Although numerical simulation in lattice field theory is one of the most effective tools to study non-perturbative properties of field theories, it faces serious obstacles coming from the sign problem in some theories such as finite density QCD and lattice field theory with the θ term. We reconsider this problem from the point of view of the maximum entropy method

Imachi, M.

Shinno, Y.

Yoneyama, H.

Symmetry Integrability and Geometry Methods and Applications

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 018, 7 pagesLattice Field Theory with the Sign Problemand the Maximum Entropy Method?Masahiro IMACHI †, Yasuhiko SHINNO ‡ and Hiroshi YONEYAMA §† Kashiidai, Higashi-ku, Fukuoka, 813-0014, Japan‡ Takamatsu National College of Technology, Takamatsu 761-8058, Japan§ Department of Physics, Saga University, Saga, 840-8502, JapanE-mail: yoneyama@cc.saga-u.ac.jpReceived September 30, 2006, in final form January 19, 2007; Published online February 05, 2007Original article is available at http://www.emis.de/journals/SIGMA/2007/018/Abstract. Although numerical simulation in lattice field theory is one of the most effectivetools to study non-perturbative properties of field theories, it faces serious obstacles comingfrom the sign problem in some theories such as finite density QCD and lattice field theorywith the θ term. We reconsider this problem from the point of view of the maximum entropymethod.Key words: lattice field theory; sign problem; maximum entropy method2000 Mathematics Subject Classification: 65C05; 65C50; 68U20; 81T25; 81T801 IntroductionLattice field theory is a powerful method to study non-perturbative aspects of quantum fieldtheory. Although numerical simulation is one of the most effective tools to study non-perturba-tive properties of field theories, it faces serious obstacles in theories such as finite density QCDand lattice field theory with the θ term. This is because the Boltzmann weights are complex andthis makes it difficult to perform Monte Carlo (MC) simulations on a Euclidean lattice. Thisis the complex action problem, or the sign problem. In the present talk, we review the analysisof the sign problem based on the maximum entropy method (MEM) [1, 2, 3]. For details, referto [4, 5, 6]. The MEM is well known as a powerful tool for so-called ill-posed problems, where thenumber of parameters to be determined is much larger than the number of data points. It hasbeen applied to a wide range of fields, such as radio astrophysics and condensed matter physics.In this talk we deal only with lattice field theory with the θ term. It is believed that theθ term could affect the dynamics at low energy and the vacuum structure of QCD, but it isknown from experimental evidence that the value of θ is strongly suppressed in Nature. Fromthe theoretical point of view, the reason for this is not clear yet. Hence, it is important to studythe properties of QCD with the θ term to clarify the structure of the QCD vacuum [7, 8, 9]. Fortheories with the θ term, it has been pointed out that rich phase structures could be realizedin θ space. For example, the phase structure of the Z(N) gauge model was investigated usingfree energy arguments, and it was found that oblique confinement phases could occur [8, 9].In CPN−1 models, which have several dynamical properties in common with QCD, it has beenshown that a first-order phase transition exists at θ = pi [10, 11, 12].In order to circumvent the sign problem, the following method is conventionally employed [11,12]. The partition function Z(θ) can be obtained by Fourier-transforming the topological charge?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative andSymmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available athttp://www.emis.de/journals/SIGMA/LOR2006.html2 M. Imachi, Y. Shinno and H. Yoneyamadistribution P (Q), which is calculated with a real positive Boltzmann weight:Z(θ) =∫[dz¯dz]e−S+iθQˆ(z¯,z)∫[dz¯dz]e−S≡∑QeiθQP (Q),whereP (Q) ≡∫[dz¯dz]Qe−S∫dz¯dze−S. (1)The measure [dz¯dz]Q in equation (1) is such that the integral is restricted to configurations ofthe field z with topological charge Q. Also, S represents the action.In the study of CPN−1 models, it is known that this algorithm works well for a small latticevolume V and in the strong coupling region [10, 11, 13, 14]. As the volume increases or inthe weak coupling region, however, this strategy too suffers from the sign problem for θ ' pi.The error in P (Q) masks the true values of Z(θ) in the vicinity of θ = pi, and this results ina fictitious signal of a phase transition [13, 14]. This is called ‘flattening’, because the freeenergy becomes almost flat for θ larger than a certain value. This problem could be remediedby reducing the error in P (Q). This, however, is hopeless, because the amount of data neededto reduce the error to a given level increases exponentially with V .Here, we are interested in whether the MEM can be applied effectively to the study of theθ term and reconsider the flattening phenomenon of the free energy in terms of the MEM. TheMEM is based upon Bayes’ theorem. It derives the most probable parameters by utilizing datasets and our knowledge about these parameters in terms of the probability. The probabilitydistribution, which is called the posterior probability, is given by the product of the likelihoodfunction and the prior probability. The latter is represented by the Shannon-Jaynes entropy,which plays an important role to guarantee the uniqueness of the solution, and the former isgiven by χ2. It should be noted that artificial assumptions are not needed in the calculations,because the determination of a unique solution is carried out according to probability theory.Our task is to determine the image for which the posterior probability is maximized.We present the results for the analysis by i) using mock data and ii) using the MC data. Forthe former, we use the Gaussian form of P (Q). The Gaussian form is realized in many cases,such as the strong coupling region of the CPN−1 model and the 2-d U(1) gauge model. For usingMC data, we simulate the CPN−1 model and apply the MEM to the obtained data. This paperis organized as follows. In the following section, we give an overview of the origin of flatteningand summarize the procedure for the analysis of the MEM. The results obtained by use of theMEM are presented in Section 3.2 Flattening and MEM2.1 FlatteningThe free energy density f(θ) is calculated by Fourier-transforming P (Q) obtained by MC simu-lation. Let us call this method the FTM. The quantity f(θ) is defined asf(θ) = − 1Vln∑QP (Q)eiθQ,where V = L2, the square of the lattice size.The MC data for P (Q) consist of the true value, P˜ (Q), and its error, ∆P (Q). When the errorat Q = 0 dominates because of the exponential damping of P (Q), f(θ) is closely approximated byf(θ) ' − 1Vln[e−V f˜(θ) +∆P (0)],Lattice Field Theory with the Sign Problem and the Maximum Entropy Method 3where f˜(θ) is the true value of f(θ). Because f˜(θ) is an increasing function of θ, ∆P (0) dominatesfor large values of θ. If |∆P (0)| ' e−V f˜(θ) at θ = θf , then f(θ) becomes almost flat for θ >∼ θf .This is called “flattening of the free energy density”, and it has been misleadingly identified asa first order phase transition, because the first derivative of f(θ) appears to jump at θ = θf . Toavoid this problem, we must carry out the order of eV measurements in the FTM.2.2 MEMIn this subsection, we briefly explain the MEM in terms of the θ term. In a parameter inference,such as the χ2 fitting, the inverse Fourier transformP (Q) =∫ pi−pidθ2piZ(θ)e−iθQ (2)is used. In the numerical calculations, we use the discretized version of equation (2); P (Q) =∑nKQ,nZn, where KQ,n is the Fourier integral kernel and Zn ≡ Z(θn). In order for the con-tinuous function Z(θ) to be reconstructed, a sufficient number of values of θ, which we denoteby Nθ, is required so that the relation Nθ > NQ holds, where NQ represents the number of datapoints in P (Q) (Q = 0, 1, . . . , NQ − 1). A straightforward application of the χ2 fitting to thecase Nθ > NQ leads to degenerate solutions. This is an ill-posed problem.The maximum entropy method is suitable to solve this ill-posed problem, yielding a uniquesolution. The MEM is based upon Bayes’ theorem, expressed asprob(Z(θ)|P (Q), I) = prob(P (Q)|Z(θ), I) prob(Z(θ)|I)prob(P (Q)|I) ,where prob(A|B) is the conditional probability that A occurs under the condition that B occurs.The posterior probability prob(Z(θ)|P (Q), I) is the probability that the partition function Z(θ)is realized when the MC data {P (Q)} and prior information I are given. The likelihood functionprob(P (Q)|Z(θ), I) is given byprob(P (Q)|Z(θ), I) = 1XLe−12χ2 ,where XL is a normalization constant and χ2 is a standard χ2 function.The probability prob(Z(θ)|I) is given in terms of an entropy S asprob(Z(θ)|I) = 1Xs(α)e−αS ,where α and XS(α) are a positive parameter and an α-dependent normalization constant, re-spectively. As S, the Shannon–Jaynes entropy is conventionally employed:S =Nθ∑n=1[Zn −mn −Zn ln Znmn].Here mn ≡ m(θn) represents a default model.The posterior probability prob(Zn|P (Q), I), thus, is given byprob(Zn|P (Q), I, α,m) = 1XLXs(α)e−12χ2+αS ≡ eW [Z]XLXs(α).For the prior information I, we impose the criterionZn > 0, so that prob(Zn ≤ 0 | I, α,m) = 0.The most probable image of Zn, denoted as Zˆn, is calculated according to the following proce-dures [3, 4].4 M. Imachi, Y. Shinno and H. Yoneyama1. Maximizing W [Z] to obtain the most probable image Z(α)n for a given α:δδZnW [Z]∣∣∣Z=Z(α)=δδZn(−12χ2 + αS) ∣∣∣Z=Z(α)= 0.2. Averaging Z(α)n to obtain the α-independent most probable image Zn:Zˆn =∫dα Z(α)n prob(α|P (Q), I,m).The range of integration is determined so that the relationprob(α|P (Q), I,m) ≥ prob(αˆ|P (Q), I,m)/10holds, where prob(α|P (Q), I,m) is maximized at α = αˆ.3. Error estimation: The error of the most probable output image Zˆn is calculated as theuncertainty of the image, which takes into account the correlations of the images Zˆn amongvarious values of θn:〈(δZˆn)2〉 ≡∫dα 〈(δZ(α)n )2〉 prob(α|P (Q), I,m). (3)Here δZˆn and δZ(α)n represent the error in Zˆn and that in Z(α)n , respectively.3 Results3.1 Mock data: GaussianFirstly, we present the results of the analysis by using mock data. For this, we use the Gaus-sian P (Q) asP (Q) = A exp[− cVQ2],where, in the case of the 2-d U(1) gauge model, c is a constant depending on the inversecoupling constant β, and V is the lattice volume. The constant A is fixed so that∑QP (Q) = 1.The distribution P (Q) is analytically transformed by use of the Poisson sum formula into thepartition functionZpois(θ) = A√piVc∞∑n=−∞exp[−V4c(θ − 2pin)2]. (4)To prepare the mock data, we add noise with variance δ × P (Q) to the Gaussian P (Q). In theanalysis, we consider sets of data with various values of δ and study the effects of δ.In Fig. 1 the Gaussian topological charge distribution and corresponding f(θ) obtained byusing the FTM are shown for various lattice volumes. For small volumes, the behavior of f(θ)is smooth. For large volume (V = 50), however, clear flattening is observed. For V = 30, somedata are missing. This is due to the fact that Z(θ) could take negative values because of largeerrors. This is also called flattening, because its origin is the same as that stated above.For the P (Q) data corresponding to small volumes without flattening, the MEM successfullyreproduces f(θ). Fig. 2 displays Z(θ) for V = 50 by using the MEM. Here, the Gaussian defaultm(θ) = exp[−γ ln 10pi2θ2]with γ = 5.5 is used. The result of the MEM does not show flatteningbut smooth behavior, in agreement with the exact result calculated by equation (4) (Poisson).In contrast, the FTM yields flattening. See [4] for details.Lattice Field Theory with the Sign Problem and the Maximum Entropy Method 5Figure 1. Gaussian topological charge distribution and corresponding f(θ) obtained by using the Fouriermethod (FTM) for various lattice volumes [4]. The parameter δ is chosen to be 1/400.Figure 2. Zˆ(θ) (crosses) with the error bars for the Gaussian default model with γ = 5.5 [4]. Here,V = 50. Compared to the result of the FTM (circles), a remarkable improvement is clearly seen.3.2 MC dataIn this subsection, we apply the MEM to real Monte Carlo data by simulating the CP3 model.(For details, see [6].) For this we used a fixed point action [15, 16] and various lattice volumesL × L. Among these, we concentrate on the data for L = 38 as the non-flattening case andL = 50 as the flattening case. We systematically studied the flattening phenomenon by adoptinga variety of default model m(θ) and prior probability of the parameter α. For the latter, g(α)dependence appearing inprob(α|P (Q), I,m) ≡ P (α) ∝ g(α)eW (α)+Λ(α),is investigated. The function g(α) represents the prior probability of α and is chosen accordingto prior information. In general, two types of g(α) are employed, one according to Laplace’srule, gLap(α) = const, and one according to Jeffrey’s rule, gJef(α) = 1/α. The latter rule isdetermined by requiring that P (α) be invariant with respect to a change in scale, because α isa scale factor. The former rule means that we have no knowledge about the prior informationof α. In general, the most probable image Zˆ(θ) depends on g(α). We investigate the sensitivity6 M. Imachi, Y. Shinno and H. Yoneyama1e-081e-071e-061e-050.00010.0012 2.2 2.4 2.6 2.8 3 3.2Z(θ) for γ=5.0θLaplaceJeffrey1e-081e-071e-061e-050.00010.0012 2.2 2.4 2.6 2.8 3 3.2Z(θ) for γ=13.0θLaplaceJeffreyFigure 3. ZˆLap(θ) and ZˆJef(θ) for θ ∈ [2.0, pi] [6].1e-081e-071e-061e-050.00010.0010.010.110 0.5 1 1.5 2 2.5 3 3.5Z(θ)θγ=3.0γ=4.0γ=5.0γ=6.0γ=8.0γ=13.0m32/50(θ)m38/50(θ)Figure 4. The most probable images Zˆ(θ) for various m(θ) [6].of Zˆ(θ) to the choice of g(α) by studying a relative difference∆(θ) ≡ |ZˆLap(θ)− ZˆJef(θ)|(ZˆLap(θ) + ZˆJef(θ))/2, (5)where ZˆLap(θ) and ZˆJef(θ) represent the most probable images according to Laplace’s rule andJeffrey’s rule, respectively.In the case without flattening (L = 38), the MEM yielded images Zˆ(θ) that are almostindependent of m(θ) and g(α). The most probable images Zˆ(θ) are in agreement with the resultof the FTM within the errors.In the case with flattening (L = 50), we found that the statistical fluctuations of Zˆ(θ) becomesmaller as the number of measurements increases except near θ = pi. We also found that Zˆ(θ)with large errors depends strongly on g(α) in the region of large θ, where the g(α) dependenceof Zˆ(θ) was estimated using the quantity ∆(θ). For θ <∼ 2.3, Zˆ(θ) agrees with the result of theFTM. For θ >∼ 2.3, Zˆ(θ) behaves smoothly, while the FTM develops flattening. In Fig. 3 , wecompare Zˆ(θ) in larger values of θ for gLap(α) and gJef(α) when two different Gaussian defaultmodels are used (γ = 5.0 and 13.0). It is noted that γ = 5.0 is the case in which the smallestvalues of ∆(θ) in equation (5) are observed in this θ region among various default models.Our results are summarized in Fig. 4. All the results obtained using the MEM behavesmoothly over the entire range of θ. Errors are estimated from uncertainties of the imagesaccording to equation (3). For larger values of θ, Zˆ(θ) depends strongly on m(θ). Each of theseimages could be a candidate for the true image. This m(θ) dependence of Zˆ(θ) may reflect theLattice Field Theory with the Sign Problem and the Maximum Entropy Method 7flattening phenomenon. If we had proper knowledge about m(θ) as prior information, we couldidentify the true image in a probabilistic sense. Such knowledge may also allow us to clarifythe relationship between the default model dependence and the systematic error, which is notincluded in the figure. This will be a task to be pursued at the next stage.The MEM provides a probabilistic point of view in the study of theories with the sign problem.It may then be worthwhile to study lattice QCD with a finite density in terms of the MEM.AcknowledgementsOne of the authors (H.Y.) is grateful to the organizers of the O’Raifeartaigh Symposium onNon-Perturbative and Symmetry Methods in Field Theory for providing him an opportunity topresent a talk. He also thanks them for their warm hospitality. The symposium reminded himwell that he had enjoyed intensive discussions with Lochlainn and other colleagues (A. Wipf,especially) at DIAS.References[1] Bryan R.K., Maximum entropy analysis of oversampled data problems, Eur. Biophys. J. 18 (1990), 165–174.[2] Jarrell M., Gubernatis J.E., Bayesian inference and the analytic continuation of imaginary-time quantumMonte Carlo data, Phys. Rep. 269 (1996), 133–195.[3] Asakawa M., Hatsuda T., Nakahara Y., Maximum entropy analysis of the spectral functions in lattice QCD,Prog. Part. Nuclear Phys. 46 (2001), 459–508, hep-lat/0011040.[4] Imachi M., Shinno Y., Yoneyama H., Maximum entropy method approach to θ term, Progr. Theoret. Phys.111 (2004), 387–411, hep-lat/0309156.[5] Imachi M., Shinno Y., Yoneyama H., True or fictitious flattening?: MEM and the θ term, hep-lat/0506032.[6] Imachi M., Shinno Y., Yoneyama H., Sign problem and MEM in lattice field theory with the θ term, Progr.Theoret. Phys. 115 (2006), 931–949, hep-lat/0602009.[7] ’t Hooft G., Topology of the gauge condition and new confinement phases in non-Abelian gauge theories,Nuclear Phys. B 190 (1981), 455–478.[8] Cardy J.L., Rabinovici E., Phase structure of Zp models in the presence of a θ parameter, Nuclear Phys. B205 (1982), 1–16.[9] Cardy J. L., Duality and the parameter in Abelian lattice models, Nuclear Phys. B 205 (1982), 17–26.[10] Seiberg N., Topology in strong coupling, Phys. Rev. Lett. 53 (1984), 637–640.[11] Bhanot G., Rabinovici E., Seiberg N., Woit P., Lattice vacua, Nuclear Phys. B 230 (1984), 291–298.[12] Wiese U.-J., Numerical simulation of lattice θ-vacua: the 2-d U(1) gauge theory as a test case, NuclearPhys. B 318 (1989), 153–175.[13] Plefka J.C., Samuel S., Monte Carlo studies of two-dimensional systems with a θ term, Phys. Rev. D 56(1997), 44–54, hep-lat/9704016.[14] Imachi M., Kanou S., Yoneyama H., Two-dimensional CP2 model with θ term and topological chargedistributions, Progr. Theoret. Phys. 102 (1999), 653–670, hep-lat/9905035.[15] Hasenfratz P., Niedermayer F., Perfect lattice action for asymptotically free theories, Nuclear Phys. B 414(1994), 785–814,hep-lat/9308004.[16] Burkhalter R., Imachi M., Shinno Y., Yoneyama H., CPN−1 models with θ term and fixed point actionProgr. Theoret. Phys. 106 (2001), 613–640, hep-lat/0103016.

Lattice Field Theory with the Sign Problem and the Maximum Entropy Method

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Lattice Field Theory with the Sign Problem and the Maximum Entropy Method

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