Phase Structures of Field Thoeries with Topological Term

采用蒙特卡罗（MC）模拟和实空间重整化群（RSRG）方法研究了具有拓扑项的晶格场理论的相结构。(1)在MC研究中，通过计算拓扑电荷分布P (Q)，克服了由于拓扑项的复杂作用而导致的复杂玻尔兹曼权值问题。配分函数是通过傅里叶变换P (Q)计算的。研究了两种情况下的CP^1模型；(i)标准作用和（ii）定点作用。(i)在强耦合区，P (Q)表现为高斯分布，其指数与1/V （V：系统体积）成正比。这显示了在=时的一阶相变。在弱耦合区域，跃迁消失。在- rhea平面上没有发现其他的相变特征。（ii）在FP作用的情况下，这种一阶相变只发生在强耦合极限的rheta=pi处。(2) RSRG研究已应用于(i) U(1)和（ii） U(2)两个维度的规范模型。为了得到RG方程的封闭形式，不可约字符展开是有用的。(i)在U(1)情况下，除了强耦合极限处的红外不动点外，还出现了非平凡的红外不动点。P (Q)是解析计算的，并表现出高斯特性。对于耦合常数的任意值，模型经历第1步。在=时的有序相变。这与MC模拟的结果一致。（ii）在U(2)情况下，研究了具有Wilson实作用和标准虚作用（阿贝尔）的模型。非平凡IR不动点出现在=pi处，如情形(i)。对于*，连续的RG变换在动作的虚部导出非阿贝尔表示。但是这些轨迹以IR固定点=0结束。

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.