Block Decimation Renormalization Group and Finite Range Scaling Method to Analyze Infinitely Long Range Interacting 1-Dimensional Systems ̃)

To study dissipative quantum mechanics we adopt the Caldeira-Leggett model where environmental harmonic oscillators are coupled to the target variable. After integrating out the environmental degrees of freedom, effective interactions of infinitely long range appear. As the simplest example we take 2-state model for the target variable, and then we investigate the 1-dimensional Ising model with long range interactions. We propose a new practical method to evaluate the critical coupling constant of the system for the spontaneous magnetization. First, we exactly calculate the system with finite range interactions by formulating the block decimation renormalization group method. Then, we assume a finite range scaling and define its exponent for the logarithm of susceptibility. Using this exponent, we can find the criticality with a high precision through the zeta function singularity. We obtain the phase diagram on the 2-dimensional plane spanned by the damping rate exponent and the total coupling constant of the power damping long range interactions.