Phase Structures of Field Thoeries with Topological Term

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Phase structures of lattice field theories with topological term are studied by means of (1) Monte Carlo (MC) simulations as well as (2) real space renormalization group (RSRG) method. (1) In MC study, the notorious problem of the complex Boltzmann weights due to the complex action of the topological term is overcome by calculating the topological charge distribution P (Q). The partition function is calculated by fourier transforming P (Q). CP^1 model is studied for two cases ; (i) standard action and (ii) fixed point (EP) action. (i) In the strong coupling regions, P (Q) behaves as gaussian with the exponent proportional to 1/V (V : volume of the system). This turns out to show the first order phase transition at rheta=pi. In the weak coupling regions, the transition disappears. In the beta-rheta plane no other signatures of phase transitions are found. (ii) In the case of the FP action, this 1st order phase transition occurs only at rheta=pi in the strong coupling limit.(2) RSRG study has applied to (i) U (1) and (ii) U (2) gauge models in two dimensions. Irreducible character expansions are useful in order to obtain the closed form of the RG equation. (i) In U (1) case, the non trivial infrared (IR) fixed point appears besides the IR fixed point at the strong coupling limit. P (Q) is calculated analytically and shows the gaussian behavior. For any value of the coupling constant, the model undergoes the 1st. order phase transition at rheta=pi. This agrees with the results of MC simulations. (ii) In U (2) case, model with the Wilson real action and the standard imaginary action (abelian) is studied. Non-trivial IR fixed point appears at rheta=pi as in the case (i). For rheta*pi, successive RG transformations induce the non-abelian representations in the imaginary part of the action. But these trajectories end up with the IR fixed point at beta=0.