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Register a new userSign problem and MEM in lattice field theory with the $theta$ term
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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 method (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.
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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 method (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.
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 $theta$ term. We reconsider this problem from the point of view of the maximum entropy method.
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 $f(theta)$ 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.
We discuss a new strategy for treating the complex action problem of lattice field theories with a $theta$-term based on density of states (DoS) methods. The key ingredient is to use open boundary conditions where the topological charge is not quantized to integers and the density of states is sufficiently well behaved such that it can be computed precisely with recently developed DoS techniques. After a general discussion of the approach and the role of the boundary conditions, we analyze the method for 2-d U(1) lattice gauge theory with a $theta$-term, a model that can be solved in closed form. We show that in the continuum limit periodic and open boundary conditions describe the same physics and derive the DoS, demonstrating that only for open boundary conditions the density is sufficiently well behaved for a numerical evaluation. We conclude our proof of principle analysis with a small test simulation where we numerically compute the density and compare it with the analytical result.
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