Complex Langevin analysis of 2D U(1) gauge theory on a torus with a θ term
Mitsuaki Hirasawa, Akira Matsumoto, Jun Nishimura, Atis Yosprakob
aa r X i v : . [ h e p - l a t ] M a y KEK-TH-2209
Complex Langevin analysis of 2D U(1) gauge theoryon a torus with a θ term Mitsuaki H irasawa ∗ , Akira M atsumoto † ,Jun N ishimura , ‡ and Atis Y osprakob § Department of Particle and Nuclear Physics,School of High Energy Accelerator Science,Graduate University for Advanced Studies (SOKENDAI),1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization,1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
Monte Carlo simulation of gauge theories with a θ term is known to be extremelydifficult due to the sign problem. Recently there has been major progress in solvingthis problem based on the idea of complexifying dynamical variables. Here we considerthe complex Langevin method (CLM), which is a promising approach for its lowcomputational cost. The drawback of this method, however, is the existence of acondition that has to be met in order for the results to be correct. As a first step, weapply the method to 2D U(1) gauge theory on a torus with a θ term, which can besolved analytically. We find that a naive implementation of the method fails becauseof the topological nature of the θ term. In order to circumvent this problem, wesimulate the same theory on a punctured torus, which is equivalent to the originalmodel in the infinite volume limit for | θ | < π . Rather surprisingly, we find that theCLM works and reproduces the exact results for a punctured torus even at large θ ,where the link variables near the puncture become very far from being unitary. ∗ E-mail address : [email protected] † E-mail address : [email protected] ‡ E-mail address : [email protected] § E-mail address : [email protected]
Introduction
The θ term provides an interesting avenue of research in quantum field theories. Due toits topological nature, its effects on physics should be genuinely nonperturbative, if presentat all. In particular, it does not affect the equation of motion, which implies that θ is aparameter that does not exist in the corresponding classical theory. For instance, the θ term in QCD is given by S θ = − iθQ , where Q is the topological charge defined by Q = 132 π ǫ µνρσ Z d x tr F µν F ρσ , (1.1)which takes integer values on a compact space. This term is renormalizable by power-counting and hence it is a term which is perfectly sensible to add in the action. However,it breaks parity and time-reversal symmetries, and hence the CP symmetry. This leadsto a non-vanishing electric dipole moment of a neutron, which is severely restricted byexperiments. The upper bound on θ thus obtained is | θ | . − [1], which is extremelysmall although there is no reason for it theoretically. This is a naturalness problem knownas the strong CP problem.A popular solution to this problem is the Peccei-Quinn mechanism [2, 3, 4, 5], whichintroduces axions as a pseudo Nambu-Goldstone boson of a hypothetical global U(1) PQ symmetry. In this mechanism, the potential for the axions induced by QCD chooses a CPinvariant vacuum automatically. Recently, gauge theories with a θ term have attractedattention also from the viewpoint of the ’t Hooft anomaly matching condition [6, 7, 8]and the gauge-gravity correspondence [9, 10, 11, 12]. In particular, there is an interestingprediction for a phase transition at θ = π , which claims that either spontaneous CP breakingor deconfinement should occur there [6, 7]. In order to investigate gauge theories from firstprinciples in the presence of a θ term motivated either by the physics related to axionsor by the recent predictions, one needs to perform nonperturbative calculations based onMonte Carlo methods. However, this is known to be extremely difficult because the θ termappears as a purely imaginary term in the Euclidean action S . The Boltzmann weight e − S becomes complex, and one cannot interpret it as the probability distribution as one doesin Monte Carlo methods.One can still use the reweighting method by treating the phase of the complex weightas a part of the observable. In the case at hand, this amounts to obtaining the histogramof the topological charge at θ = 0 and taking an average over the topological sectorscharacterized by the integer Q with the weight e iθQ . Various results obtained in this wayare nicely reviewed in Ref. [13]. Clearly, the calculation becomes extremely difficult due tohuge cancellations between topological sectors when topological sectors with | Q | ≫ π/ | θ | | θ | ∼ π or for smaller | θ | with sufficiently large volume. This is the so-called sign problem whichoccurs in general for systems with a complex action.There are actually a few promising methods that can handle systems which suffer fromthis problem such as the complex Langevin method (CLM) [14, 15, 16, 17, 18, 19], thegeneralized Lefschetz thimble method [20, 21, 22, 23, 24, 25], the path optimization method[26, 27, 28, 29] and the tensor network method [30, 31, 32, 33, 34]. Each method has itspros and cons, however. In this work, we focus on the CLM, which can be applied tovarious physically interesting models with large system size in a straightforward manner.(See, for instance, Refs. [35, 36, 37].) The only drawback of the method is that it cangive wrong results depending on the system, the parameter region, and even on the choiceof the dynamical variables. Recently, the reason for this behavior has been understoodtheoretically [16, 17, 18, 19, 38, 39]. In particular, Ref. [19] proposed a practical criterionfor correct convergence, which made the CLM a method of choice for complex action systemsas long as the criterion is met. There are indeed many successful applications to latticequantum field theory [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 35] and matrix models[52, 53, 54, 55, 36, 47, 56, 57] with complex actions.As a first step towards its application to 4D non-Abelian gauge theories, here we applyit to 2D U(1) lattice gauge theory with a θ term, which suffers from a severe sign problemdespite its simplicity. In fact, the model can be solved analytically with finite latticespacing and finite volume on an arbitrary manifold [60, 61, 62], which makes it a usefultesting ground for new methods [60, 63, 64, 65] aiming at solving the sign problem. Byusing the reweighting method [64], for instance, one can only reach θ ∼ . × θ = π by this method.Note also that the region of θ that can be explored by this method shrinks to zero as oneincreases the lattice size.We find that a naive implementation of the CLM fails. The reason for this is that theconfigurations that appear when the topology change occurs during the Langevin processnecessarily result in a large drift term. Due to this fact, the criterion [19] for the validityof the method based on the histogram of the drift term cannot be satisfied. If one tries tosuppress the appearance of the problematic configurations by approaching the continuumlimit, the criterion can be satisfied, but the topology change does not occur during theLangevin process, hence the ergodicity is lost. Abelian gauge theories have also been discussed in the context of a non-standard lattice discretization,which is not only theoretically beneficial [58] but also enables a dual simulation [59], which is free from thesign problem.
2n order to cure this problem, we introduce a puncture on the torus, which makes thebase manifold noncompact. We have obtained exact results for this punctured model aswell. Even in the continuum limit, the topological charge is no longer restricted to integervalues and the 2 π periodicity in θ does not hold. However, if we take the infinite volumelimit with | θ | < π , one cannot distinguish the model from the original non-punctured modelas far as the observables that make sense in that limit are concerned. Note that in thatlimit, the topological charge can take arbitrarily large values and therefore it does not reallymatter whether it is an integer or not.On the other hand, the situation of the complex Langevin simulation changes drasticallyfor the punctured model. The topology change occurs freely and the appearance of theproblematic configurations can be suppressed by simply approaching the continuum limit.Thus the criterion for the validity of the CLM is met without losing the ergodicity, and weare able to reproduce the exact results for the punctured model.The most striking aspect of our results is that the CLM works even if the link variablesclose to the puncture become very far from being unitary. This can happen because thedirect effect of the θ term on the complex Langevin dynamics is actually concentrated onthese link variables. While the link variables are allowed to be non-unitary in the CLM ingeneral in order to include the effects of the complex action, all the previous work suggestedthat the condition for the validity cannot be satisfied unless the non-unitarity is sufficientlysuppressed. Precisely for this reason, the gauge cooling [66, 18, 19] was invented as a crucialtechnique in applying the CLM to gauge theories. In fact, we also use the gauge cooling inour simulation, but the link variables close to the puncture nevertheless become far frombeing unitary when θ or the physical volume gets large. Yet the criterion for the drift termis not violated and the exact results are perfectly reproduced.The rest of this paper is organized as follows. In Section 2, we make a brief review on2D U(1) gauge theory on a torus with a θ term and discuss how to put it on a lattice. InSection 3, we apply the CLM to this theory and show that a naive implementation fails.In Section 4, we introduce a puncture on the torus in order to circumvent the problemencountered in Section 3, and show the equivalence of the punctured model and the originalnon-punctured model in the infinite volume limit for | θ | < π . In Section 5, we apply theCLM to the punctured model and show that it works perfectly even at large θ , where thelink variables near the puncture become very far from being unitary. Section 6 is devotedto a summary and discussions. In Appendix A, we review how one can solve the theoryanalytically for various boundary conditions, and obtain exact results for the puncturedand non-punctured models, in particular. In Appendix B, we show our results of the CLMfor the punctured model using a different lattice definition of the topological charge.3 ote added. When this paper was about to be completed, we encountered a preprint[67] on the arXiv, which investigates the same theory numerically by the density of statesmethod. The exact results are reproduced for | θ | < π up to L = 24, which goes far beyondthe reweighting method [64]. It is crucial to use an open boundary condition, which is similarin spirit to introducing a puncture in our work although the purpose is quite different. θ term In this section, we review 2D U(1) gauge theory with a θ term and discuss how to define iton a lattice.In the continuum 2D U(1) gauge theory on a Euclidean space, the action for the gaugefield A µ ( x ) ( µ = 1 ,
2) is given by S g = 14 g Z d x ( F µν ) , (2.1)where g is the gauge coupling constant and F µν is the field strength defined as F µν = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) . (2.2)We add a θ term S θ = − iθQ (2.3)in the action, where Q is the topological charge defined by Q = 14 π Z d x ǫ µν F µν , (2.4)which takes integer values if the space is compact.We put this theory on a 2D torus, which is discretized into an L × L periodic latticewith the lattice spacing a . On the lattice, we define the link variables U n,µ ∈ U(1), where n labels the lattice site as x µ = an µ . We also define the plaquette P n = U n, ˆ1 U n +ˆ1 , U − n +ˆ2 , U − n, , (2.5)which is a gauge invariant object. Here we write U − n,µ instead of U † n,µ , which will be impor-tant later in applying the CLM, where we complexify the dynamical variables respectingholomorphicity.The lattice counterpart of the field strength (2.2) can be defined as F n, = 1 ia log P n , (2.6)4here we take the principal value for the complex log; namely log z = log | z | + i arg z with − π < arg z ≤ π . Since the plaquette can then be written in terms of F n,µν as P n = e ia F n, , (2.7)the lattice counterpart of the gauge action (2.1) can be defined as S g = − β X n (cid:0) P n + P − n (cid:1) = − β X n cos (cid:0) a F n, (cid:1) , (2.8)which approaches S g ≃ g X n a ( F n,µν ) , (2.9)in the continuum limit up to an irrelevant constant with the identification β = 1( ga ) . (2.10)In the present 2D U(1) theory, the topological charge can be defined as Q log = 14 π X n a ǫ µν F n,µν = − i π X n log P n , (2.11)which gives an integer value even at finite a . This can be proved easily by noting that Q n P n = 1 since each link variable appears twice in this product with opposite directions.We call this definition (2.11) the “log definition”. As an alternative definition, we consider Q sin = − i π X n (cid:0) P n − P − n (cid:1) = 12 π X n sin (cid:0) a F n, (cid:1) , (2.12)which approaches (2.4) in the continuum limit recalling (2.7). Note, however, that thetopological charge defined on the lattice in this way can take non-integer values in generalbefore taking the continuum limit. We call this definition (2.12) the “sine definition”. Thusthe lattice theory is given by S = S g + S θ , (2.13)where S g is given by (2.8) and S θ is given by (2.3) with Q defined either by (2.11) or by(2.12).Since this theory is superrenormalizable, we can take the continuum limit a → g , which is set to unity throughout this paper without loss of generality. In this unit,the physical volume of the 2D torus is given by V phys = ( La ) = L β , (2.14)where we have used (2.10). 5 Applying the CLM to the 2D U(1) gauge theory
Since the θ term is purely imaginary in general, it makes Monte Carlo studies of gaugetheories extremely difficult due to the sign problem. We overcome this problem by usingthe complex Langevin method (CLM) [14, 15, 16, 17, 18, 19], which extends the idea ofstochastic quantization to systems with complex actions. In this section, we discuss howto apply the CLM to 2D U(1) gauge theory with a θ term and show some results, whichsuggest that a naive implementation of the method fails. The first step of the CLM is to complexify the dynamical variables. In the present caseof U(1) gauge theory, we extend the link variables U n,µ ∈ U(1) to U n,µ ∈ C \ { } , whichcorresponds to extending the gauge field A µ ( x ) ∈ R to A µ ( x ) ∈ C in the continuum theory.Then we consider a fictitious time evolution U n,µ ( t ) of the link variables governed by thecomplex Langevin equation U n,µ ( t + ∆ t ) = U n,µ ( t ) exp h i n − ∆ t D n,µ S + √ ∆ t η n,µ ( t ) oi , (3.1)where η n,µ ( t ) is a real Gaussian noise normalized by h η n,µ ( s ) η k,ν ( t ) i = 2 δ n,k δ µ,ν δ s,t . Theterm D n,µ S is the drift term defined by D n,µ S = lim ǫ → S ( e iǫ U n,µ ) − S ( U n,µ ) ǫ , (3.2)first for the unitary link variables U n,µ ( t ), and then it is defined for the complexified linkvariables U n,µ ( t ) by analytic continuation in order to respect holomorphicity. Using theaction (2.13), we obtain D n,µ S = D n,µ S g + D n,µ S θ , where the first term is given as D n, S g = − i β P n − P − n − P n − ˆ2 + P − n − ˆ2 ) ,D n, S g = − i β − P n + P − n + P n − ˆ1 − P − n − ˆ1 ) . (3.3)The second term D n,µ S θ depends on the definition of the topological charge. If oneuses the log definition (2.11), Eq. (3.2) for the θ term becomes a δ -function, which vanishesidentically except for configurations with P n = − n , reflecting the topologicalnature of the definition. Such configurations are precisely the ones that appear when thetopology change occurs within the configuration space of U n,µ . It is not straightforward toextend such a term to a holomorphic function of U n,µ .6n the other hand, if one uses the sine definition (2.12), the drift term becomes D n, S θ = − i θ π ( P n + P − n − P n − ˆ2 − P − n − ˆ2 ) ,D n, S θ = − i θ π ( − P n − P − n + P n − ˆ1 + P − n − ˆ1 ) , (3.4)which may be viewed as an approximation of the δ -function mentioned above. Moreover,it can be readily extended to a holomorphic function of U n,µ . For this reason, we use thesine definition for the non-punctured model.The criterion [19] for the validity of the CLM states that the histogram of the drift termshould fall off exponentially or faster. There are two cases in which this criterion cannot bemet. The first case occurs when the configuration comes close to the poles of the drift terms(3.3), (3.4), which correspond to configurations with P n = 0 for some n . If this happensduring the Langevin process, there is a possibility of violating the criterion. This problemis called the singular-drift problem [68, 69], which was found first in simple models [70, 71].In the present model, the same problem is caused also by approaching configurations with | P n | = ∞ for some n , which are related to the poles by the parity transformation.The second case occurs when the dynamical variables make large excursions in theimaginary directions [16]. This problem is called the excursion problem. In the presentmodel, this corresponds to the situation in which the link variables have absolute values |U n,µ | far from unity.Both the singular-drift problem and the excursion problem can occur because the linkvariables U n,µ are not restricted to be unitary in the CLM. In order to avoid these problems,it is important to perform the gauge cooling, which we explain in the next section. The idea of gauge cooling [66] is to reduce the non-unitarity of link variables as much aspossible by making gauge transformations corresponding to the complexified Lie group aftereach step (3.1) of the Langevin process. This procedure can be added without affecting theargument for justifying the CLM as demonstrated explicitly in Refs. [18, 19]. Recently, themechanism of the gauge cooling for stabilizing the complex Langevin simulation has beeninvestigated [72].The deviation of the link variables from U(1) can be defined by the unitarity norm N = 12 L X n,µ n U ∗ n,µ U n,µ + ( U ∗ n,µ U n,µ ) − − o . (3.5)7he gauge cooling reduces this quantity by a complexified gauge transformation, which isdetermined as follows.First we consider an infinitesimal gauge transformation δ U n,µ = ( ǫ n − ǫ n +ˆ µ ) U n,µ , (3.6)where ǫ n ∈ R . The change of the unitarity norm due to the transformation is given by δ N = 12 L X n,µ n ǫ n − ǫ n +ˆ µ ) U ∗ n,µ U n,µ − ǫ n − ǫ n +ˆ µ )( U ∗ n,µ U n,µ ) − o = 12 L X n ǫ n G n , (3.7)where G n is defined as G n = X µ n U ∗ n,µ U n,µ − U ∗ n − ˆ µ,µ U n − ˆ µ,µ − ( U ∗ n,µ U n,µ ) − + ( U ∗ n − ˆ µ,µ U n − ˆ µ,µ ) − o . (3.8)Therefore, we find that the unitarity norm is reduced most efficiently by choosing ǫ n ∝ − G n .Using this result, we consider a finite gauge transformation U n,µ g n U n,µ g − n +ˆ µ ; g n = e − αG n , (3.9)which makes the unitarity norm N ( α ) = 12 L X n,µ n U ∗ n,µ U n,µ e − α ( G n − G n +ˆ µ ) + ( U ∗ n,µ U n,µ ) − e α ( G n − G n +ˆ µ ) − o , (3.10)depending on α in (3.9). We search for an optimal α that minimizes (3.10). Note here thatit is typically a small number since the gauge cooling is performed after each step of theLangevin process. We therefore expand Eq. (3.10) with respect to α up to the second orderand obtain the value of α that minimizes it as α = 12 P n G n P n,µ [( G n − G n +ˆ µ ) {U ∗ n,µ U n,µ + ( U ∗ n,µ U n,µ ) − } ] . (3.11)We repeat this procedure until the unitarity norm changes by a fraction less than 10 − . When we solve the complex Langevin equation in its discretized version (3.1), it occasionallyhappens that the drift term becomes extremely large, in particular during the thermalizationprocess. This causes a large discretization error, which either makes the thermalization slow8 -6 -4 -2
1 10 100 u ( β , L) = (3, 10)( β , L) = (12, 20) Re Q sin( β , L) = (12, 20) Figure 1: The results obtained by the CLM for the non-punctured model using the sinedefinition Q sin of the topological charge. (Left) The histogram of the magnitude u of thedrift term defined by (3.12) is shown for ( β, L ) = (3 , , (12 ,
20) with θ = π . (Right) Thehistogram of Re Q sin is shown for ( β, L ) = (12 ,
20) with θ = π . The exact result obtainedfor ( β, L ) = (12 ,
20) with θ = 0 is shown by the solid line for comparison.or destabilizes the simulation. We can avoid this problem by using a small stepsize ∆ t , butthe computational cost for a fixed Langevin time increases proportionally to (∆ t ) − andthe calculation becomes easily unfeasible. The adaptive stepsize [73] is a useful technique,which amounts to reducing the stepsize only when the drift term becomes large.In our simulation, we measure the magnitude of the drift term defined as u = max n,µ | D n,µ S | (3.12)at each step, and choose the Langevin stepsize ∆ t in (3.1) as∆ t = ( ∆ t for u < v ,v u ∆ t otherwise , (3.13)where ∆ t is the default stepsize, and v is the threshold for the magnitude of drift term.In the present work, the default stepsize is set to ∆ t = 10 − , and the threshold is setto v = 2 β , considering a bound u ≤ β for θ = 0, where the CLM reduces to the realLangevin method. The measurement of the observables should be made with the sameinterval in terms of the Langevin time but not in terms of the number of steps. In this section, we present our results obtained by the CLM, which is implemented naivelyusing the non-punctured model explained above as opposed to the punctured model, which9e use later. As for the definition of the topological charge, we adopt the sine definition(2.12) for the reason given in Section 3.1.We have performed simulations at various θ for ( β, L ) = (3 , , (12 ,
20) correspondingto a fixed physical volume V phys ≡ L /β = 10 /
3. Below we show our results only for θ = π ,where the sign problem becomes severest, but the situation is the same for all values of θ .In Fig. 1(Left), we show the histogram of the magnitude u of the drift term. Thedistribution falls off rapidly for ( β, L ) = (12 , β, L ) = (3 , β, L ) = (12 , β, L ) = (3 ,
10) due to the large drifts.In Fig. 1(Right), we plot the histogram of Re Q sin obtained by the CLM for ( β, L ) =(12 ,
20) with θ = π , which has a sharp peak at Re Q sin ∼
0. In the same figure, we alsoplot the exact result for ( β, L ) = (12 ,
20) with θ = 0 for comparison, which exhibits a fewsharp peaks at integer values within the range − . Re Q sin .
2. From these two plots, weconclude that the transitions between different topological sectors are highly suppressed inthe simulation, which causes a problem with the ergodicity.This occurs also at θ = 0 for large β , and it is called the “topology freezing problem” inthe literature. In fact, the results one obtains by simulations suffering from this problemcorrespond to the expectation values restricted to the topological sector specified by theinitial configuration. This is true for both θ = 0 and θ = 0. In this case, however, theeffect of the θ term cancels between the numerator and the denominator of the expectationvalues, and the calculation essentially reduces to that of the real Langevin method at θ = 0.For ( β, L ) = (3 ,
10) with θ = π , on the other hand, the histogram of Re Q sin obtainedby the CLM has broad peaks that overlap with each other, which looks similar to the exactresult for ( β, L ) = (3 ,
10) with θ = 0. This implies that the topology freezing problem doesnot occur for ( β, L ) = (3 , w = 1 V ∂∂β log
Z . (3.14)Hereafter, V denotes the number of plaquettes in the action, which is V = L for the non-punctured model and V = L − V h Q i = − i V ∂∂θ log
Z , (3.15)which is zero at θ = 0 and purely imaginary for θ = 0. Finally, the topological susceptibility10 .810.820.830.84 0 0.5 1 1.5 2 w θ / π ( β , L) = (3, 10) 0.95760.95800.9584 0 0.5 1 1.5 2 w θ / π ( β , L) = (12, 20)-0.02-0.010.000.010.02 0 0.5 1 1.5 2 I m 〈 Q s i n 〉 / V θ / π ( β , L) = (3, 10) -0.006-0.0030.0000.0030.006 0 0.5 1 1.5 2 I m 〈 Q s i n 〉 / V θ / π ( β , L) = (12, 20)-0.06-0.04-0.020.000.02 0 0.5 1 1.5 2 χ θ / π ( β , L) = (3, 10) -0.016-0.012-0.008-0.0040.0000.004 0 0.5 1 1.5 2 χ θ / π ( β , L) = (12, 20) Figure 2: The results for various observables obtained by the CLM for the non-puncturedmodel with the sine definition Q sin . The average plaquette (Top), the imaginary part ofthe topological charge density (Middle), the topological susceptibility (Bottom) are plottedagainst θ for ( β, L ) = (3 ,
10) (Left) and (12 ,
20) (Right). The exact results for the same( β, L ) are shown by the dashed lines for comparison.11s defined by χ = 1 V (cid:0) h Q i − h Q i (cid:1) = − V ∂ ∂θ log Z , (3.16)which is real for all θ . In fact, the topological susceptibility χ is related to the topologicalcharge density (3.15) through χ = − i V ∂∂θ h Q i . (3.17)Note, however, that this relation can be violated if the CLM fails to calculate the expectationvalues correctly.In Fig. 2, we show the results obtained by the CLM for the non-punctured model.We also plot the exact results for comparison, which are derived in Appendix A.4. Inthe left column, we present our results for ( β, L ) = (3 , β, L ) = (12 , θ = 0 agree with the exact results for( β, L ) = (3 ,
10) but not for ( β, L ) = (12 , β even at θ = 0, where the sign problem is absent.Thus we find that the CLM with the naive implementation fails for both ( β, L ) = (3 , β, L ) = (12 ,
20) for different reasons. For ( β, L ) = (3 , β, L ) = (12 , In this section, we provide more in-depth discussions on the relationship between the ap-pearance of large drifts and the topology change in the non-punctured model. Let us firstrecall that the drift terms are given by (3.3) and (3.4), which depend on P n . When β islarge, the gauge action S g favors configurations with P n ∼ n , which implies thatthe drift terms are small.On the other hand, the notion of topological sectors can be defined by the real part of(2.11), which takes integer values, even for complexified configurations that are generatedin the CLM. In order for a transition between different topological sectors to occur, one ofthe plaquettes has to cross the branch cut; namely the phase of the plaquette has to jumpfrom − π to π or vice versa. When this occurs, large drift terms can appear as can be seen12 u t -2-1012 R e Q l og Figure 3: The results obtained by the CLM for the non-punctured model with the sinedefinition Q sin for ( β, L ) = (3 ,
10) with θ = π . The upper plot shows the history of thetopological charge Q log with the log definition, whereas the lower plot shows the history ofthe magnitude u of the drift term in the log scale.from Fig. 3, where we plot the histories of Re Q log and the magnitude of the drift term(3.12). We observe clear correlation between the large drift term and the topology change.We have also confirmed that the large drift term appears for the link variables composingthe plaquette that crosses the branch cut.In order to understand this observation better, we focus on a particular link variable U k, , and consider the corresponding drift term, which depends on the plaquettes P k and P k − ˆ2 sharing the link. For simplicity, we set P k − ˆ2 = 1 and consider the drift term v as afunction of P k v = β sin φ − i θ π (cos φ − , (3.18)where we have defined a complex parameter φ by φ = − i log P k . A large drift appearswhen | Im φ | → ∞ . In Fig. 4(Left), we plot the drift term as a flow diagram for β = θ = 1.Considering that the contribution of the drift term v to the change of φ at a Langevin stepis given by ∆ φ = − v ∆ t , we actually plot ( − v ) in the complex φ plane.In what follows we assume that β > θ/ π . Then we find from Eq. (3.18) that there aretwo fixed points corresponding to v = 0. One is φ = 0 and the other is φ = i log[( θ/ π + β ) / ( θ/ π − β )], which is close to ± π for β ≫ θ/ π . As one can see from Fig. 4(Left), thefixed point φ = 0 is attractive, which confirms that P k tends to become unity when β islarge. The other fixed point φ ∼ ± π is repulsive, and the magnitude | v | grows exponentially13 π - π π π - - ϕ I m ϕ | v ( φ ) | Im φ Figure 4: (Left) A flow diagram representing − v defined by (3.18) is shown as a functionof φ for β = θ = 1. (Right) The absolute value | v ( φ ) | is plotted against Im φ for Re φ = π .as one flows away in the imaginary direction; See Fig.4(Right). As we mentioned above,when the transition between topological sectors occurs, one of the plaquettes crosses thebranch cut, which corresponds to Re φ = ± π in the flow diagram. When this happens, theconfiguration can flow in the imaginary direction, which causes a large drift. Since the problem we encounter in the previous section occurs due to the topological natureof the θ term, a simple remedy would be to change the topology of the base manifold toa noncompact one. Here we consider introducing a puncture on the 2D torus. Once weintroduce a puncture, the drift term D n,µ S θ with the log definition of the topological chargehas nonzero contributions for the link variables surrounding the puncture, which enable usto include the effect of the θ term correctly in the CLM as we will see in Section 5. Therefore,for the rest of this paper, we basically use the log definition to simplify our discussions.Unlike the non-punctured model, the topological charge is no more restricted to integervalues, and it can be changed freely.Since the puncture affects the theory only locally, its effect is expected to die out in theinfinite volume limit for | θ | < π as we demonstrate explicitly in this section using the exactresults. Thus unless we are interested in a theory with a finite volume, the punctured modelis as good as the original model, the difference simply being a different choice of “boundaryconditions”. In fact, we will see that the non-punctured model has slow convergence to theinfinite volume limit for θ ∼ π , which is not the case in the punctured model.14 .1 defining the punctured model on the lattice There are various ways to introduce a puncture on the periodic lattice. Here we considerremoving a plaquette as a simple choice. More precisely, we define the punctured model byremoving one plaquette, let say P K , from the sum appearing in (2.8) and (2.11) when wedefine the action (2.13).As an alternative method, we have also tried introducing a slit at a particular link,which amounts to duplicating the corresponding link variable and including each of themin the plaquettes that share the link. The results turn out to be qualitatively the sameas the ones obtained by removing a plaquette. There are, of course, many others, but inany case, one can obtain exact results for a finite lattice as we explain in Appendix A, andusing them, one can demonstrate explicitly that the punctured model is equivalent to theoriginal non-punctured model in the infinite volume limit for | θ | < π . In this section, we show the equivalence of the non-punctured model and the puncturedmodel in the infinite volume limit. Here we use the log definition of the topological charge,but a similar statement holds as far as the same definition is used for the two models. The partition function for the non-punctured model is given by (See Appendix A.2 forderivation.) Z nonpunc = + ∞ X n = −∞ [ I ( n, θ, β )] V (4.1)for finite V = L , where the function I ( n, θ, β ) is defined by I ( n, θ, β ) = 12 π Z π − π dφ e β cos φ + i ( θ π − n ) φ . (4.2)Let us take the infinite volume limit V → ∞ , in which the sum over n in (4.1) is dominatedby the term that gives the largest absolute value |I ( n, θ, β ) | . This corresponds to the n that minimizes | θ π − n | . Thus in the infinite volume limit, the free energy is obtained aslim V →∞ V log Z nonpunc = log I (0 , ˜ θ, β ) , (4.3)where ˜ θ is defined by ˜ θ = θ − πk with the integer k chosen so that − π < ˜ θ ≤ π . In the case of the sine definition, the equivalence of the two models in the infinite volume limit holdsfor | θ | < θ c ( β ), where θ c ( β ) ∼ π { / (2 β ) } for large β . .9570.9580.9590.9600.961 0 0.5 1 1.5 2 w θ / π ( β , L) = (12, 10) nonpunctured L = 10nonpunctured L → ∞ punctured w θ / π ( β , L) = (12, 20) nonpunctured L = 20nonpunctured L → ∞ punctured -0.010-0.0050.0000.0050.0100.015 0 0.5 1 1.5 2 I m 〈 Q l og 〉 / V θ / π ( β , L) = (12, 10) nonpunctured L = 10nonpunctured L → ∞ punctured -0.010-0.0050.0000.0050.0100.015 0 0.5 1 1.5 2 I m 〈 Q l og 〉 / V θ / π ( β , L) = (12, 20) nonpunctured L = 20nonpunctured L → ∞ punctured -0.016-0.012-0.008-0.0040.0000.004 0 0.5 1 1.5 2 χ θ / π ( β , L) = (12, 10) nonpunctured L = 10nonpunctured L → ∞ punctured -0.016-0.012-0.008-0.0040.0000.004 0 0.5 1 1.5 2 χ θ / π ( β , L) = (12, 20) nonpunctured L = 20nonpunctured L → ∞ punctured Figure 5: The exact results for various observables obtained by using the log definition Q log of the topological charge. The average plaquette (Top), the imaginary part of thetopological charge density (Middle), the topological susceptibility (Bottom) obtained forthe non-punctured (solid line) and punctured (dashed line) models are plotted against θ for L = 10 (Left) and L = 20 (Right) with the same β = 12. Note that the results forthe punctured model are actually independent of L . For the non-punctured model, we alsoplot the results in the infinite volume limit L → ∞ with β = 12 by the dash-dotted linesfor comparison. 16n the other hand, the partition function for the punctured model is given by (SeeAppendix A.3 for derivation.) Z punc = [ I (0 , θ, β )] V (4.4)for finite V = L −
1, which implies that the free energy1 V log Z punc = log [ I (0 , θ, β )] (4.5)is actually V independent. Hence all the observables that can be derived from it has nofinite size effects. Note also that this model does not have the 2 π periodicity in θ . Bycomparing (4.3) and (4.5), one can see that the two models are equivalent in the infinitevolume limit for | θ | < π .The observables defined in Section 3.4 can be calculated for the two models using (4.1)and (4.4) by numerical integration (See Appendix A.4 for the details.). In Fig. 5, we plotthe average plaquette (Top) defined by (3.14), the imaginary part of the topological chargedensity (Middle) defined by (3.15) and the topological susceptibility (Bottom) defined by(3.16) for L = 10 (Left) and L = 20 (Right), respectively, with the same β = 12. Theresults for the two models tend to agree as L increases for | θ | < π .We can evaluate the free energy (4.5) for the punctured model more explicitly for large β , which is relevant in the continuum limit. By integrating over φ in Eq. (4.2) as I ( n, θ, β ) ≃ √ πβ e β − β ( θ π − n ) , (4.6)we get 1 V log Z punc ≃ β −
12 log 2 πβ − θ π β . (4.7)From this, we can obtain various observables for the punctured model as w ≃ − β + θ π β , (4.8) h Q i V ≃ iθ π β , (4.9) χ ≃ π β (4.10)for finite V , which explains the θ dependence observed in Fig. 5.From Fig. 5, we also find that the results for the non-punctured model have sizable finitevolume effects, in particular around θ ∼ π , which is absent in the punctured model. Whilethe volume independence of the punctured model may well be peculiar to the present 2Dgauge theory case, the advantage of the punctured model compared with the non-puncturedmodel from the viewpoint of finite volume effects may hold more generally.17 Application of the CLM to the punctured model
In this section, we apply the CLM to the punctured model using the log definition Q log of thetopological charge. Our results reproduce the exact results discussed in the previous sectionas long as we are close enough to the continuum limit. We also show that the topologyfreezing problem is circumvented without causing large drifts thanks to the puncture. We have discussed the drift terms in the non-punctured model in Section 3.1. For the punc-tured model, we only have to modify the drift terms for the four link variables surroundingthe puncture; i.e., U K, , U K +ˆ2 , , U K, and U K +ˆ1 , . Thus we obtain D n, S = − i β ( P n − P − n − P n − ˆ2 + P − n − ˆ2 ) for n = K, K + ˆ2 , − i β ( − P K − ˆ2 + P − K − ˆ2 ) + i θ π for n = K , − i β ( P K +ˆ2 − P − K +ˆ2 ) − i θ π for n = K + ˆ2 , (5.1) D n, S = − i β ( − P n + P − n + P n − ˆ1 − P − n − ˆ1 ) for n = K, K + ˆ1 , − i β ( P K − ˆ1 − P − K − ˆ1 ) − i θ π for n = K , − i β ( − P K +ˆ1 + P − K +ˆ1 ) + i θ π for n = K + ˆ1 , (5.2)where we have ignored the issue of δ -function discussed in Section 3.1. This is justified if allthe plaquettes in the action never cross the branch cut; i.e., | Im log P n | ≤ π − ǫ for ∀ n = K with a strictly positive ǫ during the Langevin simulation. We will see that this assumptionis justified at sufficiently large β in Section 5.3.Note that the drift term from the θ term appears only for the link variables surroundingthe puncture, and it is actually a constant independent of the configuration. While theseproperties are peculiar to the log definition Q log , similar properties hold also for the sinedefinition Q sin at large β , where all the plaquettes P n approach unity except for P K , whichcorresponds to the puncture. We discuss the case with the sine definition in Appendix B,where we see that the obtained results are qualitatively the same as those obtained withthe log definition. θ dependence of the partition function As we have seen in Section 4.2, the punctured model is equivalent to the non-puncturedmodel in the infinite volume limit for | θ | < π , beyond which the equivalence ceases to hold.In particular, the punctured model does not have the 2 π periodicity in θ , which exists inthe non-punctured model. 18 .00.10.20.30.40.5-3 -2 -1 0 1 2 3 Re Q log( β , L) = (3, 10) 0.00.10.20.30.40.5-3 -2 -1 0 1 2 3 Re Q log( β , L) = (12, 20) Figure 6: The topological charge distribution for θ = 0 obtained by the CLM for thepunctured model using the log definition Q log is plotted for ( β, L ) = (3 ,
10) (Left) and( β, L ) = (12 ,
20) (Right). The solid lines represent the exact results obtained by evaluating(5.4) using the partition function (4.4).In order to understand this point better, we discuss the θ dependence of the partitionfunction in this section. Let us first note that the partition function for arbitrary θ is relatedto the topological charge distribution ρ ( q ) for θ = 0 through Fourier transformation as Z ( θ ) = Z dU e − S g [ U ]+ iθQ [ U ] = Z dU e − S g [ U ] Z dq e iθq δ ( Q [ U ] − q )= Z (0) Z dq e iθq ρ ( q ) . (5.3)Therefore, the absence of the 2 π periodicity in θ in the punctured model is directly relatedto its property that the topological charge can take non-integer values even if we use thelog definition Q log . Going beyond the fundamental region − π < θ ≤ π simply amounts toprobing the fine structure of the topological charge distribution ρ ( q ), which is irrelevant inthe infinite volume limit.By making an inverse Fourier transform, we can obtain the topological charge distribu-tion ρ ( q ) for θ = 0 as ρ ( q ) = 1 Z (0) Z ∞−∞ dθ π Z ( θ ) e − iθq . (5.4)We calculate this quantity for the punctured model by the CLM for θ = 0. In Fig. 6,we show the results for ( β, L ) = (3 ,
10) (Left) and ( β, L ) = (12 ,
20) (Right), which agreewell with the exact results obtained by evaluating (5.4) using the partition function (4.4).Note that the calculation actually reduces to that of the real Langevin method due to the19bsence of the sign problem for θ = 0. We therefore have no concerns about the criterionfor correct convergence here.While the sign problem is absent for θ = 0, the topology freezing problem can still bean issue for large β . The agreement we see for ( β, L ) = (12 ,
20) confirms that this problemis resolved in the punctured model at least for θ = 0. In this section, we discuss the validity of the CLM for the punctured model. Fig. 7(Left)shows the histogram of the drift term for ( β, L ) = (3 ,
10) and ( β, L ) = (12 ,
20) with θ = π ,which are the parameters used in Section 3.4 for the non-punctured model. We find thatthe criterion is satisfied for ( β, L ) = (12 ,
20) but not for ( β, L ) = (3 , Q log obtained by theCLM for ( β, L ) = (12 ,
20) with θ = π . (The result for ( β, L ) = (3 ,
10) looks quite similarto this plot.) It is widely distributed within the range − . Re Q log .
3, which is insharp contrast to the plot in Fig. 1(Right) for the same ( β, L ) = (12 ,
20) in the case of thenon-punctured model. In fact, it turns out to be close to the exact result obtained for thesame ( β, L ) = (12 ,
20) with θ = 0, which is plotted in the same figure. Thus we find thatthe topology freezing problem at large β is circumvented in the punctured model and yetthe CLM remains valid.Next we discuss the reason why the punctured model can avoid the topology freezingproblem without causing large drifts. The difference from the non-punctured model is thatone of the plaquettes, P K , is removed from the action. Note that the topological charge Q log for the punctured model is given by Q log = − i π X n log P n + i π log P K , (5.5)where the first term is nothing but the topological charge defined for the non-puncturedmodel, whose real part takes integer values. The second term has a real part which lieswithin the interval [ − , ). Therefore it makes sense to define the “topology change” in thepunctured model as the situation in which the real part of the first term changes by ± Note, however, that precise agreement is not expected here since the histogram of Re Q log is a non-holomorphic quantity, for which the CLM does not allow a clear interpretation. -6 -4 -2
1 10 100 u ( β , L) = (3, 10)( β , L) = (12, 20) Re Q log( β , L) = (12, 20) Figure 7: The results obtained by the CLM for the punctured model using the log definition Q log of the topological charge. (Left) The histogram of the magnitude u of the drift termdefined by (3.12) is shown for ( β, L ) = (3 ,
10) and (12 ,
20) with θ = π . (Right) Thehistogram of Re Q log is shown for ( β, L ) = (12 ,
20) with θ = π . The exact result obtainedfor ( β, L ) = (12 ,
20) with θ = 0 is shown by the solid line for comparison.When β is large, this process is highly suppressed for all the plaquettes that are includedin the action. In the non-punctured model, the topology freezing problem occurs preciselyfor this reason. However, in the punctured model, the particular plaquette P K is removedfrom the action, and therefore it can change freely even for large β .This is demonstrated in Fig. 8, where we plot the probability distribution of the phaseof the plaquette P K as well as that of the other plaquettes P n ( n = K ) for ( β, L ) = (3 , β, L ) = (12 ,
20) (Right). We find that the phase of the removed plaquette P K is almost uniformly distributed for both ( β, L ). On the other hand, the distributionof the phase of the other plaquettes depends on ( β, L ). It has a compact support for( β, L ) = (12 ,
20) but not for ( β, L ) = (3 , P n ( n = K )does not occur at all. In the latter case, there is a small but finite distribution at the branchcut, which means that the value of β is not large enough to suppress the branch cut crossingof the plaquettes P n ( n = K ) completely.This is consistent with the fact that the histogram of the drift term has fast fall-off for( β, L ) = (12 ,
20) but not for ( β, L ) = (3 ,
10) considering the discussion given in Section 3.5.While the flow diagram in Fig. 4(Left) is obtained for the sine definition of the topologicalcharge, it looks similar for the log definition, which simply corresponds to setting θ = 0 in(3.18). Therefore, large drifts can appear when one of the plaquettes P n ( n = K ) crossesthe branch cut, which indeed occurs for ( β, L ) = (3 ,
10) also for the punctured model. For21 -3 -2 -1 -1 -0.5 0 0.5 1 Im( log P n ) / π ( β , L) = (3, 10) n = Kn ≠ K -6 -4 -2 -1 -0.5 0 0.5 1 Im( log P n ) / π ( β , L) = (12, 20) n = Kn ≠ K Figure 8: The distribution of the phase of the plaquettes is plotted for the puncturedmodel with the log definition (2.11) of the topological charge for ( β, L ) = (3 ,
10) (Left)and ( β, L ) = (12 ,
20) (Right) with θ = π . We show the results for the plaquette ( n = K )removed from the action and those for all the other plaquettes ( n = K ) separately.( β, L ) = (12 , P K to cross the branch cut freely, but all the plaquettes that areincluded in the action are forced to stay close to unity because of large β . This justifies ourassumption that the issue of δ -function can be neglected in deriving the drift terms (5.1)and (5.2). Since the plaquette P K does not appear in the drift terms, it does not cause largedrifts even if it crosses the branch cut. This makes it possible for the punctured model toavoid the topology freezing problem without causing large drifts.Let us next discuss how the unitarity norm (3.5) behaves in our complex Langevinsimulations. As we can see from (5.1) and (5.2), the link variables surrounding the puncturehave a drift term in the imaginary direction coming from the θ term. At each Langevinstep, two of the link variables are multiplied by e θ ∆ t/ π and the other two are multipliedby e − θ ∆ t/ π so that the removed plaquette is multiplied by e θ ∆ t/π due to this drift term.Therefore, there is a danger that the magnitude of these four link variables increases ordecreases exponentially and hence the unitarity norm (3.5) grows exponentially with theLangevin time.In Fig. 9, we plot the history of the unitarity norm (3.5) for various θ with ( β, L ) =(5 , β, L ). (Here and for the rest of this subsec-tion, we restrict ourselves to the parameter sets, for which the histogram of the drift termhas fast fall-off.) Indeed we observe an exponential growth at early stage, but the unitaritynorm actually saturates to a constant depending on θ at sufficiently long Langevin time.This saturation occurs since the non-unitarity of the four link variables surrounding the22 -6 -4 -2
0 60 120 180 240 300 360 N t θ = 1.0 πθ = 0.8 πθ = 0.6 πθ = 0.4 πθ = 0.2 π Figure 9: The history of the unitarity norm N is plotted for the punctured model with thelog definition (2.11) of the topological charge for various θ with ( β, L ) = (5 , S g , which tries to make each plaquette except the removed oneclose to unity. We find that thermalization of various observables can be achieved onlyafter the saturation of the unitarity norm.In fact, we find that the unitarity norm is not distributed uniformly on the lattice dueto the existence of the puncture, as is also expected from the above discussion. In order tosee this, we define the “local unitarity norm” by N ( n ) = 14 X ( k,µ ) ∈ P n n U ∗ k,µ U k,µ + ( U ∗ k,µ U k,µ ) − − o , (5.6)which is an average of the unitarity norm for the four link variables surrounding eachplaquette P n . The unitarity norm defined by (3.5) is simply an average of N ( n ) over allthe plaquettes including the removed one; namely N = L P n N ( n ). In Fig. 10(Left), weplot this quantity N ( n ) against n = ( n , n ) for ( β, L ) = (12 ,
20) with θ = π , where thepuncture is located at n = K = (10 , N ( K ) ∼ × . The plaquettes adjacent to the puncture have a local unitaritynorm ∼ . × . This implies that the unitarity norm is mostly dominated by the fourlink variables surrounding the puncture.The local unitarity norm N ( K ) at the puncture depends not only on θ but also on β and L . In Fig. 10(Right), we plot this value against x = | θ | V phys = | θ | L /β for various θ , β and L . All the data can be fitted to a single curve N ( K ) = x p exp( c x + c ), which reveals23
0 5 10 15 0 5 10 1510 -4 -2 n n N ( n )
0 50 100 150 200 N ( K ) | θ | L / β ( β , L) = (5, 16)( β , L) = (10, 16)( β , L) = (6, 20) x p exp( c x + c ) Figure 10: (Left) The local unitarity norm N ( n ) for each plaquette P n defined by (5.6) isplotted against n = ( n , n ) for the punctured model with the log definition (2.11) of thetopological charge for ( β, L ) = (12 ,
20) with θ = π . The removed plaquette corresponds to n = K = (10 ,
10) in this figure. (Right) The local unitarity norm N ( K ) for the removedplaquette P K obtained for various β , L and θ is plotted against x = | θ | L /β . The solid linerepresents a fit to x p exp( c x + c ) with c = 0 . c = − p = 0 .
0 5 10 15 0 5 10 150.920.940.960.981.00 n n | P n | Figure 11: The absolute value of the plaquette P n is plotted against n = ( n , n ) for thepunctured model with the log definition (2.11) of the topological charge for ( β, L ) = (12 , θ = π . The removed plaquette corresponds to n = K = (10 ,
10) in this figure.24n exponential behavior at large x .What actually matters for the validity of the CLM is not so much the local unitaritynorm N ( n ) as the absolute value of each plaquette P n , which we plot in Fig. 11 against n =( n , n ) for the same parameters as in Fig. 10(Left). The absolute value of P K correspondingto the removed plaquette is close to ( p N ( K )) ∼ . × , which implies that |U K, | , |U K +ˆ1 , | , |U − K +ˆ2 , | and |U − K, | are close to p N ( K ). Except for this removed plaquette, theabsolute value of the plaquette deviates only slightly from unity due to large β .In fact, this deviation of | P n | from unity for n = K has a physical meaning sinceIm h Q log i = − π P n = K h log | P n |i as one can see from (5.5). From the exact result (4.9)obtained at large β , we find that | P n | ∼ e − θ/ (2 πβ ) for n = K , which is ∼ .
96 for θ = π and β = 12 in agreement with the value observed in Fig. 11. If we flip the sign of θ , whichcorresponds to the parity transformation, we find that | P n | 7→ | P n | − for all n .Note also that P K does not appear in the drift term, which implies that its absolutevalue can become large without causing large drifts. We have confirmed that the criterionfor correct convergence is satisfied for sufficiently large β , and indeed the exact results forvarious observables can be reproduced correctly as we will see in the next section. Thisremains to be the case even for large θ and/or large V phys , where the unitarity norm becomeslarge. Thus the present model provides a counterexample to the common wisdom that theCLM fails when the unitarity norm becomes large.
In this section, we calculate the observables for the punctured model by the CLM andcompare our results with the exact results derived in Appendix A.4. Let us recall that, inthe definitions (3.14), (3.15) and (3.16), V denotes the number of plaquettes in the action,which is V = L − V phys by Eq. (2.14) not only for the non-punctured model but also for the punctured model,which simplifies the relationship between β and L for fixed V phys .In Fig. 12, we show our results for the average plaquette w (Top), the topologicalcharge (Middle) and the topological susceptibility (Bottom) against θ for ( β, L ) = (3 , ,
20) in the left and right columns, respectively, which correspond to a fixed physicalvolume V phys ≡ L /β = 10 /
3. The exact results obtained for the same model with the sameparameter sets are also shown for comparison. We find from our results for the average We find, however, that the fluctuation of the local unitarity norm is small even for the one N ( K ) atthe puncture, which implies that the distribution of each link variable has fast fall-off. Therefore, it issuggested that the problem due to the boundary terms discussed in Refs. [16, 17, 38, 39] does not occur. .820.840.86 0 0.5 1 w θ / π ( β , L) = (3, 10) 0.9570.9580.959 0 0.5 1 w θ / π ( β , L) = (12, 20)0.000.010.020.030.040.05 0 0.5 1 I m 〈 Q l og 〉 / V θ / π ( β , L) = (3, 10) 0.0000.0020.0040.0060.0080.010 0 0.5 1 I m 〈 Q l og 〉 / V θ / π ( β , L) = (12, 20)0.0000.0050.0100.0150.020 0 0.5 1 χ θ / π ( β , L) = (3, 10) 0.0000.0010.0020.0030.004 0 0.5 1 χ θ / π ( β , L) = (12, 20) Figure 12: The results for various observables obtained by the CLM for the puncturedmodel with the log definition Q log . The average plaquette (Top), the imaginary part ofthe topological charge density (Middle), the topological susceptibility (Bottom) are plottedagainst θ for ( β, L ) = (3 ,
10) (Left) and (12 ,
20) (Right). The exact results for the same( β, L ) are shown by the dashed lines for comparison.26laquette that the exact results are reproduced for ( β, L ) = (12 , β, L ) = (3 , β, L ) = (12 ,
20) but not for ( β, L ) = (3 , β, L ) = (12 ,
20) but also for ( β, L ) = (3 , β, L ) = (3 ,
10) is accidental,though, since the condition for correct convergence is not satisfied. The fact that the resultsof the CLM for the punctured model with ( β, L ) = (3 ,
10) is not as bad as those for thenon-punctured model with the same ( β, L ) shown in Fig. 2(Left) can be understood byconsidering that the effect of the θ term is included correctly by the drift terms for thelink variables composing the removed plaquette, but it is only the infrequent branch cutcrossing of the other plaquettes that spoils the validity of the CLM. In this paper, we have made an attempt to apply the CLM to gauge theories with a θ term.As a first step, we applied the CLM to the 2D U(1) case, which is exactly solvable on afinite lattice with various boundary conditions. We find that a naive implementation of themethod fails due to the topological nature of the θ term.While the gauge configurations are complexified in the CLM, one can still define thenotion of topological sectors by Re Q log ∈ Z . When a transition between different topo-logical sectors occurs, one of the plaquettes has to cross the branch cut inevitably, whichcauses the appearance of large drift terms. This indeed happens at small β , where we findthat the criterion for correct convergence of the CLM is not satisfied. Increasing β makesall the plaquettes close to unity. The large drift terms do not appear in this case, and thecriterion for correct convergence of the CLM is satisfied. However, the topology changedoes not occur during the simulation and the ergodicity is violated. This is analogous tothe topology freezing problem, which is known to occur for θ = 0. The results obtainedin this case correspond to the expectation values for an ensemble restricted to a particulartopological sector specified by the initial configuration.In order to avoid this problem, we have considered the punctured model, which canbe obtained by removing one plaquette from the action, both from the gauge action andfrom the θ term. While the quantity Re Q log is no more restricted to integer values, we canstill formally classify the complexified configurations into “topological sectors” by addingback the contribution of the removed plaquette to Re Q log . Even for large β , the removedplaquette can cross the branch cut easily, which results in frequent transitions between27ifferent “topological sectors”. Note also that, as far as β is sufficiently large, all the otherplaquettes are close to unity, and hence large drift terms do not appear. Thus the criterionfor correct convergence of the CLM can be satisfied by simply approaching the continuumlimit without causing the topology freezing problem. Indeed our results obtained by theCLM for the punctured model reproduce the exact results even at large θ .In the case of the punctured model, the drift term from the θ term appears only forthe link variables composing the removed plaquette, and it is given by ± i θ π , which causesmultiplication by a constant factor e ∓ ∆ t θ π to these link variables at each Langevin step. Thelocal unitarity norm of these link variables grows exponentially at early Langevin times,but it saturates at some point to some constant, which increases exponentially for large | θ | V phys . We have seen that the CLM works perfectly even in this situation as far as β issufficiently large. This provides a counterexample to the common wisdom that the CLMfails when the unitarity norm becomes large. Thus our results also give us new insightsinto the method itself.The punctured model is actually equivalent to the non-punctured model in the infinitevolume limit for | θ | < π . In that limit, the topological charge can take arbitrarily largevalues, so the discretization of Q to integers is no more important. This equivalence hasbeen confirmed explicitly by obtaining exact results for the punctured model. In fact, theexact results also reveal the absence of finite volume effects in the punctured model asopposed to the non-punctured model, which exhibits sizable finite volume effects around θ ∼ π . It is conceivable that the smearing of the topological charge somehow results in thereduction of finite volume effects. If so, a similar conclusion should hold more generally.We are currently working on the application of the CLM to the 4D gauge theory witha θ term. (See Ref. [74] for an earlier attempt.) Some preliminary results obtained in theSU(2) case look promising. We plan to investigate, in particular, the interesting phasestructure around θ = π predicted in Ref. [6]. Acknowledgements
We would like to thank H. Fukaya, S. Hashimoto, K. Hatakeyama, M. Honda, Y. Ito andS.M. Nishigaki for valuable discussions. The authors are also grateful to R. Kitano andN. Yamada for carefully reading our manuscript. The computations were carried out onthe PC clusters in KEK Computing Research Center and KEK Theory Center. Part of thiswork was completed through discussions during the workshop “Discrete Approaches to theDynamics of Fields and Space-Time 2019” held at Shimane University, Matsue, Japan.28
Derivation of the exact result
In 2D lattice gauge theory, we can obtain the partition function explicitly on any manifold atfinite lattice spacing and finite volume [62], from which various observables can be obtained.In this section, we review the derivation using the so-called K-functional [61].
A.1 the K-functional
Let us consider a lattice gauge theory with a θ term on a 2D lattice manifold M . Here wetake the gauge group to be U( N ), which is a generalization of U(1) considered so far. Notethat the topology of the gauge field becomes trivial for SU( N ) in 2D gauge theories.As a building block for evaluating the partition function, we define the K-functional K A for the region A ⊂ M defined by [61] K A (Γ) = Z Y U i ∈ A \ C dU i e − S A , (A.1)where the integral goes over the link variables inside A leaving out those on the boundary C . (See Fig. 13.) The action S A in Eq. (A.1) is given by S A = X P i ∈ A Tr (cid:20) − β (cid:0) P i + P − i (cid:1) − θ π log P i (cid:21) , (A.2)where the sum goes over the plaquettes P i included in the region A . Here we use the logdefinition (2.11) of the topological charge, but the results for the sine definition (2.12) canbe obtained in a similar manner as we mention at the end of Section A.4.The K-functional depends on the link variables on the boundary C = ∂A , but due tothe gauge invariance, it actually depends only onΓ = Y U i ∈ C U i , (A.3)which is a consecutive product of link variables along the loop C . The choice of the startingpoint of the loop C does not matter since a different choice simply corresponds to makinga gauge transformation of Γ, which leaves the K-functional invariant.We can calculate the K-functional for any A by gluing the K-functional for a singleplaquette P , which is nothing but K ( P ) = exp Tr (cid:20) β (cid:0) P + P − (cid:1) + θ π log P (cid:21) . (A.4)29 C Figure 13: An example of the region A , which has a boundary C = ∂A . The K-functionalfor this region is defined by integrating out the link variables represented by the dashedlines. The result depends on Γ defined by (A.3) for the loop C represented by the solid linewith arrows.Note here that (A.4) is a function of the group element P ∈ U( N ), which is invariant under P → gP g − ; g ∈ U( N ) . (A.5)It is known that any function having this property can be expressed by the so-called char-acter expansion K ( P ) = X r λ r χ r ( P ) , (A.6)which is analogous to the Fourier expansion for periodic functions. Here χ r ( P ) is the groupcharacter, which is defined by the trace of P for an irreducible representation r , and itsatisfies the orthogonality relation Z dU χ r ( U − ) χ r ( U ) = δ r ,r . (A.7)Using this relation, the coefficient λ r in the expansion (A.6) can be readily obtained as λ r = Z dU χ r ( U − ) K ( U ) . (A.8)As an example, let us obtain the K-functional K × for a 2 × P = U Ω and P = Ω − U as shown in Fig. 14. The group elements U and U are the products of three link variables, and Ω represents the link variable shared30 P Ω U U Figure 14: The K-functional K × for a 2 × P = U Ω and P = Ω − U , which are glued togetherby integrating out the shared link variable Ω.by P and P . Integrating out the shared link variable Ω, we get K × ( U U ) = Z d Ω K ( P ) K ( P )= X r ,r λ r λ r Z d Ω χ r ( U Ω) χ r (Ω − U )= X r d r (cid:18) λ r d r (cid:19) χ r ( U U ) , (A.9)where d r = χ r (1) is the dimension of the representation r and we have used a formula Z d Ω χ r ( U Ω) χ r (Ω − U ) = 1 d r χ r ( U U ) δ r ,r . (A.10)Iterating this procedure, we obtain the K-functional for any simply connected region A as K A (Γ) = X r d r (cid:18) λ r d r (cid:19) | A | χ r (Γ) , (A.11)where | A | is the number of plaquettes in A , and Γ is defined by (A.3).In the U(1) case, the representation can be labeled by the charge n ∈ Z , and thedimension of the representation is d n = 1 for ∀ n ∈ Z . Since the character for the plaquette P = e iφ is given by χ n ( P ) = e inφ , the K-functional for a single plaquette (A.6) reduces to K ( P ) = + ∞ X n = −∞ λ n e inφ , (A.12)where the coefficient λ n is a function of θ and β given explicitly as λ n ≡ I ( n, θ, β )= 12 π Z π − π dφ e − inφ K (cid:0) P = e iφ (cid:1) = 12 π Z π − π dφ exp (cid:20) β cos φ + i (cid:18) θ π − n (cid:19) φ (cid:21) (A.13)31sing (A.4) with P = e iφ . This function reduces to the modified Bessel function of the firstkind for θ = 0.The character expansion in the U( N ) case is more complicated, so we only show theend results referring the reader, for instance, to the appendix of Ref. [75] for the details.The representation of the U( N ) group is labeled by N integers ρ = ( ρ , ρ , · · · , ρ N ) ∈ Z N (A.14)satisfying ρ i ≥ ρ i +1 , and the dimension of the representation ρ can be calculated by d ρ = χ ρ (1) = N Y i>j (cid:18) − ρ i − ρ j i − j (cid:19) . (A.15)The coefficient λ ρ in (A.6) that corresponds to the representation ρ is expressed as a deter-minant λ ρ = det M ( ρ, θ, β ) , (A.16)where the matrix M ( ρ, θ, β ) is given as M jk ( ρ, θ, β ) = 12 π Z π − π dφ exp (cid:20) β cos φ + i (cid:18) θ π + ρ k + j − k (cid:19) φ (cid:21) , (A.17)which may be viewed as a generalization of (A.13). A.2 partition function for the non-punctured model
Let us evaluate the partition function for the 2D U( N ) lattice gauge theory on a torus. Forthat, we first consider the K-functional K L × L for a rectangle composed of V = L × L plaquettes, which can be expressed as (A.11). As is shown in Fig. 15, we identify the topand bottom sides represented by U − and U , respectively, and identify the left and rightsides represented by W − and W , respectively. Integrating out the group elements U and W , we obtain the partition function for the non-punctured model as Z nonpunc = Z dU dW K L × L ( U W U − W − )= X r d r (cid:18) λ r d r (cid:19) V Z dU dW χ r ( U W U − W − )= X r (cid:18) λ r d r (cid:19) V Z dU χ r ( U ) χ r ( U − )= X r (cid:18) λ r d r (cid:19) V , (A.18)32 UW − U − Figure 15: The partition function for the 2D U( N ) gauge theory on a torus is obtainedfrom the K-functional for the rectangle by integrating out the group elements U and W corresponding to the identified sides.where we have used the orthogonality relation (A.7) and a formula Z d Ω χ r ( U Ω W Ω − ) = 1 d r χ r ( U ) χ r ( W ) . (A.19)In the U(1) case, the partition function (A.18) reduces to Z nonpunc = + ∞ X n = −∞ [ I ( n, θ, β )] V . (A.20)As one can see from (A.13), the integral I ( n, θ, β ) has a property I ( n, θ + 2 πk, β ) = I ( n − k, θ, β ) for ∀ k ∈ Z , which guarantees the 2 π periodicity of (A.20) in θ .Let us consider taking the V → ∞ and β → ∞ limits simultaneously with fixed V phys ≡ V /β , which corresponds to the continuum limit. In this limit, the integral (A.13) can beevaluated as I ( n, θ, β ) ≃ √ πβ e β − β ( θ π − n ) . (A.21)Plugging this into (A.20), we obtain Z nonpunc ≃ (cid:18) e β √ πβ (cid:19) V + ∞ X n = −∞ exp " − V β (cid:18) θ π − n (cid:19) ∼ + ∞ X n = −∞ exp " − V phys (cid:18) θ π − n (cid:19) , (A.22)omitting the divergent constant factor. 33 .3 partition function for the punctured model Let us extend the calculation in the previous section to the punctured model. First, wecalculate the K-functional for a rectangle with a hole shown in Fig. 16, which we divideinto two regions A and A by cutting along two segments Ω and Ω . The outer and innerboundaries of the rectangle are divided into two segments ( U , U ) and ( ω , ω ), respectively.Then, the K-functional for each region is given, respectively, as K A ( U Ω ω Ω ) = X r d r (cid:18) λ r d r (cid:19) | A | χ r ( U Ω ω Ω ) , (A.23) K A (Ω − ω Ω − U ) = X r d r (cid:18) λ r d r (cid:19) | A | χ r (Ω − ω Ω − U ) . (A.24)By gluing the two regions A and A together at Ω and Ω , we obtain the K-functionalfor the rectangle with a hole as K A ∪ A = Z d Ω d Ω K A ( U Ω ω Ω ) K A (Ω − ω Ω − U )= X r ,r d r d r (cid:18) λ r d r (cid:19) | A | (cid:18) λ r d r (cid:19) | A | Z d Ω d Ω χ r ( U Ω ω Ω ) χ r (Ω − ω Ω − U )= X r d r (cid:18) λ r d r (cid:19) | A | (cid:18) λ r d r (cid:19) | A | Z d Ω χ r ( U Ω ω ω Ω − U )= X r (cid:18) λ r d r (cid:19) V χ r ( U U ) χ r ( ω ω ) , (A.25)where we have defined V = | A ∪ A | = | A | + | A | .Let us introduce the group elements U and W for the outer boundary as we did in Fig. 15so that U U = U W U − W − , and define ω = ω ω for the inner boundary. Integrating outthe group elements U and W , we obtain the K-functional for the punctured torus as K punc ( ω ) = X r (cid:18) λ r d r (cid:19) V χ r ( ω ) Z dU dW χ r ( U W U − W − )= X r (cid:18) λ r d r (cid:19) V χ r ( ω ) Z dU d r χ r ( U ) χ r ( U − )= X r d r (cid:18) λ r d r (cid:19) V χ r ( ω ) . (A.26)Finally, we integrate out the link variables surrounding the puncture to get the partition34 Ω ω Ω ω U A A Figure 16: The K-functional for a rectangle with a hole is obtained by gluing the two regions A and A . From this, the K-functional for the punctured torus is obtained similarly towhat we did in Fig. 15. Integrating out the link variables surrounding the puncture, weobtain the partition function for the 2D U( N ) gauge theory on a punctured torus.function for the punctured model as Z punc = Z dω K punc ( ω ) = X r d r (cid:18) λ r d r (cid:19) V δ r, = ( λ ) V , (A.27)where r = 0 corresponds to the trivial representation, which has d = 1.In the U(1) case, the partition function reduces to Z punc = [ I (0 , θ, β )] V , (A.28)which does not have the 2 π periodicity in θ .Let us consider taking the V → ∞ and β → ∞ limits simultaneously with fixed V phys ≡ V /β , which corresponds to the continuum limit. Similarly to the case of the non-puncturedmodel discussed in Section A.2, we obtain Z punc ≃ (cid:18) e β √ πβ (cid:19) V exp " − V β (cid:18) θ π (cid:19) ∼ exp " − V phys (cid:18) θ π (cid:19) , (A.29)omitting the divergent constant factor. This coincides with (A.22) in the V phys → ∞ limitfor | θ | < π . Note, however, that the equivalence between the punctured and non-puncturedmodels does not hold for finite V phys . 35 .4 evaluation of the observables We can evaluate the expectation values of various observables defined in Section 3.4 from thepartition function derived above, namely (A.20) for the non-punctured model and (A.28)for the punctured model. Since the latter case is easier due to the absence of an infinitesum, we only discuss the former case in what follows.The average plaquette w defined by (3.14) is given as w = 1 Z nonpunc + ∞ X n = −∞ A ( n, θ, β ) [ I ( n, θ, β )] V , (A.30)where we have defined A ( n, θ, β ) = ∂∂β log I ( n, θ, β )= 1 I ( n, θ, β ) 12 π Z π − π dφ cos φ exp (cid:20) β cos φ + i (cid:18) θ π − n (cid:19) φ (cid:21) = I ( n − , θ, β ) + I ( n + 1 , θ, β )2 I ( n, θ, β ) . (A.31)Similarly, the topological charge density defined by (3.15) can be obtained from h Q i = − i VZ nonpunc + ∞ X n = −∞ B ( n, θ, β ) [ I ( n, θ, β )] V , (A.32)where we have defined B ( n, θ, β ) = 1 I ( n, θ, β ) ∂∂θ I ( n, θ, β )= i I ( n, θ, β ) 14 π Z π − π dφ φ exp (cid:20) β cos φ + i (cid:18) θ π − n (cid:19) φ (cid:21) . (A.33)Finally, the topological susceptibility defined by (3.16) can be obtained from h Q i = − VZ nonpunc + ∞ X n = −∞ (cid:2) C ( n, θ, β ) + ( V − B ( n, θ, β ) (cid:3) [ I ( n, θ, β )] V , (A.34)where we have defined C ( n, θ, β ) = 1 I ( n, θ, β ) ∂ ∂θ I ( n, θ, β )= − I ( n, θ, β ) 18 π Z π − π dφ φ exp (cid:20) β cos φ + i (cid:18) θ π − n (cid:19) φ (cid:21) . (A.35)36ote that I ( n, θ, β ) and the functions (A.31), (A.33) and (A.35) derived from it are allreal-valued, and we can calculate them by numerical integration with sufficient precision.Also, when we evaluate the infinite sum in the expressions (A.30), (A.32) and (A.34), wehave to truncate it at some n . Note here that |I ( n, θ, β ) | vanishes quickly as | θ/ π − n | increases. We can therefore evaluate the infinite sum with sufficient precision by keepingonly a few terms when the lattice volume V is sufficiently large.In this section, we have derived the exact results for the log definition (2.11) of thetopological charge. As is clear from the derivation, we can obtain the exact results for thesine definition (2.12) by simply replacing I ( n, θ, β ) with˜ I ( n, θ, β ) = 12 π Z π − π dφ exp (cid:20) β cos φ + i θ π sin φ − inφ (cid:21) . (A.36) B The punctured model with the sine definition Q sinIn Sections 4 and 5, we have discussed the punctured model with the log definition (2.11)of the topological charge for simplicity. In fact, we can also use the sine definition (2.12) inthe punctured model. Here we discuss what happens in this case.The drift terms for the sine definition are given already for the non-punctured modelin Section 3.1. When we consider the punctured model, the only modification from thenon-punctured model appears in the drift terms for the four link variables surrounding thepuncture; i.e., U K, , U K +ˆ2 , , U K, and U K +ˆ1 , . Thus the drift terms are given as D n, S = − i β ( P n − P − n − P n − ˆ2 + P − n − ˆ2 ) − i θ π ( P n + P − n − P n − ˆ2 − P − n − ˆ2 )for n = K, K + ˆ2 , − i β ( − P K − ˆ2 + P − K − ˆ2 ) + i θ π ( P K − ˆ2 + P − K − ˆ2 ) for n = K , − i β ( P K +ˆ2 − P − K +ˆ2 ) − i θ π ( P K +ˆ2 + P − K +ˆ2 ) for n = K + ˆ2 , (B.1) D n, S = − i β ( − P n + P − n + P n − ˆ1 − P − n − ˆ1 ) − i θ π ( − P n − P − n + P n − ˆ1 + P − n − ˆ1 )for n = K, K + ˆ1 , − i β ( P K − ˆ1 − P − K − ˆ1 ) − i θ π ( P K − ˆ1 + P − K − ˆ1 ) for n = K , − i β ( − P K +ˆ1 + P − K +ˆ1 ) + i θ π ( P K +ˆ1 + P − K +ˆ1 ) for n = K + ˆ1 . (B.2)At large β , all the plaquettes except P K , namely the one that is removed, approach unity.The drift term from the θ term therefore vanishes for all the link variables except for thosesurrounding the puncture, which have constant drifts ± i θ π . Thus in the continuum limit,the drift terms for the sine definition agree with those for the log definition given by (5.1)and (5.2). This connection makes it easier to understand why we can safely ignore the issueof δ -function in the drift term for the log definition described in Section 3.1.37 -6 -4 -2
1 10 100 u ( β , L) = (3, 10)( β , L) = (12, 20) Re Q sin( β , L) = (12, 20) Figure 17: The results obtained by the CLM for the punctured model using the sine defi-nition of the topological charge. (Left) The histogram of the magnitude u of the drift termis shown for ( β, L ) = (3 ,
10) and (12 ,
20) with θ = π . (Right) The histogram of Re Q sin forthe punctured model is shown for ( β, L ) = (12 ,
20) with θ = π . The exact result obtainedfor ( β, L ) = (12 ,
20) with θ = 0 is shown by the solid line for comparison.It is therefore expected that the results of the CLM for the sine definition are essentiallythe same as those for the log definition for large β . In Fig. 17, we show our results for thepunctured model with the sine definition for the same ( β, L ) as those in Fig. 7 with the logdefinition. For ( β, L ) = (12 , u of the driftterm falls off rapidly, and that the histogram of Re Q sin obtained by the CLM is widelydistributed within the range − . Re Q sin .
3. Hence the topology freezing problem iscircumvented without causing large drifts similarly to the situation with the log definition.On the other hand, for ( β, L ) = (3 , u of the drift term falls off fast and that the condition for the validity of the CLM issatisfied unlike the case of the log definition. As a result, all the observables are in completeagreement with the exact results for all values of θ even with ( β, L ) = (3 , β, L ) as the ones used in Fig. 12. For ( β, L ) = (1 . , V phys ≡ L /β , however, we actually find that the histogram hasa power-law tail similarly to the case of the log definition. Therefore, the difference betweenthe two definitions is merely a small shift in the validity region of the CLM.We also show the exact results for the punctured model with the log and sine definitions,which tend to agree as β is increased with fixed V phys ≡ L /β , which corresponds to thecontinuum limit. 38 .820.840.86 0 0.5 1 w θ / π ( β , L) = (3, 10) logsin w θ / π ( β , L) = (12, 20) logsin I m 〈 Q 〉 / V θ / π ( β , L) = (3, 10) logsin I m 〈 Q 〉 / V θ / π ( β , L) = (12, 20) logsin χ θ / π ( β , L) = (3, 10) logsin χ θ / π ( β , L) = (12, 20) logsin Figure 18: The results for various observables obtained by the CLM for the puncturedmodel with the sine definition Q sin . The average plaquette (Top), the imaginary part ofthe topological charge density (Middle), the topological susceptibility (Bottom) are plottedagainst θ for ( β, L ) = (3 ,
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20) (Right). The exact results for the puncturedmodel with the log and sine definitions are shown for the same ( β, L ) by the dashed linesand the dash-dotted lines, respectively, for comparison.39 eferences [1] C. Baker et al.,
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