetrygroup is U( N f ) instead of O( N f ). The asymptotic zero-temperature behavior of these models have been studiedin Ref. [6]. Therefore, in the following we focus on theasymptotic low-temperature behavior for N c ≥ N c ) gauge theories. In this workwe focus on the two-dimensional case. According to theMermin-Wagner theorem [9], lattice SO( N c ) gauge theo-ries are not expected to show finite-temperature transi-tions with a low-temperature phase in which the globalO( N f ) symmetry is broken. Therefore, there are onlytwo possibilities: either the system is always disorderedfor any β or a finite-temperature transitions occurs witha low-temperature phase in which there is no long-rangeorder, but correlations decay algebraically with the dis-tance. We expect the first behavior whenever the globalsymmetry group is nonabelian, the second one wheneverthe symmetry group is isomorphic to U(1).For N f ≥
3, the global O( N f ) symmetry group is non-abelian. Therefore, we expect a nontrivial critical be-havior only in the zero-temperature limit, analogous tothat occurring in the nonlinear O( N ) σ model or in theCP N − model, see, e.g., Ref. [5]. Infinite-volume correla-tion functions are characterized by a length scale ξ thatdiverges as ξ ∼ β p e cβ . (7)For N f = 2 and N c ≥
3, the model has an abelian O(2)global symmetry. It is therefore possible that it under-goes a finite-temperature Berezinskii-Kosterlitz-Thoulesstransition [10–14], with a spin-wave low-temperaturephase characterized by correlation functions decaying al-gebraically. For N f = 2 and N c = 2, due to the map-ping to the Abelian-Higgs model (6), the global symme-try group, the U(2) group, is nonabelian. Therefore, themodel is only critical for β → ∞ . The low-temperaturebehavior belongs to the universality class of the 2D CP model [6], which is equivalent to that of the nonlinearO(3) σ model.The global symmetry group of the model is O( N f ),which is not a simple group. Therefore, in principle, onemay have both the breaking the Z subgroup and of theSO( N f ) subgroup. However, on the basis of the resultsfor the same model in three dimensions [15] we do notexpect the Z subgroup to play any role (a similar de-coupling occurs in the unitary case [7]). The critical low-temperature behavior is therefore associated with the or-der parameter for the breaking of the SO( N f ) subgroup,which is the bilinear operator Q fg x = X a φ af x φ ag x − N f δ fg , (8)which is a symmetric and traceless N f × N f matrix.In the following sections we provide numerical evidencethat, for N c ≥ N f ≥
3, the asymptotic zero-temperature limit of the SO( N c ) gauge model (2) is thesame as that of the 2D RP N f − models, which are also in-variant under O( N f ) transformations. The RP N − mod-els are defined by associating a real N -component unit-length vector ϕ x with each lattice site and consideringactions that are invariant under global O( N ) rotationsof the fields and local Z transformations ϕ x → s x ϕ x ( s x = ± N − modelis defined by the lattice action S RP = − t X x ,µ ( ϕ x · ϕ x +ˆ µ ) . (9)Alternatively, one may introduce an explicit link variable σ x ,µ = ±
1, and consider the lattice action S RP σ = − t X x ,µ ϕ x · σ x ,µ ϕ x +ˆ µ . (10)The nature of their low-temperature behavior for N ≥ N − model belongs to the same universality class as the O( N )vector model. We refer to Ref. [23] for a thorough discus-sion of this point. There, we report extensive numericalresults that indicate that the long-distance universal be-havior of the 2D RP N − models differs from that of the2D O( N ) vector models: In the low-temperature limitthey appear as distinct universality classes.In this work we will show that renormalization-groupinvariant quantities defined in terms of Q fg in the non-abelian gauge theory have the same universal behavior as the corresponding RP N f − quantities defined in termsof the local gauge-invariant operator P fg x = ϕ f x ϕ g x − N f δ fg . (11)Such correspondence can be established using the samearguments we used for unitary models in Ref. [7]. Asdiscussed in the Appendix, for β → ∞ the φ configu-rations can be parametrized by a single N f -dimensionalunit vector ϕ f . Modulo gauge transformations, we have φ af = 0 a < N c φ af = ϕ f a = N c (12)which implies that the bilinear Q x becomes equivalentin this limit to the RP N f − operator P x . Since the Z global symmetry does not play any role, in the zerotemperature-limit the gauge model can be described byan effective theory only in terms of the SO( N f ) orderparameter P x . The natural candidate for the action is H eff = − κ X x ,µ Tr P x P x +ˆ µ , (13)which gives (9) apart from an irrelevant constant. Wehave thus obtained the RP N f − model. III. UNIVERSAL FINITE-SIZE SCALING
We exploit FSS techniques [24–27] 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 focus on the correlations of the gauge-invariant variable Q x defined in Eq. (8). The correspond-ing two-point correlation function is defined as G ( x − y ) = h Tr Q x Q y i , (14)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 ) , (15)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 , (16)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 . (17)Indeed, in the FSS limit we have (see, e.g., Refs. [6]) U ( β, L ) ≈ F ( R ξ ) , (18)where F ( x ) is a universal scaling function that completelycharacterizes the universality class of the transition. Theasymptotic values of F ( R ξ ) for R ξ → R ξ → ∞ correspond to the values that U takes in the small- β andlarge- β limits. For R ξ → R ξ → U = 1 + 4( N f − N f + 2) . (19)independently of the value of N c . In the large- β limit wehave U → L . In the case of asymptotically freemodels, such as the 2D CP N − and O( N ) vector models,corrections decrease as L − , multiplied by powers of ln L [28]. However we note that sometimes, when the avail-able data are not sufficiently asymptotic, the approachto the asymptotic behavior may appear slower, and cor-rections apparently decay as L − p with p < 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 forthe study of the asymptotic zero-temperature behaviorof 2D models was employed in Refs. [6, 7]. IV. NUMERICAL RESULTS
In this section we study the large- β critical behaviorof the lattice scalar gauge model (2) for some values of N f ≥ N c ≥
3. We perform MC simulations, usingthe same upgrading algorithm employed in three dimen-sions [15]. We show that the FSS curves (18) of theBinder parameter U versus R ξ computed in the model(2) agree with those computed in RP N − models (we usethe results reported in Ref. [23]). These results providenumerical evidence that, for N c ≥
3, the critical behaviorbelongs to the universality class of the 2D RP N f − fieldtheory, in agreement with the arguments of the previoussection.We first mention that the data of R ξ ≡ ξ/L corre-sponding to different lattice sizes, see Fig. 1, do not in-tersect, confirming the absence of a phase transition atfinite β , as expected on the basis of the Mermin-Wagnertheorem [9]. In Fig. 2, we show the estimates of the corre-lation length for the three-flavor SO(3) and SO(4) gaugetheories (2) with γ = 0, up to lattice sizes L = 256 and β R ξ L=16L=32L=64L=128 N c =4 N f =3 β R ξ L=16L=32L=64L=128L=256 N c =3 N f =3 FIG. 1: R ξ ≡ ξ/L for the three-flavor SO(3) and SO(4) gaugetheories (2) with γ = 0. We show data up to L = 256 for N c = 3 (bottom) and up L = 128 for N c = 4 (top). Datafor different sizes do not show evidence of crossing points.Statistical errors are hardly visible on the scale of the figure. L = 128, respectively. When data for different latticesizes match, they can be considered as a good approxi-mation of the correlation length in thermodynamic limitat the given inverse temperature β . The data in thisregime are substantially consistent with an exponentialdependence of ξ on β , see Eq. (7), as expected for asymp-totically free models.In Fig. 3 we plot U versus R ξ for the three-flavor SO(3)and SO(4) gauge theories with γ = 0, up to L = 256 and L = 128, respectively. We observe that the data of U appear to approach a FSS curve in the large- L limit, inagreement with the FSS prediction (18). In the samefigure we also report data for the standard RP latticemodel with action (9), and for the RP gauge model withaction (10) (as shown in Ref. [23], the data for L = 320provide a good approximation of the asymptotic curve).The RP results are consistent with the asymptotic FSScurve for the SO( N c ) gauge model, confirming our claimthat the RP model and the SO( N c ) gauge model with N f = 3 and any N c ≥ β → ∞ . β ξ L=32L=64L=128 N c =4 N f =3 β ξ L=32L=64L=128L=256 N c =3 N f =3 FIG. 2: The correlation length ξ versus β for N f = 3, N c = 3(bottom) and N f = 3 N c = 4 (top). We set γ = 0. Whendata for different values of L match, they may be considered asgood approximations of the infinite-volume correlation length,within their errors. The behavior of the infinite-volume datais consistent with an exponential dependence on β (we use alogarithmic scale on the vertical axis). We have also performed MC simulations for nonvan-ishing values of γ . Fig. 4 reports data for the three-flavorSO(3) gauge theory (2) with γ = ±
1, up to L = 128.They appear to approach the asymptotic FSS curve ofthe RP universality class, demonstrating that the uni-versal features of the asymptotic low-temperature behav-ior are independent of the inverse gauge coupling γ , atleast in a wide interval around γ = 0. Data up to L = 64for γ = ± curveas L increases. As discussed in Sec. II, the asymptoticFSS curves must change if we take the limit γ → ∞ andthen the limit β → ∞ , In this case the SO(3) and SO(4)gauge theories turn into the O(9) and O(12) model, re-spectively.These results should be considered as a robust evidencethat the asymptotic low-temperature behavior of thethree-flavor lattice gauge theory with SO(3) and SO(4)gauge symmetry belongs to the universality class of the2D RP universality class, in a large interval of values of R ξ U L=16L=32L=64L=128L=320L=320 N c =4 N f =3 RP standard RP gauge R ξ U L=16L=32L=64L=128L=256L=320L=320 N c =3 N f =3 RP standard RP gauge FIG. 3: Plot of U versus R ξ for the three-flavor SO(3) (bot-tom) and SO(4) (top) gauge theory at γ = 0. The dataapproach the asymptotic curve of the 2D RP models (9)and (10) (labelled as standard and gauge , respectively; thecorresponding data for L = 320 are taken from Ref. [23]).Statistical errors are so small to be hardly visible. γ around γ = 0.As an additional check of the arguments presented inSec. II, we have performed simulations of the model (2)for N f = 4, N c = 3 and γ = 0. The results for the Binderparameter are plotted versus R ξ in Fig. 5. For compari-son we also report results for the RP gauge model. TheSO(3) gauge data show a significant size dependence, butwith a clear trend towards the RP data. In particular,the SO(3) gauge data corresponding to L = 128 are es-sentially consistent with the RP data, confirming againthe asymptotic equivalence of the universal large-distancebehavior of the SO(3) gauge model and of the RP model. V. CONCLUSIONS
We have studied a class of 2D lattice nonabelianSO( N c ) gauge models with multicomponent scalar fields,focusing on the role that global and local nonabeliangauge symmetries play in determining the universal fea-tures of the asymptotic low-temperature behavior. Such R ξ U L=16L=32L=64L=128L=320L=320 N c =3 N f =3 γ =1 RP standard RP gauge R ξ U L=16L=32L=64L=128L=320L=320 N c =3 N f =3 γ =-1 RP standard RP gauge FIG. 4: Plot of U versus R ξ for N f = 3, N c = 3, γ = − γ = 1 (upper panel). Data approach thesame universal FSS curve obtained for the γ = 0 SO( N c )gauge model and the RP models (9) and (10) (see Fig. 3). R ξ U L=16L=32L=64L=128L=32L=64L=128 N c =3 N f =4 RP gauge FIG. 5: Plot of U versus R ξ for N f = 4, N c = 3, and γ = 0.Data approach the same universal FSS curve obtained for theRP models (9) and (10) . lattice gauge models are obtained by partially gaug-ing a maximally O( M )-symmetric multicomponent scalarmodel, M = N f N c , using the Wilson lattice approach.For N c ≥
3, the resulting theory is locally invariant un-der SO( N c ) gauge transformations and globally invari-ant under O( N f ) transformations. For N c = 2, theselattice gauge models are instead equivalent to the 2DAbelian-Higgs model and therefore have a larger U( N f )global invariance group. The fields belong to the coset S M − /SO( N c ), where M = N f N c and S M − is the( M − M -dimensional space.Since for N c = 2 these lattice gauge models are equiv-alent to the 2D Abelian-Higgs models, already studied inRef. [6], we only consider N c ≥
3. Moreover, we will onlyconsider models with N f ≥
3. In this case the globalsymmetry group is nonabelian, and thus one expects thesystem to develop a critical behavior only in the zero-temperature limit. For N f = 2 the behavior is expectedto be different, since the global abelian O(2) symme-try may allow finite-temperature Berezinskii-Kosterlitz-Thouless transitions.The universal features of the zero-temperature behav-ior are determined by means of MC simulations. Weconsider the lattice SO( N c ) gauge models (2) for N c = 3,4 and for N f = 3 , N f − models, characterized by the same global O( N f )symmetry and by a local Z gauge symmetry. The nu-merical results are supported by theoretical argumentsthat show that RP N f − models and SO( N c ) gauge the-ories with N f flavors have the same ground-state (zero-temperature) properties. Moreover, the gauge degrees offreedom decouple as β → ∞ .These results provide further support to the conjectureput forward in Ref. [7], that the renormalization-groupflow determining the asymptotic low-temperature behav-ior is generally controlled by the 2D statistical theoriesassociated with the symmetric spaces that have the sameglobal symmetry. For models with complex fields andU( N f ) global invariance—for instance, the multicompo-nent lattice Abelian-Higgs model and the multiflavor lat-tice scalar chromodynamics considered in Ref. [7]—theuniversal behavior is described by the 2D CP N f − fieldtheory. For the lattice SO( N c ) gauge models with N c ≥ N f ≥
3, instead, the RP N f − field theory is the rel-evant one. 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 (2). The analysis is very similarto that presented for unitary models in Ref. [7]. We referthe reader to this work for additional details.
TABLE I: Estimates of several observables on the minimum-energy configurations for γ = 0, for two lattice sizes L = 4 , β numerical data (we usethe same procedure discussed in the appendix of Ref. [7]).( N c , N f ) L h Tr Π x i /N c S g / (2 V N f ) U h Tr Q x i (3 ,
3) 4 0.3504(2) − − ,
4) 4 0.3600(3) − − ,
3) 4 0.2564(1) − − ,
4) 4 0.2599(1) − − We start by considering the simplest case γ = 0. Theminimum-energy configurations are those that satisfy thecondition Tr [ φ t x V x ,µ φ x +ˆ µ ] = 1 (A1)for each link. This condition is satisfied if φ x +ˆ µ = V t x ,µ φ x , and therefore Q x = Q x +ˆ µ , thus entailing thebreaking of the global symmetry for β → ∞ .The previous relation implies the consistency condition φ x = Π x φ x , where Π x is the plaquette operator (3). For N c ≥
3, such consistency condition has several classes ofdifferent solutions. The plaquette Π x must satisfyΠ x = A ⊕ A
00 1 ! (A2)where A is an SO( N c −
1) matrix, modulo a gauge trans-formation. The corresponding configurations of the fields φ x depend on the structure of the matrix A . If A is ageneric unitary matrix which does not have unit eigen-values, the field φ is necessarily given by φ af = 0 a < N c ,φ af = v f a = N c , (A3) where v f is a unit N f -dimensional vector. Different φ configurations are only possible if A has some unit eigen-values. For instance, if A = A ⊕
1, with A belonging tothe SO( N c −
2) subgroup, then the φ field configurationsof the form φ af = 0 a < N c − ,φ af = w f a = N c − ,φ af = v f a = N c , (A4)( v f and w f are generic N f -dimensional vectors) satisfythe condition φ x = Π x φ x . To understand which typeof configurations dominate, we have again resorted tonumerical simulations on small lattices. The results arereported in Table I. For the plaquette operator Π x , seeEq. (3), results are consistent with h Tr Π x i = 1 , (A5)in the large- L limit. This relation is consistent withEq. (A2) only if we assume that the matrix A is arandomly chosen SO( N c −
1) matrix. For instance, if A = A ⊕ A ∈ SO( N c − h Tr Π x i = 2. This result con-straints the field φ to be of the form (A3). If this isthe case, the operator Q x , defined in Eq. (8), takes theform Q fg x = v f v g − δ fg /N f in the large- β regime. There-fore, Q x becomes equivalent to the operator P x definedin the RP N f − theory. As an additional check that therelevant configurations are those of the form (A3), wecompute the Binder parameter, which should convergeto 1. The numerical results reported in Table I are ingood agreement.When γ = 0 the analysis of the minimum-energy con-figurations becomes more complicated, as is also the casefor lattice SU( N c ) gauge theories (see the appendix ofRef. [7]). We do not repeat here the arguments of Ref. [7].They apply to SO( N c ) theories as well, as we have explic-itly verified numerically for γ = − γ = 1. We onlymention that, as in the case of SU( N c ) gauge theories,the gauge parameter γ is relevant for gauge properties,but not for the behavior of the φ correlations, which dom-inate the large- β limit. [1] K.G. Wilson, Confinement of quarks, Phys. Rev. D ,2445 (1974).[2] S. Weinberg, The Quantum Theory of Fields , (CambridgeUniversity Press, 2005).[3] S. Sachdev, Topological order, emergent gauge fields,and Fermi surface reconstruction, Rep. Prog. Phys. ,014001 (2019).[4] P. W. Anderson, Basic Notions of Condensed MatterPhysics , (The Benjamin/Cummings Publishing Com-pany, Menlo Park, California, 1984).[5] J. Zinn-Justin,
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