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Langevin Simulation of the Chirally Decomposed Sine-Gordon Model L. Moriconi and M. Moriconi Instituto de F´ısica, Universidade Federal do Rio de Janeiro, C.P. 68528, 21945-970, Rio de Janeiro, RJ, Brazil 5 Alargeclassofquantumandstatisticalfieldtheoreticalmodels,encompassingrelevantcondensed 0 matter and non-abelian gauge systems, are defined in terms of complex actions. As the ordinary 0 Monte-Carlomethodsareuselessindealingwiththesemodels,alternativecomputationalstrategies 2 havebeenproposedalongtheyears. TheLangevintechnique,inparticular,isknowntobefrequently n plaguedwithdifficultiessuchasstrongnumericalinstabilitiesorsubtleergodicbehavior. Regarding a thechirallydecomposedversionofthesine-Gordonmodelasaprototypicalcaseforthefailureofthe J Langevinapproach,wedeviseatruncationprescriptioninthestochasticdifferentialequationswhich 1 yieldsnumericalstabilityandisassumednottospoiltheBerezinskii-Kosterlitz-Thoulesstransition. 3 This conjecture is supported by a finite size scaling analysis, whereby a massive phase ending at a line of critical points is clearly observed for thetruncated stochastic model. ] h PACSnumbers: 05.50.+q,05.10.Gg,11.10.Wx c e m I. INTRODUCTION tions. Since the field variables are promoted to complex - t numbersinthe stochasticdifferentialequations,the con- a figurational phase space is enlarged, and unstable man- t Several field theory models devoted to concrete con- s ifolds appear along the imaginary direction, containing densed matter and high energy applications are given . t in terms of complex-valued bosonic (“c-number”) ac- the classical vacuum solutions. The time evolution of a a phase space point (that is, a field configuration) will m tions. In the condensed matter context, for instance, depart, in brownian-like motion, from the determinis- they are the focus of much attention in low-dimensional - tic trajectory that would be followed in the absence of d systems related to Luttinger liquids and quantum Hall n phenomena [1, 2, 3, 4]. On the high-energy front, on the the stochastic noise. Thus, as soon as the phase space o otherhand,afundamentalopenissuereferstothe phase point happens to visit a region of “intense stream” in c theneighborhoodofanunstablemanifold,thenumerical diagram of SU(3)-QCD with non-zero fermion density, [ wherethegaugeactionnotoriouslyacquiresanimaginary convergence of the Langevin equation is under risk. 1 part, due to the non-vanishing quark chemical potential Whileinpolynomialmodelsliketheλφ4 withcomplex v [5, 6, 7, 8, 9, 10, 11]. couplingconstantstheLangevinsimulationismostlysuc- 7 Numerical non-perturbative efforts to study all of the cessful in the cases of physical interest, difficulties arise 3 above problems are considerably restricted by the fact inotherwell-knownmodelswheretheself-interactionpo- 7 1 that the Monte-Carlo method cannot be straightfor- tential is non-polynomial and invariant under compact 0 wardlyapplied, since the weightsin the path-integration Lie groups. The Langevin equations will contain, due to 5 measures are not positive-definite quantities anymore. the extension of fields to the complex plane, “explosive” 0 To overcome this difficulty, a number of alternative nu- hyperbolic functions, leading,in general,to unstable nu- t/ mericalmethodshavebeensuggested,fromtimetotime, merical simulations [18]. That is precisely the situation a such as complex Langevin simulations [12, 13], analyti- found in many of the bosonized condensed matter sys- m cal expansions [14], kernel mappings [15], microcanon- tems or in the finite Baryon density QCD realm. - ical averaging prodecures [16], path-integral factoriza- In order to investigate the problem of numerical d n tion schemes [17], etc. To date, there are not yet well- Langevin stability, we have regarded the exactly solv- o established results on how general and efficient these able euclidean sine-Gordon (SG) model as an ideal test- c tools are. Among them, the Langevin approach – the ing ground prototype, aiming, in this simpler context, v: central subject of discussion in this paper – has as at- to antecipate applications for a large class of systems. i tractivefeaturesitsgreatsimplicityanddirectnumerical As briefly reviewed in Sec. II, the SG partition func- X implementation. However, as it has been learned from tion is alternatively written in terms of interacting left r purely empirical investigations, the stochastic evolution and right chiral bosons, governed by a complex-valued a provided by the Langevin equations is affected, in many action. In Sec. III, we introduce the stochastic quanti- important cases, by bad numerical convergence and the zation procedure for the chirally decomposed SG model, existence of blow-up trajectories [9, 18]. Also, some spe- which is observed to produce numerical blow-up solu- cialcaremaybeneededwhenwritingtheLangevinequa- tions. It turns out that the instabilities are not fixed by tions for actions which have cuts in the complex plane, a reasonable time step resizing, or even with the help in order to assure good ergodic properties of the phase of more precise evolution algorithms (as a fourth-order space flow [7, 19]. Runge-Kutta scheme). We propose, then, a pragmatical It is not hard to understand in an essential way the solutionofthenumericalconvergenceproblem,througha originofeventualinstabilitiesincomplexLangevinequa- straightforward“truncation prescription” applied to the 2 stochastic equations. The basic idea is to modify the As the jacobian of the mapping (φ,π) → (φ ,φ ) does R L Langevin equations without spoiling its thermodynami- notdependonthefields,thepartitionfunction(2.2)may cal phases. In fact, as a strong piece of evidence sup- be expressed, by means of a direct substitution, as porting our method, the Berezinskii-Kosterlitz-Thouless (BKT)phasetransition[20,21]ofthe originalSGmodel Z = Dφ Dφ exp(−S) , (2.5) isverifiedbymeansofafinitesizescalingstudy. Finally, Z R L in Sec. IV, we comment on our findings and discuss di- where rections of further research. S = d2x[∂ φ (∂ +i∂ )φ Z 1 R 1 0 R II. THE CHIRALLY DECOMPOSED SG MODEL g + ∂ φ (∂ −i∂ )φ − cos(β(φ +φ ))](2.6) 1 L 1 0 L R L β Asitiswell-known,theusualeuclideanSGcorrelation functions may be obtained, in path-integral language, is, thus, defined as the action of the CDSG model. It is from the zero temperature partition function [22], clear that correlation functions of the SG model can be computed, in principle, from the above partition func- 1 g tion, through the substitution φ → φ +φ . Observe, Z = Dφexp{− d2x[ (∂φ)2− cos(βφ)]} . (2.1) R L Z Z 2 β however, that the CDSG action is complex, and the numerical chiral correlation functions cannot be obtained It is worth summarizing here some important facts on with the help of ordinary Monte-Carlo techniques. As the above model [23]. In the g → 0 limit, the SG spec- an alternative numerical method, we will investigate the trum is massless for β2 ≥ 8π ≡ β2, defining a line of CDSGmodel,inthenextsection,fromthe pointofview c critical points, and massive for β2 < β . At β = β the ofLangevinequations,basedontheformalismofstochas- c c SGmodelhasaBKTphasetransition. Thesamebehav- tic quantization. ior holds for finite g, with β =β (g) (β grows with g). c c c TheSGmodelmaybe regardedasthe bosonizedversion of the massless Thirring model perturbed by a Gross- III. STOCHASTIC SIMULATIONS Neveu interaction, defined in terms of a Dirac fermion field ψ [24]. Its phases can be distinguished by the mass The Parisi-Wu stochastic quantization method [25], order parameter m ≡ hψ¯ψi, which is proportional to states,underverygeneralhypothesis,thateuclideanfield Φ ≡ hcos(βφ/2)i. For β → βc from below, the exact theories can be modelled through Langevin (= stochas- solutionoftheSGmodelyieldshψ¯ψi∼exp[c/(β−βc)]. tic) differential equations. Actually, the CDSG model is Forβ >βctheSGinfraredfixedpointisjustthegaussian equivalentlygivenbythe followingsetofLangevinequa- modelandtheorderparametervanishesinthethermody- tions: namicallimit. Atβ =β theorderparameterhasscaling c δS dimension ∆(Φ)=−1/2, that is, for a finite size system ∂ φ =− +η , with linear dimension L, it follows that hΦi∼L−1/2. τ R δφR R The chirally decomposed version of the SG model δS ∂ φ =− +η , (3.1) (CDSG)canbereadlyderivedinafewelementarysteps. τ L δφ L L Introducing the conjugate momentum π =π(x ,x ), we 0 1 may rewrite (2.1) as whereτ istheauxiliary“stochastictime”. Intheasymp- totic τ → ∞ limit, we expect that correlation functions π involvingthechiralfields,φ (τ,x ,x )andφ (τ,x ,x ), Z = DφDπexp{− d2x[ +iπ∂ φ R 0 1 L 0 1 Z Z 2 0 taken at equal stochastic times, will converge to the 1 g CDSGones. BothηR andηL aregaussianrandomfields, + (∂φ)2− cos(βφ)]} . (2.2) satisfying to 2 β hη i=hη i=0 , Define now the right and left chiral fields, R L ′ ′ hη (τ,x)η (τ ,x)i=0 , R L φR = 12[φ+Z0x1dξπ(x0,ξ)] , hhηηR((ττ,,xx))ηηR((ττ′′,,xx′′))ii==22δδ((ττ−−ττ′′))δδ22((xx−−xx′′))., (3.2) 1 x1 L L φL = 2[φ−Z dξπ(x0,ξ)] . (2.3) Above, x and x′ denote points in the physical two- 0 dimensional space. Substituting the action (2.6) into It is straightforwardto invert (2.3). We find (3.1) we get φ=φ +φ , ∂ φ =2∂ (∂ +i∂ )φ −gsin(βφ)+η , R L τ R 1 1 0 R R π =∂ φ −∂ φ . (2.4) ∂ φ =2∂ (∂ −i∂ )φ −gsin(βφ)+η , (3.3) 1 R 1 L τ L 1 1 0 L L 3 whereφ=φ +φ . Sincetheseequationsarecomplex,it Theintegralcurvesforthisdynamicalsystem,thatisthe R L is convenient, for simulation purposes, to split their real vector field (ψ˙ ,ψ˙ ), are indicated in the top of Fig. 1. 1 2 andimaginarycomponents. DefineφR =φ(R1)+iφ(R2) and Due to the large “speeds” far from the ψ2 = 0 axis, nu- φ =φ(1)+iφ(2). Theresultingequationsare,therefore, merical solutions exhibit bad uniform convergence prop- L L L erties: for any choice of time step, a point close enough ∂ φ(1) = 2∂2φ(1)−2∂ ∂ φ(2) to one of the unstable manifolds would evolve towards a τ R 1 R 1 0 R region of difficult numerical control. − gcosh(βφ(2))sin(βφ(1))+η , R ∂ φ(2) = 2∂2φ(2)−2∂ ∂ φ(1) τ R 1 R 1 0 R − gsinh(βφ(2))cos(βφ(1)) , ∂ φ(1) = 2∂2φ(1)+2∂ ∂ φ(2) τ L 1 L 1 0 L 4 − gcosh(βφ(2))sin(βφ(1))+η , L 3 ∂τφ(L2) = 2∂12φ(L2)+2∂1∂0φ(L1) 2 − gsinh(βφ(2))cos(βφ(1)) . (3.4) 1 ) It is a hopeless task to solve the above coupled differ- m( 0 I entialequationsviadirectnumericalsimulations. Ingen- -1 eral,forβ closetoβ (butnotnecessarilywithinthecrit- c -2 ical region), the numerical convergence is ruined by the -3 existence of blow-up configurations. That kind of phe- nomenonisakintotheonefoundinsimulationsoflattice -4 -8 -6 -4 -2 0 2 4 6 8 QCD, when the quark chemical potential is set around the threshold for fermion production [9]. After several Re( ) atempts, we concluded it is unlikely that the numerical 2 evolution can be stabilized by simple time step reduc- tions or the use of improved higher order Runge-Kutta 1 schemes. However,from a purely mathematical perspec- tive, the source of instability in the Langevin simulation ) m( 0 of the CDSG model is clear: it is related to the hyper- I bolicfunctionsdefinedinequations(3.4). Theireventual -1 fast growing, likely to happen close to phase transition points where field fluctuations are more intense, cannot -2 be tamed by any finite precision numerical algorithm. Relying upon the idea of universality classes, we pro- -8 -6 -4 -2 0 2 4 6 8 poseawayoutoftheinstabilityproblem. Onemayargue Re( ) that as far as the correlation length is much larger than the lattice spacing, the main interest is not the precise location, in the phase diagram, of the phases of the SG model. Instead, it is important to retain, in the contin- FIG. 1: The vector fields for the dimensionally-reduced dy- uum limit, the “topology” of the phase diagram and the namical systems (3.6) and (3.8). correct singularity order of the transitions between different phases. In loose words, we suggest to change the Asimplemodificationof(3.6)leadstoadynamicalsys- model, but not the large scale physics. We have found, tem with uniform convergence properties, and the same in fact, that it is possible to deform the above set of asymptoticsolutions. Togetthe alternativesetofdiffer- equations, rendering them numerically stable, while pre- ential equations, we perform, in (3.5), the substitution: serving the BKT transition they are assumed to imply. To understand how the Langevin equations are to be sinψ =sin(ψ +iψ ) modified, it is instructive to consider the following ordi- 1 2 narydifferentialequation,directlyinspiredfromtheform =coshψ sinψ +isinhψ cosψ 2 1 2 1 of (3.3): =coshψ (sinψ +itanhψ cosψ ) 2 1 2 1 ψ˙ =−sinψ . (3.5) →sinψ1+itanhψ2cosψ1 . (3.7) Extending the field ψ to the complex plane, by means of Now, instead of (3.6), we will have ψ =ψ +iψ , equation (3.5) becomes 1 2 ψ˙ = −sinψ coshψ , ψ˙ = −sinψ , 1 1 2 1 1 ψ˙ = −cosψ sinhψ . (3.6) ψ˙ = −cosψ tanhψ . (3.8) 2 1 2 2 1 2 4 Theintegralcurvesfortheabovemodelarewell-behaved, of6simulationswasconsidered,relatedtodifferentseeds as it may be seen from the picture shown at the bot- for the pseudorandom number generator. Each simula- tom of Fig. 1. Indeed, the vector field is bounded, for tion consisted of 107 iterations per lattice site. The SG ||(ψ˙ ,ψ˙ )||2 =sin2(ψ )+cos2(ψ )tanh2(ψ )<1 andnu- couplingconstantwassettog =10.0. Thetimestepwas 1 2 1 1 2 mericalconvergenceisneveraprobleminthisframework. taken to be ǫ=2×10−3. Averages were computed over It is worth noting that the truncation prescription (3.7) a set of approximately decorrelated (in stochastic time) keeps the original structure of the stable and unstable 2000 configurations per simulation. manifolds, as well as the positions of the fixed points In Fig. 2, the mass order parameter Φ is plotted for in the (ψ ,ψ ) plane. Its role is basically to soften the β = 0.5,1.0,...,7.5. In more detail, we have evaluated, 1 2 vector field for the purpose of having stable numerical for a lattice of linear size L, simulations. The same stabilization trick may be applied without 1 2000 6 L βφl(i,j) Φ= | cos( k )| , any further complication to the CDSG Langevin equa- 6×2000×L2 2 XX X k=1 l=1 i,j=1 tions (3.3). We get (3.10) ∂ φ(1) = 2∂2φ(1)−2∂ ∂ φ(2) where (k,l) labels a given complex configuration of φ = τ R 1 R 1 0 R φ(i,j) in the statistical sample. There is clearly a phase − gsin(βφ(1))+ηR , transition taking place around βc = 2.5. The inset of ∂ φ(2) = 2∂2φ(2)−2∂ ∂ φ(1) Fig. 2 givesthe best profilefor the transitionamongour τ R 1 R 1 0 R simulations, for the lattice with size 20×20. − gtanh(βφ(2))cos(βφ(1)) , ∂ φ(1) = 2∂2φ(1)+2∂ ∂ φ(2) τ L 1 L 1 0 L − gsin(βφ(1))+η , L ∂ φ(2) = 2∂2φ(2)+2∂ ∂ φ(1) τ L 1 L 1 0 L 0.20 − gtanh(βφ(2))cos(βφ(1)) . (3.9) 5x5 Numericalsimulationswere successfuly carriedoutfor theabovesetofLangevinequations,usinganelementary ar) 0.15 10x10 b 15x15 Euler evolution algorithm, for lattices with sizes 5×5, r 20x20 o 10×10,15×15,and20×20. Periodicboundaryconditions r r e were imposed on the field configurations. ( 0.10 0.05 1.0 1.0 0.00 0.8 0 1 2 3 4 5 6 7 0.8 0.6 0.4 0.2 0.6 os 0.0 FIG. 3: The standard deviation analysis for the set of aver- c 0 1 2 3 4 5 6 7 aged evaluations of themass order parameter Φ. 0.4 In addition, we have carried out an estimate of the 5x5 error bars related to the curves shown in Fig. 2. They 0.2 10x10 are defined as 15x15 20x20 6 1 0.0 δΦ=vu6 (Φl)2−Φ2 , (3.11) 0 1 2 3 4 5 6 7 u X t l=1 where 1 2000 L βφl(i,j) FIG.2: Themass orderparameter Φisplotted as afunction Φ = | cos( k )| . (3.12) of β for different lattice sizes. The inset shows the data for l 2000×L2 X X 2 thelattice with size 20×20. k=1 i,j=1 The results are shown in Fig. 3. As expected, fluctu- Foreachlatticesizeandagivenvalueofβ,anensemble ations are larger around the phase transition, and get 5 smallerasthe latticesizegrows. Forthelattice withsize tion, revealed when one examines the results of simula- 20×20, errors are just of the order of a few percent. tions for the truncated CDSG model: while the scaling Different lattice sizes were studied in order to estab- dimensionofthemassorderparameterdependsquadrat- lish a finite size scaling analysis of the transition, re- icallyonβ inthe masslessphase ofthe SG model,in the ported in Fig. 4. Each one of the five straight lines truncated context, it is approximately constant. That is draw in that picture corresponds to a fixed value of β notasurprise,however. Oncewehavedeformedtheorig- (β =3.5,4.5,5.5,6.5,7.5),indicating that the scaling re- inalmodel, wecannotattribute the samescalingproper- lation ties to formally identical operators. Also, non-universal parameters, like β are not supposed to be the same in c Φ∼L∆ (3.13) the alternativemodel. Actually, we havefound β ≃2.5, c which is smaller than the value predicted for the SG holdsattherightsideofthetransition. Accordingtothe model. ideas of finite size scaling analysis [22], equation (3.13) provides strong evidence for the fact that the whole region β > 3.5 is a massless phase of the model under investigation. No mass scale is involved; the only relevant IV. CONCLUSIONS scale turns to be the lattice linear size. Incidentally, as ∆ ≃ −1 in the extended critical region, Φ plays indeed The alternative, but equivalent, chirally decomposed the role of a mass order parameter, suggesting it is di- version of the SG model yields an interesting stage for rectly related to the spectrum of massive excitations for the test of numerical methods devoted to the analysis of β < βc. We note that the phase transition separates a systems governed by complex-valued actions. We have massivephase, with ∆=0, froma line ofcriticalpoints. beenabletoshowhowaslightmodificationoftheCDSG This is the hallmark of the BKT phase transition, re- modelatthelevelofthecomplexLangevinequationspro- alized by the SG model. Therefore, we put forward the vides a stable numerical framework,allowing us to iden- conjecturethatthetruncationprescription,implemented tify expected properties of the SG model, as the BKT on the Langevin equations, is able to preserve the SG transition. The essential idea of the truncation prescrip- thermodynamical phases. tion, performed on the “bad behaved” hyperbolic terms contained in the complex Langevin equations, is to re- duce the degree of divergence of the unstable manifolds defined in the phase space, while preserving the thermodynamical phases of the original model. -1.2 Since the CDSG model is similar to several rele- -1.4 3.5 -0.79 vant models found in condensed matter and high energy 4.5 -1.00 -1.6 5.5 -1.01 physics, it is likely that the rather elementary method 6.5 -0.98 ofnumericalstabilizationaddressedinthis paper will be -1.8 7.5 -0.96 of utility in these contexts. Low-dimensional condensed ) -2.0 matter bosonized models and the case of finite fermion ( og -2.2 density QCD are the ideal instances where an analogous L -2.4 line of numerical investigation could be applied. -2.6 0.0 Furthermore, alongside with the numerical approach, -0.2 a dynamical renormalization group analysis of the trun- -2.8 -0.4 -0.6 cated stochastic differential equations is in order, to put -3.0 -0.8 theresultsestablishedhereonafirmerground. Inpartic- -1.0 -3.2 0 1 2 3 4 5 6 7 ular,itisoffundamentalimportancetorederivetheBKT -3.4 renormalization group flow directly from the Langevin 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 equations (3.9), a task which is under current investigation. Log(L) FIG. 4: Finite size scaling results obtained from the analysis oflattices with sizes5×5, 10×10, 15×15 and20×20. The Acknowledgments inset shows the scaling dimension of the order parameter as a function of β. 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