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Register a new userLattice Field Theory with the Sign Problem and the Maximum Entropy Method
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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.
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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.
In Monte Carlo simulation, lattice field theory with a $theta$ term suffers from the sign problem. This problem can be circumvented by Fourier-transforming the topological charge distribution $P(Q)$. Although this strategy works well for small lattice volume, effect of errors of $P(Q)$ becomes serious with increasing volume and prevents one from studying the phase structure. This is called flattening. As an alternative approach, we apply the maximum entropy method (MEM) to the Gaussian $P(Q)$. It is found that the flattening could be much improved by use of the MEM.
We study the mass spectra of excited baryons with the use of the lattice QCD simulations. We focus our attention on the problem of the level ordering between the positive-parity excited state N(1440) (the Roper resonance) and the negative-parity excited state N^*(1535). Nearly perfect parity projection is accomplished by combining the quark propagators with periodic and anti-periodic boundary conditions in the temporal direction. Then we extract the spectral functions from the lattice data by utilizing the maximum entropy method. We observe that the masses of the N and N^* states are close for wide range of the quark masses (M_pi=0.61-1.22 GeV), which is in contrast to the phenomenological prediction of the quark models. The role of the Wilson doublers in the baryonic spectral functions is also studied.
We investigate the sign problem in field theories by using the path optimization method with use of the neural network. For theories with the sign problem, integral in the complexified variable space is a promising approach to obtain a finite (non-zero) average phase factor. In the path optimization method, the imaginary part of variables are given as functions of the real part, $y_i=y_i({x})$, and are optimized to enhance the average phase factor. The feedforward neural network can be used to give and to optimize functions with many variables. The combined framework, the path optimization with use of the neural network, is applied to the complex $phi^4$ theory at finite density, the 0+1 dimensional QCD at finite density, and the Polyakov loop extended Nambu-Jona-Lasinio (PNJL) model, all of which have the sign problem. In these cases, the average phase factor is found to be enhanced significantly. In the complex $phi^4$ theory, it is demonstrated that the number density is calculated at a high precision. On the optimized path, the imaginary part is found to have strong correlation with the real part on the temporal nearest neighbor site. In the 0+1 dimensional QCD, we compare the results in two different treatments of the link variable: optimization after the diagonal gauge fixing and optimization without the diagonal gauge fixing. These two methods show consistent eigenvalue distribution of the link variables. In the PNJL model with homogeneous field ansatz, finite volume results approach the mean field results as expected, and the phase transition behavior can be described.
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 of $P(Q)$ 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 ${cal Z}(theta)$.
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