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 l
attice field theory with the $theta$ term. We reconsider this problem from the point of view of the maximum entropy method.
Lattice field theory with the $theta$ 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 me
thod (MEM) to Monte Carlo data obtained using the CP$^3$ model with the $theta$ term. For data without flattening, we obtain the most probable images of the partition function ${hat{cal Z}}(theta)$ 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 ${hat{cal Z}}(theta)$ are consistent with the FTM for small values of $theta$, while the behavior of ${hat{cal Z}}(theta)$ depends strongly on the default model for large values of $theta$. This behavior of ${hat{cal Z}}(theta)$ reflects the flattening phenomenon.
The weak coupling region of CP$^{N-1}$ lattice field theory with the $theta$-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 $h=-{rm Im} theta$ is studied, where $theta $ 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.
The sign problem is notorious in Monte Carlo simulations of lattice QCD with the finite density, lattice field theory (LFT) with a $theta$ term and quantum spin models. In this report, to deal with the sign problem, we apply the maximum entropy met
hod (MEM) to LFT with the $theta$ term and investigate to what extent the MEM is applicable to this issue. Based on this study, we also make a brief comment about lattice QCD with the finite density in terms of the MEM.
We study the sign problem in lattice field theory with a $theta$ term. We apply the maximum entropy method (MEM) to flattening phenomenon of the free energy density $f(theta)$, which originates from the sign problem. In our previous paper, we applied
the MEM by employing the Gaussian topological charge distribution $P(Q)$ as mock data. In the present paper, we consider models in which `true flattening of $f(theta)$ occurs. These may be regarded as good examples for studying whether the MEM could correctly detect non trivial phase structure.
We study the sign problem in lattice field theory with a $theta$ term, which reveals as flattening phenomenon of the free energy density $f(theta)$. We report the result of the MEM analysis, where such mock data are used that `true flattening of $f(t
heta)$ occurs. This is regarded as a simple model for studying whether the MEM could correctly detect non trivial phase structure in $theta$ space. We discuss how the MEM distinguishes fictitious and true flattening.
A $theta$ 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 $P(Q)$. This strategy, however, has a limitation, because errors o
f $P(Q)$ prevent one from calculating the partition function ${cal Z}(theta)$ properly for large volumes. This is called flattening. As an alternative approach to the Fourier method, we utilize the maximum entropy method (MEM) to calculate ${cal Z}(theta)$. We apply the MEM to Monte Carlo data of the CP$^3$ model. It is found that in the non-flattening case, the result of the MEM agrees with that of the Fourier transform, while in the flattening case, the MEM gives smooth ${cal Z}(theta)$.
In Monte Carlo simulations of lattice field theory with a $theta$ 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 proced
ure, however, causes flattening phenomenon of the free energy $f(theta)$, 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 $theta$ 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 $f(theta)$ 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.
