Berezinskii-Kosterlitz-Thouless transitions in two-dimensional lattice SO($N_c$) gauge theories with two scalar flavors

We study the phase diagram and critical behavior of a two-dimensional lattice SO($N_c$) gauge theory ($N_c \ge 3$) with two scalar flavors, obtained by partially gauging a maximally O($2N_c$) symmetric scalar model. The model is invariant under local SO($N_c$) and global O(2) transformations. We show that, for any $N_c \ge 3$, it undergoes finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transitions, associated with the global Abelian O(2) symmetry. The transition separates a high-temperature disordered phase from a low-temperature spin-wave phase where correlations decay algebraically (quasi-long range order). The critical properties at the finite-temperature BKT transition and in the low-temperature spin-wave phase are determined by means of a finite-size scaling analysis of Monte Carlo data.


I. INTRODUCTION
Abelian and non-Abelian gauge symmetries appear in various physical contexts.For instance, they are relevant for the theories of fundamental interactions [1][2][3] and in the description of some emerging phenomena in condensed matter physics [2,4,5].The main features of these theories, such as the spectrum, the phase diagram, and the critical behavior at thermal and quantum transitions, crucially depend on the interplay between global and local gauge symmetries.
These issues have been recently investigated in several two-dimensional (2D) lattice gauge models, considering: (i) the multicomponent lattice Abelian-Higgs model [6], characterized by a global SU(N f ) symmetry (N f ≥ 2) and a local U(1) gauge symmetry; (ii) the multiflavor lattice scalar quantum chromodynamics [7], characterized by a global SU(N f ) symmetry and a local SU(N c ) gauge symmetry; (iii) lattice SO(N c ) gauge models with N f ≥ 3 real scalar flavors [8], characterized by a non-Abelian O(N f ) global symmetry.In agreement with the Mermin-Wagner theorem [9], all these 2D lattice gauge models do not have finite-temperature transitions.A critical behavior is only observed in the zero-temperature limit: for T → 0 the correlation length increases exponentially, as in the 2D O(N ) σ model wih N ≥ 3 and in the 2D CP N −1 model with N ≥ 2 [2].The interplay of global non-Abelian symmetries and local gauge symmetries determines the large-scale properties of the system in the zero-temperature limit, and therefore, the field theory realized in the corresponding continuum limit.
The results for the above-mentioned lattice gauge models support the following general conjecture, originally put forward in Ref. [7].The universal features, i.e., the universality class, of the asymptotic low-temperature behavior of lattice gauge models is the same as that of the 2D field theories defined on the symmetric spaces [2,10] that have the same global symmetry.According to this conjecture, the zero-temperature critical behavior of multiflavor Abelian-Higgs models and lattice scalar chromodynamics with N f scalar flavors belongs to universality class of the 2D CP N f −1 model, as both models have the same global SU(N f ) symmetry.Analogously, lattice SO(N c ) gauge theories with N f ≥ 3 real scalar flavors have the same critical behavior as RP N f −1 models [11] with the same global O(N f ) symmetry.These predictions have been numerically verified in Refs.[6][7][8].We note that all cases considered so far involve systems with global non-Abelian symmetries, which are not expected to show finite-temperature transitions in two dimensions [9].
In this paper we investigate a system that undergoes a finite-temperature transition and show that also in this case the conjecture holds.We consider a lattice SO(N c ) gauge model with two real scalar flavors, obtained by partially gauging a maximally O(2N c ) symmetric scalar theory.For N c ≥ 3 this model is characterized by a global Abelian O(2) symmetry (for N c = 2 the global symmetry group is SU(2) [8], which is non-Abelian, and therefore we only expect a zero-temperature critical behavior).If the general conjecture extends to systems with global Abelian symmetries, we expect this model to have the same critical behavior as the O(2)invariant XY lattice model.Therefore, for N c ≥ 3, 2D lattice SO(N c ) gauge models with two scalar flavors may undergo a finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition [12][13][14][15][16][17][18][19][20], between the hightemperature disordered phase and a low-temperature spin-wave phase characterized by quasi-long range order (QLRO) with vanishing magnetization.We recall that BKT transitions are characterized by an exponentially divergent correlation length: we have ξ ∼ exp(c/ √ T − T c ) approaching the BKT critical temperature T c from the high-temperature phase.
To verify the general conjecture for the two-flavor SO(N c ) gauge theory, we perform Monte Carlo simula-tions of a lattice model for several N c ≥ 3. We anticipate that the numerical results confirm the presence of a low-temperature QLRO phase, separated by a BKT transition from the high-temperature disordered phase.These results extend the validity of the conjecture to 2D lattice non-Abelian gauge theories with global Abelian symmetries.
The paper is organized as follows.In Sec.II we define the lattice SO(N c ) gauge model and the gauge-invariant observables that we compute in the simulation.We also describe the finite-size scaling (FSS) analysis we use to investigate the phase diagram and to determine the nature of the critical behavior.Sec.III reports the numerical results for N c = 3, 4, 5.We show that QLRO holds in the low-temperature phase and that the transition between the high-temperature and the low-temperature QLRO phase is a BKT one, as in the standard XY model.Finally, in Sec.IV we summarize and draw our conclusions.

II. 2D LATTICE SO(Nc) GAUGE MODELS
We consider a multiflavor lattice SO(N c ) gauge model defined on square lattices of linear size L with periodic boundary conditions.It is obtained [21] by partially gauging a maximally O(M ) symmetric model with M = N f N c , defined in terms of real unit-length matrix variables φ af x , with a = 1, .., N c and f = 1, ..., N f (we will refer to these two indices as color and flavor indices, respectively), such that Tr φ t x φ x = 1.Using the Wilson approach [1], we introduce gauge variables associated with each link of the lattice.The Hamiltonian reads [21] where V x,µ ∈ SO(N c ), and Π x is the plaquette operator The plaquette parameter γ plays the role of inverse gauge coupling.The partition function reads Note that, for γ → ∞, the link variables V x become equal to the identity modulo gauge transformations.Thus, one recovers the O(M )-symmetric nearest-neighbor M -vector model, which does not have a finite-temperature transition and becomes critical only in the zero-temperature limit [2,22].
For N c ≥ 3 the global symmetry group of model ( 1) is O(N f ).For N c = 2 the global symmetry is actually larger [21], since the model can be exactly mapped onto the two-component lattice Abelian-Higgs model, which is invariant under local U(1) and global U(N f ) transformations.Therefore, for N f = N c = 2 the model has a zerotemperature critical behavior belonging to the universality class of the CP 1 field theory [6], which is equivalent to that of the nonlinear O(3) σ model.In the following we consider only the case N c ≥ 3.
For N f = 2 and N c ≥ 3 the theory is characterized by a global Abelian O(2) symmetry.The conjecture we have discussed in the introduction suggests therefore that the two-flavor gauge model has a finite-temperature transition analogous to that occurring in 2D O(2)-invariant spin models, which undergo a BKT transition from the disordered phase to a low-temperature QLRO phase [2].As we shall see, this conjecture is supported by the numerical results.
To determine the nature of the transitions, we will perform a FSS analysis [22][23][24][25] of the numerical data.We focus on the correlations of the gauge-invariant bilinear operator Note that, for N f = 2, Q x has only two independent real components.We consider the two-point function where the translation invariance of the system has been taken into account.We define the susceptibility χ = x G(x) and the correlation length where G(p) = x e ip•x G(x) is the Fourier transform of G(x), and p m = (2π/L, 0).We also consider universal RG invariant quantities, such as the Binder parameter where V = L 2 , and the ratio In the FSS limit we have (see, e.g., Ref. [6]) where F U (x) is a universal scaling function that completely characterizes the universality class of the transition.In particular, universality is expected at BKT transitions and in the whole low-temperature spin-wave phase, see, e.g., Refs.[16,17,20,[26][27][28].
Because of the universality of relation ( 9), we use the plots of U versus R ξ to identify the models that have the same universal behavior.If the estimates of U for two different systems fall onto the same curve when plotted versus R ξ , the transitions in the two models belong to the same universality class.Therefore, we will compare the FSS curves for the lattice SO(N c ) gauge model with the analogous ones for the 2D XY model.If the data for the two models have the same scaling behavior, we will conclude that the gauge model undergoes a BKT transition as the XY model.The same strategy was employed in Refs.[6][7][8], to characterize the asymptotic zerotemperature behavior of 2D lattice gauge models with non-Abelian global symmetry group.

III. NUMERICAL RESULTS
A. The conjecture for systems with O(2) global symmetry We wish to verify numerically the general conjecture originally put forward in Ref. [7].In the present case it predicts that, for any N c ≥ 3, the lattice model with Hamiltonian (1) with two flavors undergoes a transition analogous to that of the paradigmatic 2D O(2) invariant XY model defined by the Hamiltonian where ψ x are complex phase variables, |ψ x | = 1, associated with each site of the square lattice.This model undergoes a BKT transition at β c = 1.1199(1) [16,19], with a low-temperature phase that shows QLRO with vanishing magnetization.The correspondence can be justified using the arguments presented in Ref. [8].If the conjecture holds, the lattice model (1) with N f scalar flavors should be related to the 2D RP N f −1 model, defined by the Hamiltonian where ϕ x is a unit-length N f -component real field.Indeed, the RP N −1 space is a symmetric space that has the same global O(N f ) symmetry.The model has also a local Z 2 symmetry, which effectively appears because the order parameter Q x is invariant under the local Z 2 transformations φ x → s x φ x , s x = ±1.In the RP N −1 model the order parameter is which is the counterpart of Q f g x defined in the lattice SO(N c ) gauge theory.In the two-flavor case, N f = 2, one can easily show that, for the computation of Z 2 gaugeinvariant quantities, the RP 1 model can be mapped onto the XY model.Under this mapping, the order parameter q f g x (which has only two independent real components) is mapped onto the complex field ψ x of the XY model.Therefore, the critical behavior of the correlation function of the operator Q x , defined in Eq. ( 5), is expected to correspond to that of the two-point function We also report the universal asymptotic large-L curve (full line) computed in the spin-wave theory, for a system with square geometry and periodic boundary conditions [19,27].
in the XY model.Using G XY , one can then define the correlation length ξ, the Binder parameter U , and the ratio R ξ , using again Eqs. ( 6), (7), and (8), respectively.

B. The low-temperature spin-wave phase
To gain evidence of the existence of a low-temperature QLRO phase, we show that spin-wave relations hold asymptotically for sufficiently low temperatures.The spin-wave theory is expected to describe the critical behavior of the XY model along the line of fixed points that runs from T = 0 up to the BKT point T c .Conformal field theory, see, e.g., Ref. [29], exactly provides the large-L limit of the two-point function in the spinwave model.In particular, it allows us to compute the universal asymptotic relation between the ratio R ξ and the exponent η. Results for square lattices with periodic boundary conditions are reported in Refs.[19,27] (see, in particular, the formulas reported in App.B of Ref. [27]).The exponent η characterizes the temperature-dependent power-law decay of the two-point function in the QLRO phase Alternatively, we can define it by considering the large-L behavior of the susceptibility In the QLRO phase, η(T ) varies from η(T c ) = 1/4 to η(T → 0) → 0, and We recall that, at T c , the RG theory appropriate for the BKT transition predicts the asymptotic large-L behavior [19,20,27] 16) where Λ is a model-dependent constant, and R ξ (T c ) and C R ξ are universal.Using the spin-wave theory, one obtains R ξ (T c ) = 0.750691... and C R ξ = 0.212431.... Analogous results can be obtained for the Binder parameter U [19].
To study the low-temperature behavior, we have performed simulations for N c = 3 at values of β such that R ξ > R ξ (T c ), using periodic boundary conditions.We have determined the large-L extrapolations of R ξ and η, by fitting the data of χ and R ξ at fixed β, typically for L up to L ≈ 100, to the Ansätze respectively, where ε is the exponent associated with the expected leading corrections: [27,30]  In Fig. 1 we plot R ξ versus η together with the universal curve computed in the spin-wave theory.The results for R ξ and η are in excellent agreement with the spin-wave predictions.This shows the existence of a lowtemperature phase with QLRO, analogous to that occurring int XY model.

C. FSS at the BKT transition
In Sec.III B we showed that the SO(3) gauge model has a low-temperature phase with the same features of the low-temperature phase of the XY model.Now, we focus on the finite-temperature transition that ends the hightemperature phase, to check whether the FSS behavior is the same as that observed at the BKT transition of the XY model.
To begin with, in Fig. 2 we show the estimates of the correlation length ξ for N c = 3 and γ = 0.They show a sudden increase around β 3.5, as expected in the presence of a finite-temperature BKT transition.To characterize the nature of the transition, we plot the Binder parameter U versus the ratio R ξ , In the FSS limit data should belong to a curve that only depends on the universality class.In Fig. 3 we report our numerical results for N c = 3 and for three values of γ,γ = 0, ±1.In all cases, data appear to approach a universal FSS curve with corrections that decrease quite rapidly with the size.We also report data for the 2D XY model, that have been obtained by standard MC simulations for lattice sizes L = 100, 200.They are apparently sufficient to provide a good approximation of the asymptotic FSS behavior (the differences between the L = 100 and L = 200 scaling curves are very small and hardly visible in Fig. 3).It is quite clear that the data for the gauge model fall on top of the XY scaling curve, confirming that the transition has the same universal features: the gauge SO(3) model undergoes a BKT transition as the XY model.Analogous results are obtained for N c = 4 and N c = 5, as shown in Fig. 4, where we report data for γ = 0.In both cases, the data for the gauge model converge toward the FSS curve of the XY model.We note that the approach to the asymptotic FSS behavior ( 9) is apparently quite fast in all lattice gauge models considered, and in also in the 2D XY model.In the range of values of L we consider, scaling corrections effectively decrease as powers of L −1 .There is no evidence of the logarithmic corrections generally expected at BKT transitions, see for instance Eq. ( 16) and Refs.[16,17,20,26,27].Accurate estimates of the critical BKT temperatures are hard to obtain, since their determination is generally affected by logarithmic corrections, see Eq. ( 16).The problem of the logarithmic corrections can be overcome by the so-called matching method put forward in Refs.[16,17,19] (see also Refs.[27,28] for applications to some 2D quantum lattice gas models).Here, we do not pursue this analysis further, since we are not particularly interested in obtaining precise estimates of the  In conclusion, the FSS analysis has allowed us to determine the nature of the finite-temperature transitions occurring in the lattice SO(N c ) gauge model (1) with two flavors.For N c = 3, 4, 5 we find that the transition belongs to the BKT universality class, as in the classical XY model.This occurs at least in an interval of values of γ around the infinite gauge-coupling value γ = 0.

IV. CONCLUSIONS
We have studied a class of 2D lattice non-Abelian SO(N c ) gauge models with two real scalar fields, defined by the Hamiltonian (1).Such lattice gauge models are obtained by partially gauging a maximally O(2N c )symmetric multicomponent real scalar model, using the Wilson lattice approach.For N ≥ 3, the resulting theory is locally invariant under SO(N c ) gauge transformations and globally invariant under Abelian O(2) transformations.This study extends previous work on 2D models with a local gauge invariance and a global non-Abelian symmetry, [6][7][8], in which a critical behavior can only be observed in the zero-temperature limit.Ih the models considered here, instead, the global Abelian O(2) symmetry may allow finite-temperature BKT transitions between the disordered phase and the low-temperature spin-wave phase.
The universal features of the transitions have been determined by performing FSS analyses of Monte Carlo data.We present results for the two-flavor lattice SO(N c ) gauge models (1) with N c = 3, 4, 5.They show that these systems undergo a finite-temperature BKT transition that separates the disordered phase from the lowtemperature phase.Moreover, we have verified that the low-temperature phase is completely characterized by spin waves, as in the standard XY model.
These results provide additional evidence in favor of the conjecture that the critical behavior of 2D lattice gauge models, defined using the Wilson approach [1], belongs to the universality class of the field theories as-sociated with the symmetric spaces that have the same global symmetry.This conjecture assumes that gauge correlations are not critical and decouple in the critical limit.Therefore, the conjecture may fail when the gauge correlations are critical, giving rise to a more complex behavior.A similar phenomenon has been observed in the three-dimensional lattice Abelian-Higgs model with noncompact gauge fields, see, e.g., Ref. [31] and references therein.
We finally mention that the interplay between global and gauge symmetries has also been studied in three dimensional models, see Refs.[21,[31][32][33].

FIG. 1 :
FIG.1: Plot of the large-L extrapolations of R ξ versus η (computed from the finite-size behavior of the susceptibility χ) for the lattice SO(3) gauge model.We report results for γ = 0 and β = 4.0, 4.2, 4.4, and for γ = 1, β = 4.4.We also report the universal asymptotic large-L curve (full line) computed in the spin-wave theory, for a system with square geometry and periodic boundary conditions[19,27].

FIG. 2 :
FIG. 2: Estimates of the correlation length ξ versus β for the lattice SO(3) gauge model (1) with γ = 0, for several values of L, up to L = 128.When the results for different values of L agree, they can be considered as good approximations of the infinite-volume correlation length, within errors.The vertical lines indicate the interval of values of β in which the BKT transition occurs.