Berezinskii-Kosterlitz-Thouless transitions in two-dimensional lattice SO( N c ) gauge theories with two scalar flavors
Claudio Bonati, Alessio Franchi, Andrea Pelissetto, Ettore Vicari
aa r X i v : . [ h e p - l a t ] J a n Berezinskii-Kosterlitz-Thouless transitions intwo-dimensional lattice SO( N c ) gauge theorieswith two scalar flavors 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: January 5, 2021)We study the phase diagram and critical behavior of a two-dimensional lattice SO( N c ) gauge the-ory ( N c ≥
3) with two scalar flavors, obtained by partially gauging a maximally O(2 N c ) symmetricscalar model. The model is invariant under local SO( N c ) and global O(2) transformations. Weshow that, for any N c ≥
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 decayalgebraically (quasi-long range order). The critical properties at the finite-temperature BKT tran-sition and in the low-temperature spin-wave phase are determined by means of a finite-size scalinganalysis of Monte Carlo data.
I. INTRODUCTION
Abelian and non-Abelian gauge symmetries appear invarious physical contexts. For instance, they are rele-vant for the theories of fundamental interactions [1–3]and in the description of some emerging phenomena incondensed matter physics [2, 4, 5]. The main features ofthese theories, such as the spectrum, the phase diagram,and the critical behavior at thermal and quantum tran-sitions, crucially depend on the interplay between globaland local gauge symmetries.These issues have been recently investigated in severaltwo-dimensional (2D) lattice gauge models, considering:(i) the multicomponent lattice Abelian-Higgs model [6],characterized by a global SU( N f ) symmetry ( N f ≥ N f ) symmetry and a local SU( N c )gauge symmetry; (iii) lattice SO( N c ) gauge models with N f ≥ N f ) global symmetry. In agreement with theMermin-Wagner theorem [9], all these 2D lattice gaugemodels do not have finite-temperature transitions. Acritical behavior is only observed in the zero-temperaturelimit: for T → N ) σ model wih N ≥ N − model with N ≥ N f scalar flavors belongs to the universal-ity class of the 2D CP N f − model, as both models havethe same global SU( N f ) symmetry. Analogously, latticeSO( N c ) gauge theories with N f ≥ N f − models [11]with the same global O( N f ) symmetry. These predic-tions have been numerically verified in Refs. [6–8]. Wenote that all cases considered so far involve systems withglobal non-Abelian symmetries, which are not expectedto show finite-temperature transitions in two dimensions[9].In this paper we investigate 2D lattice non-Abeliangauge models that undergo a finite-temperature tran-sition, and show that also in this case the conjectureholds. We consider a 2D lattice SO( N c ) gauge modelwith two real scalar flavors, obtained by partially gaug-ing a maximally O(2 N c ) symmetric scalar theory. For N c ≥ N c = 2 the global symmetry groupturns out to be SU(2) [8], which is non-Abelian, andtherefore we only expect a zero-temperature criticalbehavior). If the general conjecture extends to sys-tems with global Abelian symmetries, we expect thismodel to have the same critical behavior as the O(2)-invariant XY lattice model. Therefore, for N c ≥ N c ) gauge models with two scalar flavorsmay undergo a finite-temperature Berezinskii-Kosterlitz-Thouless (BKT) transition [12–20], between the high-temperature disordered phase and a low-temperaturespin-wave phase characterized by quasi-long range or-der (QLRO) with vanishing magnetization. We recallthat BKT transitions are characterized by an exponen-tially divergent correlation length ξ at a finite criticaltemperature. Indeed, ξ behaves as ξ ∼ exp( c/ √ T − T c )approaching the BKT critical temperature T c from thehigh-temperature phase.To verify the general conjecture for the lattice SO( N c )gauge theory with two scalar flavors, we present finite-size scaling (FSS) analyses of Monte Carlo simulationsfor several N c ≥
3. We anticipate that the numericalresults confirm the presence of a low-temperature QLROphase, separated by a BKT transition from the high-temperature disordered phase. These results extend thevalidity of the conjecture to 2D lattice non-Abelian gaugetheories with global Abelian symmetries.The paper is organized as follows. In Sec. II we de-fine the lattice SO( N c ) gauge model with scalar fields,and the gauge-invariant observables that we consider inour numerical study. We also describe the FSS analy-sis we use to investigate the phase diagram and to de-termine the nature of the critical behavior. Sec. III re-ports the numerical results for N c = 3 , ,
5. We showthat QLRO holds in the low-temperature phase and thatthe transition between the high-temperature and the low-temperature QLRO phase is a BKT one, as in the stan-dard XY model. Finally, in Sec. IV we summarize anddraw our conclusions.
II. 2D LATTICE SO( N c ) GAUGE MODELS We consider a multiflavor lattice SO( N c ) gauge modeldefined on square lattices of linear size L with peri-odic boundary conditions. It is obtained [21] by par-tially gauging a maximally O( M ) symmetric model with M = N f N c , defined in terms of real unit-length matrixvariables φ af x , with a = 1 , .., N c and f = 1 , ..., N f (we willrefer to these two indices as color and flavor indices, re-spectively), such that Tr φ t x φ x = 1. Using the Wilson ap-proach [1], we introduce gauge variables associated witheach link of the lattice. The Hamiltonian reads [21] H = − N f X x ,µ Tr φ t x V x ,µ φ x +ˆ µ − γN c X x Tr Π x , (1)where V x ,µ ∈ SO( N c ), Π x is the plaquette operatorΠ x = V x , V x +ˆ1 , V t x +ˆ2 , V t x , . (2)We set the lattice spacing equal to 1, so that all lengthsare measured in units of the lattice spacing. The plaque-tte parameter γ plays the role of inverse gauge coupling.The partition function reads Z = X { φ,V } e − β H , β ≡ /T . (3)Note that, for γ → ∞ , the link variables V x become equalto the identity modulo gauge transformations. Thus, onerecovers the O( M )-symmetric nearest-neighbor M -vectormodel, which does not have a finite-temperature transi-tion and becomes critical only in the zero-temperaturelimit [2, 22].For N c ≥ N f ). For N c = 2 the global symmetry is actually larger [21], since the model can be exactly mapped ontothe two-component lattice Abelian-Higgs model, whichis invariant under local U(1) and global SU( N f ) trans-formations. Therefore, for N f = N c = 2 the model has azero-temperature critical behavior belonging to the uni-versality class of the CP field theory [6], which is equiv-alent to that of the nonlinear O(3) σ model. In the fol-lowing we consider only the case N c ≥ N f = 2 and N c ≥ Q fg x = X a φ af x φ ag x − δ fg . (4)Note that, for N f = 2, Q x has only two independent realcomponents. We consider the two-point function G ( x − y ) = h Tr Q x Q y i , (5)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 ) , (6)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, UU = h µ ih µ i , µ = 1 V X x , y Tr Q x Q y , (7)where V = L (note that χ = V h µ i ), and the ratio R ξ ≡ ξ/L . (8)In the FSS limit we have (see, e.g., Ref. [6]) U ( β, L ) ≈ F U ( R ξ ) , (9)where F U ( x ) is a universal scaling function that com-pletely characterizes the universality class of the tran-sition. In particular, universality is expected at BKTtransitions and in the whole low-temperature spin-wavephase, see, e.g., Refs. [16, 17, 20, 26–28].Because of the universality of relation (9), we use theplots of U versus R ξ to identify the models that have thesame universal behavior. If the estimates of U for twodifferent systems fall onto the same curve when plottedversus R ξ , the transitions in the two models belong tothe same universality class. Therefore, we will comparethe FSS curves for the lattice SO( N c ) gauge model withthe analogous ones for the 2D XY model. If the datafor the two models have the same scaling behavior, wewill conclude that the gauge model undergoes a BKTtransition as the XY model. The same strategy was em-ployed in Refs. [6–8], to characterize the asymptotic zero-temperature behavior of 2D lattice gauge models withnon-Abelian global symmetry group. III. NUMERICAL RESULTSA. The conjecture for systems with O(2) globalsymmetry
We wish to verify numerically the general conjectureoriginally put forward in Ref. [7]. In the present caseit predicts that, for any N c ≥
3, the lattice model withHamiltonian (1) with two flavors undergoes a transitionanalogous to that of the paradigmatic 2D O(2) invariant XY model defined by the Hamiltonian H XY = − X x ,µ Re ψ ∗ x ψ x +ˆ µ , (10)where ψ x are complex phase variables, | ψ x | = 1, asso-ciated with each site of the square lattice. This modelundergoes a BKT transition at β c = 1 . N f scalar flavors should be relatedto the 2D RP N f − model, defined by the Hamiltonian H RP = − t X x ,µ ( ϕ x · ϕ x +ˆ µ ) , (11)where ϕ x is a unit-length N f -component real field. In-deed, the RP N − space is a symmetric space that hasthe same global O( N f ) symmetry. The model has alsoa local Z symmetry, which effectively appears becausethe order parameter Q x is invariant under the local Z transformations φ x → s x φ x , s x = ±
1. In the RP N − model the order parameter is q fg x = ϕ f x ϕ g x − N f δ fg , (12)which is the counterpart of Q fg x defined in the latticeSO( N c ) gauge theory. In the two-flavor case, N f = 2, onecan easily show that, for the computation of Z gauge-invariant quantities, the RP model can be mapped ontothe XY model. Under this mapping, the order parameter q fg 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 func-tion of the operator Q x , defined in Eq. (5), is expectedto correspond to that of the two-point function G XY ( x , y ) = h ψ ∗ x ψ y i , (13)in the XY model. Using G XY , one can then define thecorrelation length ξ , the Binder parameter U , and theratio R ξ , using again Eqs. (6), (7), and (8), respectively. B. Monte Carlo simulations
In the following we report numerical results for the2D lattice SO( N c ) gauge theories with two scalar flavors,cf. Eq. (1). We consider square lattices of linear size L with periodic boundary conditions. To update the gaugefields we use an overrelaxation algorithm implemented `a la Cabibbo-Marinari [29], considering the SO(2) sub-groups of SO( N c ). We use a combination of Metropolisupdates and microcanonical steps [31] in the ratio 3:7. Inthe Metropolis update, link variables are randomly gen-erated, and then accepted or rejected by a Metropolisstep [30], with an acceptance rate of approximately 30%.For the scalar fields a combination of Metropolis and mi-crocanonical updates is used, with the Metropolis steptuned to have an acceptance rate of approximately 30%.Errors are estimated using a standard blocking and jack-knife procedure, to take into account autocorrelations,which are expected to increase roughly as L . Typicalstatistics of our runs, for a given value of the parametersand of the size of the lattice, are approximately 10 latticesweeps (in a sweep we update once all lattice variables).For the larger lattice sizes the autocorrelation times ofthe observables considered were of order 10 sweeps atmost, even at T c , thus obtaining a sufficiently large num-ber of independent measures. C. The low-temperature spin-wave phase
To gain evidence of the existence of a low-temperatureQLRO phase, we show that spin-wave relations holdasymptotically for sufficiently low temperatures. Thespin-wave theory is expected to describe the critical be-havior of the XY model along the line of fixed pointsthat runs from T = 0 up to the BKT point T c . Con-formal field theory, see, e.g., Ref. [32], exactly providesthe large- L limit of the two-point function in the spin-wave model. In particular, it allows us to compute theuniversal asymptotic relation between the ratio R ξ andthe exponent η . Results for square lattices with periodicboundary conditions are reported in Refs. [19, 27] (see, inparticular, the formulas reported in App. B of Ref. [27]).The exponent η characterizes the temperature-dependentpower-law decay of the two-point function in the QLROphase G ( x ) ∼ | x | − η ( T ) . (14)
20 30 40 50 60 70 80 90 100 L ln χ β = 4.4β = 4.2
20 30 40 50 60 70 80 90 100 L R ξ β = 4.4β = 4.2 FIG. 1: Data of R ξ (bottom) and ln χ (top) in the low-temperature spin-wave phase of the model with N c = 3 and γ = 0, at β = 4 . β = 4 .
4. The dashed lines are obtainedby fitting the data (results for the smallest lattice sizes havebeen discarded) to the Ans¨atze (17) and (18).
Alternatively, we can define it by considering the large- L behavior of the susceptibility χ ( L, T ) ∼ L − η ( T ) . (15)In the QLRO phase, η ( T ) varies from η ( T c ) = 1 / η ( T → →
0, and R ξ from R ξ ( T c ) = 0 . ... to R ξ ( T → → ∞ .We recall that, at T c , the RG theory appropriate forthe BKT transition predicts the asymptotic large- L be-havior [19, 20, 27] R ξ ( L, T c ) = R ξ ( T c ) + C R ξ w ( L ) + O ( w − ) , (16) w ( L ) = ln L Λ + 12 ln ln L Λ , where Λ is a model-dependent constant, and R ξ ( T c ) and C R ξ are universal. Using the spin-wave theory, one ob-tains R ξ ( T c ) = 0 . ... and C R ξ = 0 . ... . Anal-ogous results can be obtained for the Binder parameter U [26], in particular U ( T c ) = 1 . η R ξ spin wave N c =3 γ=0 N c =3 γ=1 FIG. 2: 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 . , . , .
4, and for γ = 1, β = 4 .
4. Wealso report the universal asymptotic large- L curve (full line)computed in the spin-wave theory, for a system with squaregeometry and periodic boundary conditions [19, 27]. To study the low-temperature behavior, we have per-formed simulations for N c = 3 at values of β such that R ξ > R ξ ( T c ), using periodic boundary conditions. Wehave determined the large- L extrapolations of R ξ and η ,by fitting the data of χ and R ξ at fixed β to the Ans¨atzeln χ ( L ) = a + (2 − η ) ln L + bL − ε , (17) R ξ ( L ) = R ξ + aL − ε , (18)respectively, where ε is the exponent associated with theexpected leading corrections [27, 33]: ε = Min(2 − η, ω ) , ω = 1 /η − O [(1 /η − ] . (19)For N c = 3 and γ = 0 the quality of the fits canbe assessed from the results shown in Fig. 1. Fits toEqs. (17) and (18) are very good, as it also supportedby the values of χ / dof ( χ is here the sum of thefit residuals and dof is the number of degrees of free-dom of the fit), which are smaller than 1, if a few re-sults for the smallest lattice sizes are discarded. For N c = 3 and γ = 0 we obtain the large- L extrapo-lations η = 0 . , . , . R ξ =0 . , . , . β = 4 . , . , .
4, re-spectively. We have also performed a detailed studyfor γ = 1 and β = 4 .
4. We obtain η = 0 . R ξ = 0 . ω , see Eq. (19), is knownprecisely only for η close to 1/4. In the fits we use ω as obtained from Eq. (19), and therefore ω gives theleading correction-to-scaling exponent for η & .
17. Insuch cases, to estimate the error due to the uncertaintyon ω , we checked the variation of the results of the fitswhen varying ω in around the approximation obtainedfrom Eq. (19), within a reasonable interval of about 10%. β ξ L=16L=32L=64L=128 N c =3 FIG. 3: Estimates of the correlation length ξ versus β for thelattice SO(3) gauge model (1) with γ = 0, for several valuesof L , up to L = 128. When the results for different valuesof L agree, they can be considered as good approximationsof the infinite-volume correlation length, within errors. Thevertical lines indicate the interval of values of β in which theBKT transition occurs. This has allowed us to estimate how η and R ξ vary withchanges of ω . Such variation has been included in thefinal error.In Fig. 2 we plot R ξ versus η together with the uni-versal curve computed in the spin-wave theory. The re-sults for R ξ and η are in excellent agreement with thespin-wave predictions. This shows the existence of a low-temperature phase with QLRO, analogous to that occur-ring in the XY model. D. FSS at the BKT transition
In Sec. III C we showed that the SO(3) gauge model hasa low-temperature phase with the same features of thelow-temperature phase of the XY model. Now, we focuson the finite-temperature transition that ends the high-temperature phase, to check whether the FSS behavior isthe same as that observed at the BKT transition of the XY model.To begin with, in Fig. 3 we show the estimates of thecorrelation length ξ for N c = 3 and γ = 0. They show asudden increase around β & .
5, as expected in the pres-ence of a finite-temperature BKT transition. To char-acterize the nature of the transition, we plot the Binderparameter U versus the ratio R ξ , In the FSS limit datashould belong to a curve that only depends on the univer-sality class. In Fig. 4 we report our numerical results for N c = 3 and for three values of γ , which are γ = 0 , ± XY model, thathave been obtained by standard Monte Carlo simulationsfor lattice sizes L = 100 , R ξ U L=16L=32L=64L=128L=100L=200 N c =3 γ=1 XY R ξ U L=16L=32L=64L=128L=100L=200 N c =3 γ=0 XY R ξ U L=16L=32L=64L=128L=100L=200 N c =3 γ=−1 XY FIG. 4: We plot data of U versus R ξ for N c = 3, γ = 1(top), γ = 0 (middle), and γ = − XY model (10). We observea nice agreement, supporting the conjecture that the lat-tice SO( N c ) gauge model with two scalar flavors undergoesa finite-temperature BKT transition for generic values of γ .The horizontal and vertical lines indicate the universal valuesof U and R ξ at the BKT transition, i.e. U ( T c ) = 1 . R ξ ( T c ) = 0 . ... , respectively [19, 26]. cient to provide a good approximation of the asymptoticFSS behavior (the differences between the L = 100 and L = 200 scaling curves are very small and hardly visiblein Fig. 4). It is quite clear that the data for the gaugemodel fall on top of the XY scaling curve, confirmingthat the transition has the same universal features: thegauge SO(3) model undergoes a BKT transition as the XY model. Analogous results are obtained for N c = 4and N c = 5, as shown in Fig. 5, where we report datafor γ = 0. In both cases, the data for the gauge modelconverge toward the FSS curve of the XY model.We note that the approach to the asymptotic FSS be-havior (9) is apparently quite fast in all lattice modelsconsidered, including the 2D XY model. In particular,the scaling corrections for the lattice SO( N c ) gauge mod-els appear to effectively decrease roughly as L − in thelimited range of L that we consider, up to L = 128.At BKT transitions, logarithmic corrections are gener-ally expected [16, 17, 20, 26, 27]. However, our rangeof values of L is too small to allow us to detect loga-rithmic changes of the estimates. In the range we con-sider power-law corrections effectively dominate. Signifi-cantly larger sizes are needed to allow us to perform fitsthat include both logarithmic and power-law corrections.Even though our analyses are not sensitive to the slowly-decaying logarithmic corrections, we can argue that thesystematic error they induce is small (we only refer hereto the behavior of U versus R ξ ; we are not claimingthat logarithmic corrections are always negligible). In-deed, the coefficients of the logarithmic corrections arenot universal, and therefore we expect different logarith-mic corrections in the XY model and in the gauge modelswe consider here. Thus, assuming that all models havea common universal asymptotic behavior, we can inferthe size of the logarithmic correction by looking at thedifferences between the results obtained in the differentmodels. As apparent from Figs. 4 and 5, differences aretiny, indicating that these elusive corrections play littlerole here.Accurate estimates of the critical BKT temperaturesare hard to obtain, since their determination is gener-ally affected by logarithmic corrections, see Eq. (16).The problem of the logarithmic corrections can be over-come by the so-called matching method put forward inRefs. [16, 17, 19] (see also Refs. [27, 28] for applicationsto some 2D quantum lattice gas models). Here, we donot pursue this analysis further, since we are not par-ticularly interested in obtaining precise estimates of thecritical temperatures. We only mention some rough esti-mates of the transition temperatures obtained by lookingat the β -values where R ξ ( β, L ) ≈ R ξ ( T c ) = 0 . ... .For N c = 3 we find β c ≈ .
82 for γ = 0, β c ≈ .
77 for γ = 1, and β c ≈ .
92 for γ = −
1. Moreover, we estimate β c ≈ .
80 for N c = 4 and β c ≈ .
76 for N c = 5, at γ = 0.In conclusion, the FSS analysis has allowed us to de-termine the nature of the finite-temperature transitionsoccurring in the lattice SO( N c ) gauge model (1) with twoflavors. For N c = 3 , , R ξ U L=16L=32L=64L=100L=200 N c =5XY R ξ U L=16L=32L=64L=128L=100L=200 N c =4XY FIG. 5: Plot of U versus R ξ for N c = 4 (bottom) and N c = 5(top), at γ = 0. We also report data for the 2D XY model(10). The FSS curve of the XY model is clearly approachedby the data for the lattice SO( N c ) models with increasing L . The horizontal and vertical lines indicate the BKT val-ues U ( T c ) = 1 . R ξ ( T c ) = 0 . ... , respec-tively [19, 26]. longs to the BKT universality class, as in the classical XY model. This occurs at least in an interval of valuesof γ around the infinite gauge-coupling value γ = 0. IV. CONCLUSIONS
We have studied a class of 2D lattice non-AbelianSO( N c ) gauge models with two real scalar fields, de-fined by the Hamiltonian (1). Such lattice gauge modelsare obtained by partially gauging a maximally O(2 N c )-symmetric multicomponent real scalar model, using theWilson lattice approach. For N c ≥
3, the resulting the-ory is locally invariant under SO( N c ) gauge transforma-tions and globally invariant under Abelian O(2) trans-formations. This study extends previous work on 2Dmodels with a local gauge invariance and a global non-Abelian symmetry, [6–8], in which a critical behavior canonly be observed in the zero-temperature limit. In themodels considered here, instead, the global Abelian O(2)symmetry may allow finite-temperature BKT transitionsbetween the disordered phase and the low-temperatureQLRO phase.The universal features of the transitions have been de-termined by performing FSS analyses of Monte Carlodata. We present results for the two-flavor lattice SO( N c )gauge models (1) with N c = 3 , ,
5. They show thatthese systems undergo a finite-temperature BKT transi-tion that separates the disordered phase from the low-temperature phase. Moreover, we have verified that thelow-temperature phase is characterized by spin waves,analogously to the standard XY model.These results provide additional evidence in favor ofthe conjecture that the critical behavior of 2D latticegauge models, defined using the Wilson approach [1], be-longs to the universality class of the field theories as-sociated with the symmetric spaces that have the same global symmetry. This conjecture assumes that gaugecorrelations are not critical and decouple in the criticallimit. Therefore, the conjecture may fail when the gaugecorrelations are critical, giving rise to a more complex be-havior. A similar phenomenon has been observed in thethree-dimensional lattice Abelian-Higgs model with non-compact gauge fields, see, e.g., Ref. [34] and referencestherein.We finally mention that the interplay between globaland gauge symmetries has also been studied in three di-mensional models, see Refs. [21, 34–36]. Acknowledgement . Numerical simulations have beenperformed on the CSN4 cluster of the Scientific Comput-ing Center at INFN-PISA. [1] K. G. Wilson, Confinement of quarks, Phys. Rev. D ,2445 (1974).[2] J. Zinn-Justin, Quantum Field Theory and Critical Phe-nomena , fourth edition (Clarendon Press, Oxford, 2002).[3] S. Weinberg,
The Quantum Theory of Fields , (CambridgeUniversity Press, 2005).[4] S. Sachdev, Topological order, emergent gauge fields,and Fermi surface reconstruction, Rep. Prog. Phys. ,014001 (2019).[5] P. W. Anderson, Basic Notions of Condensed MatterPhysics , (The Benjamin/Cummings Publishing Com-pany, Menlo Park, California, 1984).[6] C. Bonati, A. Pelissetto and E. Vicari, Two-dimensionalmulticomponent Abelian-Higgs lattice models, Phys.Rev. D , 034511 (2020).[7] C. Bonati, A. Pelissetto, and E. Vicari, Universal low-temperature behavior of two-dimensional lattice scalarchromodynamics, Phys. Rev. D , 054503 (2020).[8] C. Bonati, A. Franchi, A. Pelissetto, and E. Vicari,Asymptotic low-temperature critical behavior of two-dimensional mulitiflavor lattice SO( N c ) gauge theories,Phys. Rev. D , 034512 (2020).[9] N. D. Mermin and H. Wagner, Absence of ferromag-netism or antiferromagnetism in one- or two-dimensionalisotropic Heisenberg models, Phys. Rev. Lett. , 1133(1966).[10] E. Br´ezin, S. Hikami, and J. Zinn-Justin, Generalizednon-linear σ -models with gauge invariance, Nucl. Phys.B , 528 (1980).[11] C. Bonati, A. Franchi, A. Pelissetto, and E. Vi-cari, Asymptotic low-temperature behavior of two-dimensional RP N − models, Phys. Rev. D , 034513(2020).[12] J. M. Kosterlitz and D. J. Thouless, Ordering, metasta-bility and phase transitions in two-dimensional systems,J. Phys. C: Solid State , 1181 (1973).[13] V. L. Berezinskii, Destruction of Long-range Order inOne-dimensional and Two-dimensional Systems havinga Continuous Symmetry Group I. Classical Systems, Zh.Eksp. Theor. Fiz. , 907 (1970) [Sov. Phys. JETP ,493 (1971)]. [14] J. M. Kosterlitz, The critical properties of the two- di-mensional xy model, J. Phys. C , 1046 (1974).[15] J. V. Jos´e, L. P. Kadanoff, S. Kirkpatrick, and D.R. Nelson, Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planarmodel, Phys. Rev. B , 1217 (1977).[16] M. Hasenbusch, M. Marcu, and K. Pinn, High precisionrenormalization group study of the roughening transi-tion, Physica A , 124 (1994).[17] M. Hasenbusch and K. Pinn, Computing the rougheningtransition of Ising and solid-on-solid models by BCSOSmodel matching, J. Phys. A , 63 (1997).[18] J. Balog, Kosterlitz-Thouless theory and lattice artifacts,J. Phys. A , 5237 (2001).[19] M. Hasenbusch, The two dimensional XY model at thetransition temperature: a high precision numerical study,J. Phys. A , 5869 (2005).[20] A. Pelissetto and E. Vicari, Renormalization-group flowand asymptotic behaviors at the Berezinskii-Kosterlitz-Thouless transitions, Phys. Rev. E , 032105 (2013).[21] C. Bonati, A. Pelissetto, and E. Vicari, Three-dimensional phase transitions in multiflavor scalarSO( N c ) gauge theories, Phys. Rev. E , 062105 (2020).[22] A. Pelissetto and E. Vicari, Critical phenomena andrenormalization group theory, Phys. Rep. , 549(2002).[23] M. E. Fisher and M. N. Barber, Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. ,1516 (1972).[24] M. N. Barber, in Phase Transitions and Critical Phenom-ena , edited by C. Domb and J. L. Lebowitz (AcademicPress, New York, 1983), Vol. 8.[25] V. Privman ed.,
Finite Size Scaling and Numerical Simu-lation of Statistical Systems (World Scientific, Singapore,1990).[26] M. Hasenbusch, The Binder cumulant at the Kosterlitz-Thouless transition, J. Stat. Mech.: Theory Expt.P08003 (2008).[27] G. Ceccarelli, J. Nespolo, A. Pelissetto, and E. Vicari,Universal behavior of two-dimensional bosonic gases atBerezinskii-Kosterlitz-Thouless transitions, Phys. Rev. B , 024517 (2013).[28] F. Delfino and E. Vicari, Dimensional crossover of Bose-Einstein condensation phenomena in quantum gases con-fined within slab geometries, Phys. Rev. A , 043623(2017).[29] N. Cabibbo and E. Marinari, A New Method for Updat-ing SU(N) Matrices in Computer Simulations of GaugeTheories, Phys. Lett. , 387 (1982).[30] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth,A. H. Teller, and E. Teller, Equation of state calcula-tions by fast computing machines, J. Chem. Phys. ,1087 (1953).[31] M. Creutz, Overrelaxation and Monte Carlo Simulation,Phys. Rev. D , 515 (1987).[32] P. Di Francesco, P. Mathieu, and D. Senechal, ConformalField Theory (Springer Verlag, New York, 1997).[33] M. Hasenbusch, A. Pelissetto, and E. Vicari, Multicritical behaviour in the fully frustrated XY model and relatedsystems J. Stat. Mech.: Theory Expt. P12002 (2005).[34] C. Bonati, A. Pelissetto, and E. Vicari, LatticeAbelian-Higgs models with noncompact gauge field,arXiv:2010.06311.[35] C. Bonati, A. Pelissetto and E. Vicari, Phase dia-gram, symmetry breaking, and critical behavior of three-dimensional lattice multiflavor scalar chromodynamics,Phys. Rev. Lett. , 232002 (2019); Three-dimensionallattice multiflavor scalar chromodynamics: interplay be-tween global and gauge symmetries, Phys. Rev. D ,034505 (2020).[36] A. Pelissetto and E. Vicari, Multicomponent compactAbelian-Higgs lattice models, Phys. Rev. E100