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

Abstract

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...

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