Masahiro Imachi

CPN−1 Models with a θ Term and Fixed Point Action

The topological charge distribution P (Q) is calculatedfor lattice CP N −1 models. In order to suppress lattice cutoff 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 observedfor 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 γeff , which is an effective power of P (Q )( ∼ exp(−CQ γ eff )), andreflects the local behavior of P (Q) as a function of Q .W e study γeff for two cases, the dilute gas approximation based on the Poisson distribution of

Two-Dimensional CP2 Model with θ-Term and Topological Charge Distributions

Topological charge distributions in the two-dimensional CP 2 model with the θ-term are calculated. In strong coupling regions, the topological charge distribution is approximately given by a Gaussian form as a function of the topological charge, and this behavior leads to a first order phase transition at θ = π. In weak coupling regions, this distribution exhibits nonGaussian form, and the first order phase transition disappears. The free energy as a function of θ displays “flattening” behavior at θ = θf <π when we calculate the free energy directly from the topological charge distribution. A possible origin of this flattening phenomenon is proposed.

Maximum Entropy Method Approach to the θ Term

In Monte Carlo simulations of lattice field theory with a 0 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.

The Sign Problem and MEM in Lattice Field Theory with the θ Term

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 function ˆ Z(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 images ˆ Z(θ )a re consistent with the FTM for small values of θ, while the behavior of ˆ Z(θ) depends strongly on the default model for large values of θ .T his behavior ofˆ Z(θ) reflects the flattening phenomenon.

The θ-term, CPN-1 model and the inversion approach in the imaginary θ method

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.

CPN−1 model with the theta term and maximum entropy method

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Application of maximum entropy method to lattice field theory with a topological term

\theta

CP(N-1) model and topological term☆

term in lattice field theory causes the sign problem in Monte Carlo simulations. This problem can be circumvented by Fourier-transforming the topological charge distribution

Phase structure of CPN−1 model with topological term

P(Q)

Prior probability and the most probable image in MEM(Fundamental Problems and Applications of Quantum Field Theory)

. This strategy, however, has a limitation, because errors of