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Register a new userThe $theta$-term, CP$^{N-1}$ Model and the Inversion Approach in the Imaginary $theta$ Method
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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.
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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 ${cal Z}(theta)$.
The topological charge distribution P(Q) is calculated for lattice ${rm CP}^{N-1}$ models. In order to suppress lattice cut-off effects we employ a fixed point (FP) action. Through transformation of P(Q) we calculate the free energy $F(theta)$ as a function of the $theta$ parameter. For N=4, scaling behavior is observed for P(Q), $F(theta)$ as well as the correlation lengths $xi(Q)$. For N=2, however, scaling behavior is not observed as expected. For comparison, we also make a calculation for the ${rm CP}^{3}$ model with standard action. We furthermore pay special attention to the behavior of P(Q) in order to investigate the dynamics of instantons. For that purpose, we carefully look at behavior of $gamma_{it eff}$, which is an effective power of P(Q)($sim exp(-CQ^{gamma_{it eff}})$), and reflects the local behavior of P(Q) as a function of Q. We study $gamma_{it eff}$ for two cases, one of which is the dilute gas approximation based on the Poisson distribution of instantons and the other is the Debye-Huckel approximation of instanton quarks. In both cases we find similar behavior to the one observed in numerical simulations.
We numerically study the phase structure of the CP(1) model in the presence of a topological $theta$-term, a regime afflicted by the sign problem for conventional lattice Monte Carlo simulations. Using a bond-weighted Tensor Renormalization Group method, we compute the free energy for inverse couplings ranging from $0leq beta leq 1.1$ and find a CP-violating, first-order phase transition at $theta=pi$. In contrast to previous findings, our numerical results provide no evidence for a critical coupling $beta_c<1.1$ above which a second-order phase transition emerges at $theta=pi$ and/or the first-order transition line bifurcates at $theta eqpi$. If such a critical coupling exists, as suggested by Haldanes conjecture, our study indicates that is larger than $beta_c>1.1$.
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 procedure, 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.
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.
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