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Dtd aur ,2021) 1, January (Dated: = nraPelissetto, Andrea N 4 f the- N c . iin.Fnly esmaieadda u conclusions our V. draw Sec. and tran- in phase summarize the we of Finally, nature the sitions. ascertain to perform we that fmliao atc SO( lattice multiflavor of sthe is h aueo h rniin.I e.I erpr MC report we IV Sec. In transitions. for the results of scalar nature continuum the the and model SO( the for appropriate ory h lblO( global The n oaeinguemdl nwihtefilsbelong fields the which coset in the model, to gauge nonabelian a ing ed n h otnu utflvrsaa SO( scalar multiflavor continuum the and effective order-parameter field, gauge-invariant the a on phase based consider theory, approaches: the we LGW field-theoretical phases, of different two nature the two rank- the separates a identify which To transition, using the by symmetry. breaks condensation global characterized whose parameter, be order real two can which phases, oa ymty nSc I edfieteLWΦ LGW the and define we global III its Sec. of In discussion a symmetry. with local introduced, is model not are modes im- gauge quarks, SU(3) massless [14–16]. the of critical that limit the assuming in plicitly matter transition phase hadronic pre- finite-temperature of the to of used nature originally the was dict that approach recall We LGW analogous predictions. an LGW numeri- the the supports of results analysis cal (FSS) [7], scaling symmetry finite-size gauge lat- detailed U(1) multicomponent a with the model for Abelian-Higgs and tice 10] [9, symmetry gauge eut.A twstecs o h utflvrlat- multiflavor the for case SU( the an by was characterized it chromodynamics scalar As tice (MC) pre- Carlo Their Monte results. numerical with fields. compared gauge are dictions nonabelian explicit with theory nti ae,w hl hwta h hs diagrams phase the that show shall we paper, this In h ae sognzda olw.I e.I h lattice the II Sec. 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II. THE LATTICE MODEL under the global transformation ϕx ϕxW and Vx → ,µ → Vx with W O(N ). ,µ ∈ f We consider a 3D lattice model defined in terms of For Nc = 2 the global symmetry is actually larger than af O(Nf ). We write Vx,µ SO(2) as Nc Nf real matrix variables ϕx associated with each ∈ site×x of a cubic lattice. We start from a maximally cos θx,µ sin θx,µ symmetric model with action Vx,µ = , (8)  sin θx,µ cos θx,µ − t t Sinv = Tr ϕxϕx+ˆµ + V (Trϕx ϕx) , (1) − we define a complex Nf -dimensional vector Xx,µ Xx 1 2 f 1f 2f V (X)= rX + uX , (2) zx = ϕx + iϕx , (9) 2 which satisfies z¯x zx = 1 because of Eq. (3), and the where the first sum is over the lattice links, the second · iθx,µ ˆ ˆ ˆ U(1) link variable λx,µ e . In terms of the new one is over the lattice sites, andµ ˆ = 1, 2, 3 are unit vec- variables, the lattice action≡ (6) becomes tors along the three lattice directions. In this paper we consider unit-length variables satisfying SAH = Re [z¯x λx zx+ˆ] (10) − · ,µ µ t Xx,µ Tr ϕxϕx =1 , (3) γ Re [λx λx+ˆ λ¯x+ˆ λ¯x ] . − ,µ µ,ν ν,µ ,ν so that the action is simply xX,µ>ν

t Sinv = Tr ϕ ϕx+ˆ . (4) This is the action of the Nf -component lattice Abelian- − x µ Xx,µ Higgs model, which is invariant under local U(1) and global U(Nf ) transformations. There is therefore an Formally, the model can be obtained setting r = u, and enlargement of the global symmetry of the model: the − taking the limit u of the potential (2). Models with global symmetry group is U(N ) instead of O(N ). The → ∞ f f actions (1) and (4) are invariant under O(N) transforma- phase structure of the Abelian-Higgs model (10) has been tions with N = NcNf . This is immediately checked if we studied in detail in Ref. [7]. Therefore, in this work we express the matrices ϕx in terms of N-component real will focus on the behavior for N 3. c ≥ vectors Sx. In the new variables we obtain the standard It is interesting to note that one can consider more action of the O(N) nonlinear σ-model general Hamiltonians that have the same global and local invariance. Indeed, one can start from a Hamiltonian in S = Sx Sx+ˆ , Sx Sx =1 . (5) which the potential is any O(N )-invariant function of N − · µ · f Xx,µ ϕtϕ. For instance, if we only consider quartic potentials in ϕx, we can take We now proceed by gauging some of the degrees of free- t t t 2 dom: we associate an SO(Nc) matrix Vx,µ with each lat- V (ϕ)= V (X)+ v Tr[ϕ ϕx ϕ ϕx] (Tr[ϕ ϕx]) . g { x x − x } tice link and extend the action (4) to ensure SO(Nc) (11) gauge invariance. We also add a kinetic term for the If we consider this class of more general Hamiltonians, gauge variables in the Wilson form [13]. We thus obtain there is no enlargement of the symmetry from O(Nf ) to the model with action O(N) in the limit γ , in which gauge degrees of → ∞ freedom are frozen. Moreover, for Nc = 2, the symmetry t S = N Tr ϕ Vx ϕx+ˆ g − f x ,µ µ enlargement from O(Nf ) to U(Nf ) does not occur. x,µ X   Since the global symmetry group O(Nf ) corresponds γ (6) t t to SO(Nf ) Z2, there is the possibility of breaking sepa- Tr Vx,µ Vx+ˆµ,ν Vx+ˆν,µ Vx,ν , × − Nc rately the two different groups. In this work we will focus xX,µ>ν   on the breaking of the SO(Nf ) subgroup, which, by anal- and partition function ogy with our results for complex U(Nf ) invariant gauge models [10], is expected to be the only one occurring in Z = e−βSg . (7) the model with action (6). However, the Z2 symmetry {Xϕ,V } may play a role in more general models, for instance in those with action (11), in which the breaking of both Note that, for γ , the product of the gauge fields the Z2 and the SO(Nf ) subgroups may occur. Note that along a plaquette→ converges ∞ to one. This implies that the presence of two possible symmetry breaking patterns Vx,µ = 1 modulo a gauge transformation. Therefore, in is related to the fact that the gauge symmetry group is the γ limit we reobtain the O(N) invariant theory SO(N ). Had we considered an O(N ) gauge invariant → ∞ c c (4) we started from. For any value of Nc and Nf , Sg model, we would have only an SO(Nf ) global invariance. is invariant under the local gauge transformation ϕx The natural order parameter for the breaking of the t → Gxϕx and Vx GxVx G with Gx SO(N ), and SO(N ) global symmetry group is the gauge-invariant ,µ → ,µ x+ˆµ ∈ c f 3 real traceless and symmetric bilinear operator The previous conclusions also hold for Nc = 2 for generic actions with SO(Nf ) global symmetry, for in- δfg Qfg = ϕaf ϕag , (12) stance for the action (11). On the other hand, for our x x x − N model (6) the previous results do not hold for N = 2, Xa f c because of the symmetry enlargement to U(Nf ). In this which is a rank-2 operator with respect to the global case the LGW field is a Hermitean traceless N N ma- f × f O(Nf ) symmetry group. As we shall show, the phase trix field [23, 24] with a Lagrangian that is the analogue diagram of the model (Nc 3) presents two different of the one considered here, Eq. (13). Its RG flow predicts ≥ phases, separated by a phase-transition line associated that [7] the transition can be continuous for Nf = 2, in with the condensation of the bilinear Q. the O(3) vector universality class, while it is of first order We finally mention that, for Nf = 1, the phase diagram for N 3. f ≥ of the lattice scalar SO(Nc) gauge model (6) is expected The above-reported discussion applies to any model to show only one phase. This can be easily verified for in which the global symmetry group is SO(Nf ) and the γ = 0. In this case the Nf = 1 model is trivial and order parameter is a real operator that transforms as a cannot have any phase transition. rank-two tensor under SO(Nf ) transformations. There- fore, the results apply to other scalar models and, in par- ticular, to scalar SU(2) gauge theories with scalar fields III. EFFECTIVE FIELD THEORIES in the adjoint representation, which have been recently considered to describe the critical behavior of cuprate A. The LGW field theory superconductors for optimal doping [6, 11]. In these the- ories the fundamental fields are Nh Higgs fields trans- To characterize the finite-temperature transitions of forming under the adjoint representation of SU(2), i.e. 3 scalar SO(N ) gauge theories, we consider the LGW ap- φˆf φaf τ a, where τ a σa/2, σa are the Pauli c x ≡ a=1 x ≡ proach [17–20]. We start by considering an order pa- matricesP and f = 1, ..., Nh. In the fixed-length limit t rameter that breaks the global symmetry of the model. Tr[φxφx] = 1, the lattice action is [11, 25] For Nc 3, the global symmetry group is O(Nf ) and an ≥ f f † appropriate order parameter is the bilinear tensor Qx de- S = Tr φˆ Ux φˆ U h − x ,µ x+ˆµ x,µ fined in Eq. (12). The corresponding LGW theory is ob- xX,µ,f h i tained by considering a real symmetric traceless Nf Nf γ t t × Tr Ux Ux+ˆ U U matrix field Φ(x), which represents a coarse-grained ver- − 2 ,µ µ,ν x+ˆν,µ x,ν (15) xX,µ>ν sion of Qx. The Lagrangian is   t t 2 3 + u Tr[φxφxφxφx] , LGW = Tr ∂ Φ∂ Φ+ r Tr Φ + u3 Tr Φ µ µ Xx L 4 2 2 + u41 Tr Φ + u42 (Tr Φ ) , (13) where Ux,µ are SU(2) link variables. For any Nh, the where the potential is the most general O(Nf )-invariant action Sh is invariant under the local SU(2) gauge trans- f f † † fourth-order polynomial in the field. The Lagrangian formation φˆ Gxφˆ G and Ux GxUx G x → x x ,µ → ,µ x+ˆµ (13) is invariant under the global transformations with Gx SU(2), and under the global transformation ∈ t φx φxW with W O(Nh). The appropriate order pa- Φ W ΦW , W O(Nf ) . (14) → ∈ → ∈ rameter is again a gauge-invariant operator which trans- The renormalization-group (RG) flow of model (13) forms as a rank-two traceless real tensor with respect to the global O(N ) symmetry, has been already discussed in Ref. [21]. For Nf = 2 the h cubic term vanishes and the two quartic terms are pro- fg 4 δ portional, so that we obtain the two-component vector Φ fg = φaf φag . (16) Qx x x − N action. Therefore, for Nf = 2 the system may undergo a Xa h continuous transition in the XY universality class. For Nf > 2 the cubic term is generically present. Assum- The corresponding LGW action is again Eq. (13). There- ing that the usual mean-field arguments, valid close to fore, for Nh = 2 transitions associated with the breaking four dimensions, apply also to the three-dimensional case, of the O(Nh) symmetry may be continuous in the XY only first-order transitions are expected. A continuous universality class. For Nh 3 only first-order transitions ≥ transition is only possible if the Hamiltonian parame- are possible. ters are tuned or an additional symmetry is present, so that the cubic term vanishes. If this occurs, we obtain the LGW model discussed in Ref. [21], in the context of B. The continuum scalar SO(Nc) gauge theory the antiferromagnetic RPNf −1 model. In particular, for Nf = 3, the LGW theory is equivalent to that of the O(5) The continuum scalar SO(Nc) gauge theory provides vector model [21, 22], so that continuous transitions in another effective theory for the lattice model (6). Its the O(5) universality class may occur. Lagrangian is obtained by considering all monomials up 4 to dimension four, which can be constructed using the A more careful analysis shows that, for any Nc, a non- scalar field Φaf (with a = 1, ..., Nc and f = 1, ..., Nf ). trivial stable fixed point (with nonzero values of all cou- Gauge invariance is obtained as usual, by adding a gauge plings) exists only for sufficiently large Nf [26, 27]. This field A k T k , where the matrices T k are the gen- result is also confirmed by three-dimensional large-N µ,ab ≡ Aµ ab f erators of the SO(Nc) gauge algebra. The Lagrangian computations for fixed Nc [26]. Therefore, continuous reads [26, 27] transitions are only possible for a large number of com- ponents. 1 2 1 2 = F + (∂ Φ + g0A Φ ) (17) L 4 µν 2 µ af µ,ab bf The above results contradict the LGW predictions. For Xafµ Nf = 2 the continuum theory predicts a first-order tran- 1 2 1 2 2 sition, while, according to the LGW analysis, XY con- + r0 Φaf + u0( Φaf ) 2 4! tinuous transitions are possible. Vice versa, for large Nf Xaf Xaf continuous transitions are possible according to the con- tinuum theory, but not on the basis of the LGW analy- 1 2 2 2 2 + v0  Φaf Φbf ( Φaf )  , sis. We note that analogously contradictory results were 4! − Xabf Xaf obtained for the Abelian-Higgs model [7] and the scalar   chromodynamics [9]. where Fµν is the non-Abelian field strength associated with the gauge field Aµ,ab. Note that, unlike the LGW theory (13), the RG flow of To determine the nature of the transitions described by the continuum scalar SO(Nc) gauge theory (17) presents the continuum SO(Nc) gauge theory (17), one studies the an unstable O(N) fixed point with N = Nf Nc, which RG flow determined by the β functions of the model. In describes the critical behavior of the lattice model (6) in the ǫ-expansion framework, the one-loop MS β functions the γ limit. This is located at → ∞ control the RG flow close to four dimensions. Introducing 2 the renormalized couplings u, v, and α = g , the one-loop 6 2 β functions are (see Ref. [26] for the exact normalizations u = ǫ + O(ǫ ) , v =0 , α =0 . (21) Nf Nc +8 of the renormalized couplings)

Nf Nc +8 2 (Nf 1)(Nc 1) 2 One can easily show that this fixed point is always un- β = ǫu + u + − − (v 2uv) u − 6 6 − stable, even in the absence of gauge interactions. The 3 9 perturbation associated with the coupling v is a spin-four (N 1)uα + (N 1)α2 , (18) −2 c − 8 c − perturbation at the O(N) fixed point, which is relevant Nf + Nc 8 2 3 for any N 3 [29, 30]. Moreover, also the gauge pertur- βv = ǫv + − v +2uv (Nc 1)vα bation associated≥ with the coupling α is relevant, as it is − 6 − 2 − 2 9 associated with a negative eigenvalue λα = ǫ + O(ǫ ) of + (N 2)α2 , − 4 c − the stability matrix, see Eq. (20). N 22(N 2) β = ǫα + f − c − α2 , α − 12 where ǫ 4 d. Note that for N = 2 the β func- ≡ − c IV. NUMERICAL RESULTS tions βu and βα for v = 0 map exactly onto those of the Abelian-Higgs model [7, 28], after an appropriate change of normalization of the couplings. We recall that the RG In this section we report numerical results for the lat- flow of the Abelian-Higgs theory has a stable fixed point tice scalar SO(3) gauge theory with two and three fla- only for large Nf , i.e. Nf > 90+24√15 + O(ǫ). vors. We consider cubic lattices of linear size L with The RG flow described by the β functions (18) gener- periodic boundary conditions. To update the gauge ally predicts first-order transitions, unless the number of fields we use an overrelaxation algorithm implemented flavors is large. In particular, one can easily see that the `ala Cabibbo-Marinari [31], considering three SO(2) sub- RG flow described by the β functions (18) cannot have groups of SO(3). We use a combination of biased- stable fixed points for Metropolis updates1 and microcanonical steps [33] in the ratio 1:5. For the update of the scalar fields a combina- N < 22(N 2) , (19) f c − tion of Metropolis and microcanonical updates is used, for which the fixed points must necessarily have α = 0, with the Metropolis step tuned to have an acceptance at least sufficiently close to four dimensions. The fixed rate of approximately 30%. points with vanishing gauge coupling α = 0 are always unstable with respect to the gauge coupling, since their stability matrix Ωij = ∂βi/∂gj has a a negative eigen- value 1 In the biased-Metropolis algorithm, links are generated according to a Gaussian approximation of the action and then accepted or ∂βα 2 rejected by a Metropolis step [32]; the acceptance ratio was larger λα = = ǫ + O(ǫ ) . (20) ∂α − than 90% in all cases studied. α=0

5

A. Observables and analysis method ∗ ∗ Ref. ν η ωRξ U [35] 0.6717(1) 0.0381(2) 0.785(20) 0.5924(4) 1.2432(2) We consider the energy density and the specific heat, [36] 0.67169(7) 0.03810(8) 0.789(4) 0.59238(7) 1.24296(8) defined as [37] 0.67175(10) 0.038176(44) 0.794(8)

1 1 2 2 E = Sg , CV = 2 Sg Sg , (22) TABLE I: Some estimates of the universal critical exponents −3Nf V h i 9N V h i − h i f
for the 3D XY universality class, obtained from the analysis of high-temperature expansions supplemented by MC sim- where V = L3. To study the breaking of the O(N ) flavor f ulations [35], from MC simulations [36] and the conformal- symmetry, we consider the order parameter Qx defined bootstrap approach [37]. See Ref. [34] for earlier theo- in Eq. (12). Its two-point correlation function is defined retical estimates and experimental results. We report the by correlation-length exponent ν, the order-parameter exponent η, the exponent ω (associated with the leading scaling cor- ∗ ∗ G(x y)= Tr QxQy , (23) − h i rections), and the universal quantities Rξ and U (large-L limits of Rξ and U at the critical point, for cubic lattices with where the translation invariance of the system has been periodic boundary conditions). explicitly taken into account. We define the correspond- ing susceptibility χ and correlation length ξ as models belongs to the XY universality class, by verifying 1 G(0) G(p ) 2 m that the asymptotic FSS behavior of U versus Rξ, see χ = G(x), ξ = 2 − , (24) Xx 4 sin (π/L) e G(pme) Eq. (28), matches that obtained for the XY model. On the other hand, for Nf = 3, we will show that Eq. (28) is ip·x e where G(p) = x e G(x) is the Fourier transform of not satisfied—the data of U do not scale on a single curve G(x) and pm =P (2π/L, 0, 0). when plotted versus Rξ. This can be taken as evidence At continuouse transitions, RG-invariant quantities, that the transition is of first order, a conclusion that such as the Binder parameter will be also confirmed by the two-peak structure of the distributions of the energy. 2 µ2 1 U = h i2 , µ2 = 2 Tr QxQy , (25) µ2 V Xx,y h i B. The two-flavor lattice SO(3) gauge model and We now present numerical results for the two-flavor Rξ = ξ/L , (26) SO(3) gauge model (6), showing that it undergoes con- tinuous transitions in the 3D XY universality class, as (which we generically denote by R), are expected to scale predicted by the corresponding LGW theory. Some ac- as [34] curate results for the universal quantities of the 3D XY R(β,L)= f (X)+ L−ωg (X)+ ..., (27) universality class are reported in Table I. R R To begin with, we present results for γ = 0. In Fig. 1 where X = (β β )L1/ν and next-to-leading scaling cor- − c rections have been neglected. The function fR(X) is uni- L=8 N = 2, N = 3, γ = 0 versal up to a multiplicative rescaling of its argument, ν L=12 f c 1.5 L=16 is the critical exponent associated with the correlation L=24 length, and ω is the exponent associated with the lead- L=32 ∗ ing irrelevant operator. In particular, U fU (0) and ∗ ≡ R R f (0) are universal, depending only on the bound- ξ 1.0 ξ ≡ Rξ ary conditions and aspect ratio of the lattice. Since Rξ defined in Eq. (26) is an increasing function of β, we can write 0.5 −ω U(β,L)= FU (Rξ)+ O(L ) , (28) where FU now depends on the universality class, bound- 0.0 ary conditions and lattice shape, without any nonuniver- 1.85 1.90 1.95 2.00 2.05 sal multiplicative factor. The scaling (28) is particularly β convenient to test universality-class predictions, since it permits easy comparisons between different models with- FIG. 1: MC data of Rξ versus β for the lattice SO(3) gauge out requiring a tuning of nonuniversal parameters. model (6) with Nf = 2 and γ = 0. The dotted line corre- In the following we will show that the critical behavior sponds to Rξ = 0.5924, which is the critical value for the XY along the phase transition line of two-flavor SO(3) gauge universality class, see Table I. 6

2.0 2.0 L=8 N = 2, N = 3, γ = 0 N = 2, N = 3, γ = 3 L=8 L=12 f c f c L=12 L=16 L=16 L=24 1.8 L=24 1.5 L=32 L=32 O(2) L=16 L=24 Rξ U 1.6 1.0 1.4

0.5 1.2

0.0 1.0 −8 −4 0 4 8 0.0 0.3 0.6 0.9 1.2 1.5 1/ν R (β-β )L ξ c 2.0 γ L=8 Nf = 2, Nc = 3, = -3 R β − β L1/ν L=12 FIG. 2: MC data of ξ versus ( c) for the lattice L=16 SO(3) gauge model (6) with Nf = 2 and γ = 0. We use ν = 1.8 L=24 L=32 0.6717, the value for the XY universality class, see Table I. O(2) L=16 L=24 U 1.6 we show the estimates of Rξ for different values of L and β. The data sets for different values of L clearly display a 1.4 crossing point, which provides an estimate of the critical point. The data are consistent with the predicted XY behavior. Indeed, the data close to the crossing point 1.2 nicely fit the simple biased ansatz 1.0 ∗ 1/ν 0.0 0.3 0.6 0.9 1.2 1.5 Rξ = Rξ +a1X, X = (β βc)L , ν =0.6717, (29) − Rξ where ν is the critical exponent of the XY universality class, see Table I. Using data within the self-consistent FIG. 4: MC data of U versus Rξ for the lattice SO(3) gauge window R (β,L) [R∗(1 δ), R∗(1 + δ)] with δ = 0.2, model (6) with Nf = 2 and γ = ±3 (up to L = 32), and ξ ∈ ξ − ξ we obtain the estimate β =1.97690(7). The collapse of for the XY [O(2)] universality class (data up to L = 24 for c the standard nearest-neighbor XY model). The dotted lines the data in Fig. 2, where we report the data of Rξ versus ∗ ∗ correspond to the universal values Rξ and U for the XY X, clearly shows the effectiveness of the XY biased fit. universality class, see Table I.

2.0 N = 2, N = 3, γ = 0 L=8 f c L=12 L=16 In Fig. 3 we report results for the Binder parameter U 1.8 L=24 and the ratio Rξ = ξ/L. The data of U versus Rξ clearly L=32 O(2) L=16 approach a single curve. which matches the correspond- L=24 1.6 ing curve computed in the standard nearest-neighbor XY U model with action (5) (again we consider cubic lattices with periodic boundary conditions). This test, which 1.4 does not require any tuning of free parameters, provides the strongest evidence that the phase transition in the 1.2 gauge model belongs to the 3D XY universality class. Note that scaling corrections are significantly smaller in the gauge theory than in the standard discretizaton of 1.0 the XY model, though they are hardly visible in Fig. 3. 0.0 0.3 0.6 0.9 1.2 1.5 Rξ We have also checked that XY behavior is also ob- served for other values of γ; see Fig. 4, where we report FIG. 3: MC data of U versus R for the lattice SO(3) gauge results for γ = 3 and 3. This proves the irrelevance of ξ − model (6) with Nf = 2 and γ = 0 (data up to L = 32) and γ along the transition line. Of course, a crossover is ex- for the XY [O(2)] universality class (data up to L = 24 for pected in the limit γ , Indeed, in this limit we should the standard nearest-neighbor XY model). The dotted lines → ∞ ∗ ∗ observe an O(N) critical behavior, with N = Nf Nc = 6 correspond to the universal values Rξ and U for the XY (results for the O(6) universality class can be found in universality class, see Table I. Ref. [38]). 7

2.1 N = 3, N = 3, γ = 0 observing the linear behavior in the volume. A second f c L=8 L=12 indication of a first-order transition is provided by the L=16 L=24 plot of U versus Rξ. The absence of a data collapse is 1.8 an early indication of the first-order nature of the transi- U tion, as already advocated in Ref. [24]. In Fig. 5 we plot the Binder parameter U versus Rξ. The data, that are 1.5 obtained at values of β close to the transition tempera- ture βc 1.7707, do not show any scaling. Moreover, U displays≈ a pronounced peak, whose height increases with increasing volume. We take the absence of scaling as an 1.2 evidence that the transition is of first order. The first-order transition is also clearly supported by 0.0 0.3 0.6 0.9 1.2 1.5 the emergence of a double peak structure in the distribu- Rξ tion P (E) of the energy with increasing the lattice size around β 1.7707. This is shown in Fig. 6 where the c ≈ FIG. 5: MC data of U versus Rξ for the lattice SO(3) gauge energy histograms for L = 32 and L = 48 are compared. model (6) with Nf = 3 and γ = 0. The absence of scaling Correspondingly the specific heat CV defined in Eq. (22) indicates that the transition is not continuous, thus first order. shows more and more pronounced peaks with increas- ing L (not shown). However, the expected asymptotic 150 large-volume behaviors, such as CV,max V of the max- γ ∼ Nf = 3, Nc = 3, = 0 imum value CV,max of CV , are not clearly observed yet, L=48 presumably requiring larger lattice sizes. We have also considered the gauge-invariant two-point 100 correlation function of the local operator det ϕx (note that ϕaf is the 3 3 matrix), which may be taken as an x × order parameter for the Z2 global symmetry briefly dis- cussed in Sec. II. The correlation function does not show L=32 any qualitative change across the transition. It is always 50 short-ranged, confirming that the Z2 global symmetry is not broken and does not play any role at the transition. In conclusion, the numerical results for Nf = Nc = 3 provide a convincing evidence that the transition is of 0 first order for γ = 0. As it occurs for N = 2, we conjec- 0.495 0.500 0.505 0.510 f ture that the nature of the transition does not change in E a large interval of values of γ around γ = 0. In particu- FIG. 6: Energy distribution for the lattice SO(3) gauge model lar, we conjecture that the transition is of first order for all positive finite values of γ. Note that, for large γ, we (6) with Nf = 3 and γ = 0. We report results for L = 32 and L = 48 for β values close to the transition. The values of expect significant crossover effects, since the transition β have been selected to obtain two maxima of approximately is continuous for γ = in the universality class of the ∞ the same height. O(9) vector σ-model.

C. The three-flavor lattice SO(3) gauge model V. SUMMARY AND CONCLUSIONS

For Nf = 3 the LGW effective field theory predicts a In this paper we investigate the phase diagram of 3D first-order phase transition for any number of colors. To multiflavor lattice scalar theories in the presence of non- verify the prediction, we perform simulations for γ = 0. abelian SO(Nc) gauge interactions. We consider the lat- Some evidence in favor of a first-order transition is pro- tice scalar SO(Nc) gauge theory (6) with Nf flavors, de- vided by the analysis of the Binder parameter U. Ata fined starting from a maximally O(N)-symmetric multi- first-order transition, the maximum Umax of U behaves component scalar model (N = Nf Nc). The global O(N) as [39, 40] Umax V . On the other hand, at a contin- symmetry is partially gauged, obtaining a gauge model, ∼ N uous phase transition, U is bounded as L and the in which the fields belong to the coset S /SO(Nc), → ∞ N data of U corresponding to different values of Rξ collapse where S is the N-dimensional sphere. Note that, for onto a common scaling curve as the volume is increased. Nc = 2, the action (6) exactly maps onto that of the Nf - Therefore, U has a qualitatively different scaling behav- component lattice Abelian-Higgs model characterized by ior for first-order and continuous transitions. In prac- a U(1) gauge symmetry, whose phase diagram has been tice, a first-order transition can be identified by verifying studied in Refs. [7, 24]. We thus focus on models with that Umax increases with L, without the need of explicitly N 3. c ≥ 8

invariant order parameter, only the global symmetry group SO(Nf ) and the nature of the order parameter Q =0 (a rank-two symmetric real traceless tensor) play a role. β h i 6 The gauge degrees of freedom are absent in the effective O(N) model. A second approach is based on the continuum Q =0 SO(Nc) gauge theory, in which the gauge fields are ex- h i plicitly present. As it occurs for the lattice scalar chro- 0 γ ∞ modynamics characterized by an SU(Nc) gauge symme- try [9, 10], the numerical results agree with the LGW FIG. 7: Sketch of the phase diagram of the 3D lattice scalar predictions. The LGW framework provides the correct SO(Nc) gauge theory (6) with Nf flavors and O(Nf ) global description of the large-scale behavior of these systems, symmetry. The transition line is continuous for Nf = 2—it predicting first-order transitions for Nf = 3, and contin- belongs to the XY universality class for Nc ≥ 3 and to the uous transitions for Nf = 2, which belong to the XY O(3) universality class for Nc = 2—and is of first order for universality class for any N 3. Nf ≥ 3. We conjecture that its nature is the same for any c ≥ finite γ. The endpoint for γ → ∞ is the O(N) critical point (N = NcNf ). The results for Nf = 2 are in contradiction with the predictions of the continuum gauge model (6): since no stable FP exists for Nf = 2, one would expect a first- For Nf 2 the phase diagram is characterized by order transition. An analogous contradiction was also two phases:≥ a low-temperature phase in which the or- observed in the case of scalar chromodynamics [9]. This fg der parameter Qx defined in Eq. (12) condenses, and a apparent failure of the continuum scalar gauge theory high-temperature disordered phase. The two phases are may suggest that it does not encode the relevant modes separated by a transition line, where the SO(Nf ) sym- at the transition. Alternatively, the failure may be traced metry is broken, as sketched in Fig. 7. The line ends at back to the perturbative treatment around four dimen- the unstable O(N) transition point with N = NcNf for sions, which does not provide the correct description of γ . The gauge parameter γ, corresponding to the the 3D behavior. The 3D FP may not be related to a inverse→ ∞ gauge coupling, does not play any particular role: four-dimensional FP, and therefore it escapes any per- the nature of the transition is conjectured to be the same turbative analysis in powers of ε. This has been also ob- for any γ. We have numerically verified this conjecture served in other physical systems; see, e.g., Refs. [41, 42]. for two values of γ. Along the transition line only the cor- We finally recall that the two field-theoretical approaches fg relations of the gauge-invariant operator Qx are critical, give different results also in the large-Nf limit. The LGW while gauge modes are not critical and only represent a theory predicts a first-order transition for any Nf 3 background that gives rise to crossover effects. due to the presence of the cubic term. On the other≥ The nature of the finite-temperature transitions can be hand, continuous transitions are possible for large values investigated using different field-theoretical approaches. of Nf according to the continuum scalar SO(Nc) gauge On one side, one can use the effective LGW theory with theory, because of the presence of a stable large-Nf fixed Lagrangian (13). In this approach based on a gauge- point [26, 27].

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