The topological charge distribution P (Q) is calculated for lattice CP N−1 models. In order to suppress lattice cut-off effects, we employ a fixed point (FP) action. Through transformation of P (Q), we calculate the free energy F (θ) as a function of the θ parameter. For N=4, scaling behavior is observed for P (Q) and F (θ), as well as the correlation lengths ξ(Q). For N=2, however, scaling behavior is not observed, as expected. For comparison, we also make a calculation for the CP 3 model with a standard action. We furthermore pay special attention to the behavior of P (Q) in order to investigate the dynamics of instantons. For this purpose, we carefully consider the behavior of γ eff , which is an effective power of P (Q) (∼ exp(−CQ γ eff )), and reflects the local behavior of P (Q) as a function of Q. We study γ eff for two cases, the dilute gas approximation based on the Poisson distribution of instantons and the Debye-Hückel approximation of instanton quarks. In both cases, we find behavior similar to that observed in numerical simulations. §1. IntroductionIt is interesting to study the phase structure of asymptotic free theories such as QCD and the CP N −1 model. Non-perturbative studies of the phase structure of such theories are necessary in order to understand why effects of the topological term (θ term) are suppressed in Nature. The θ term affects the dynamics at low energy and is expected to lead to rich phase structures. 1) Actually, in the Z(N) gauge model, it has been shown by use of free energy arguments that oblique confinement phases could emerge and that an interesting phase structure may be realized. 2) In this paper we are concerned with the dynamics of the θ vacuum of CP N −1 models with a topological term, which have several dynamical properties in common with QCD. We believe that study of the two-dimensional model will be useful in acquiring information about realistic physics.From the numerical point of view, the topological term introduces a complex Boltzmann weight in the Euclidean lattice path integral formalism. The complex nature of the weight prevents one from straightforwardly applying the standard algorithm used for Monte Carlo simulations. This problem can be circumvented * )

In Monte Carlo simulations of lattice field theory with a θ term, one confronts the complex weight problem, or the sign problem. This is circumvented by performing the Fourier transform of the topological charge distribution P (Q). This procedure, however, causes flattening phenomenon of the free energy f (θ), which makes study of the phase structure unfeasible. In order to treat this problem, we apply the maximum entropy method (MEM) to a Gaussian form of P (Q), which serves as a good example to test whether the MEM can be applied effectively to the θ term. We study the case with flattening as well as that without flattening. In the latter case, the results of the MEM agree with those obtained from the direct application of the Fourier transform. For the former, the MEM gives a smoother f (θ) than that of the Fourier transform. Among various default models investigated, the images which yield the least error do not show flattening, although some others cannot be excluded given the uncertainty related to statistical error. * )

Partition function zeros provide alternative approach to study phase structure of finite density QCD. The structure of the Lee-Yang edge singularities associated with the zeros in the complex chemical potential plane has a strong influence on the real axis of the chemical potential. In order to investigate what the singularities are like in a concrete form, we resort to an effective theory based on a mean filed approach in the vicinity of the critical point. The crossover is identified as a real part of the singular point. We consider the complex effective potential and explicitly study the behavior of its extrema in the complex order parameter plane in order to see how the Stokes lines are associated with the singularity. Susceptibilities in the complex plane are also discussed.

The weak coupling region of CP N −1 lattice field theory with the θ-term is investigated. Both the usual real theta method and the imaginary theta method are studied. The latter was first proposed by Bhanot and David. Azcoiti et al. proposed an inversion approach based on the imaginary theta method. The role of the inversion approach is investigated in this paper. A wide range of values of h = −Imθ is studied, where θ denotes the magnitude of the topological term.Step-like behavior in the x-h relation (where x = Q/V , Q is the topological charge, and V is the two-dimensional volume) is found in the weak coupling region. The physical meaning of the position of the step-like behavior is discussed. The inversion approach is applied to weak coupling regions. * )

Lattice field theory with the θ term suffers from the sign problem. The sign problem appears as flattening of the free energy. As an alternative to the conventional method, the Fourier transform method (FTM), we apply the maximum entropy method (MEM) to Monte Carlo data obtained using the CP 3 model with the θ term. For data without flattening, we obtain the most probable images of the partition functionẐ(θ) with rather small errors. The results are quantitatively close to the result obtained with the FTM. Motivated by this fact, we systematically investigate flattening in terms of the MEM. Obtained imagesẐ(θ) are consistent with the FTM for small values of θ, while the behavior ofẐ(θ) depends strongly on the default model for large values of θ. This behavior ofẐ(θ) reflects the flattening phenomenon. * ) Although this approach works well for small lattice volumes and/or in the strong coupling region, it does not work for large volumes and/or in the weak coupling region, due to flattening of the free energy density f (θ). This flattening phenomenon results from the error in P (Q) obtained using Monte Carlo (MC) simulations and leads to a spurious phase transition for θ = θ f (< π). This is the sign problem. To overcome this problem requires exponentially increasing statistics.As an alternative approach to the FTM, 8), 9) we have applied the maximum entropy method (MEM) to the treatment of the sign problem. This method is based upon Bayes' theorem, and it has been widely used in various fields. The MEM gives the most probable images by utilizing sets of data and our knowledge about these images. The probability distribution function, called the posterior probability, is given by the product of the likelihood function and the prior probability. The latter represents our knowledge about the image and the former indicates how data points are distributed around the true values. The prior probability is given as an entropy term, which plays the essential role to guarantee the uniqueness of the solution.In order to investigate whether the MEM is applicable to the sign problem, we applied it to mock data, which were prepared by adding Gaussian noise to a model. In Ref. 21), a Gaussian P (Q) was used. The corresponding free energy can be calculated analytically using the Poisson sum formula. As mock data, data with flattening and without flattening were prepared. We found that in both cases, the MEM reproduced smooth f (θ). The values of obtained f (θ) are in agreement with exact ones and the errors are reasonably small compared to those resulting when using the Fourier transform. These results might lead one to believe that the MEM has the effect of smoothing data and that, for this reason, it is not a suitable technique for detecting singular behaviors, such as phase transitions. To determine whether this is indeed the case, we analyzed some toy models that exhibit singular behavior originating from the characteristics of the models themselves. 22) We found in Ref. 22) that in fact, the MEM can detect such singular beha...