An Exact Renormalization Group Transformation and Properties of Positive Symmetry Matrices

胡進錕(Chin-Kun Hu)

《Chinese Journal of Physics》 21卷3&4期 (1983/12) Pp. 24-30

並列摘要

A positive symmetric m×m matrix P can be interpreted as the transfer matrix for an one-dimensional array of particles which have m-discrete energy levels and the most general nearest neighbor interactions. It is shown that P^1=P^2 can be considered as exact renormalization group (RG) transformation applied on such one dimensional system which does not have any phase transitions for T＞0. The exact RG transformation can be iterated to reach the high temperature fixed point. From the known asymptotic behaviour of RG transformations near the high temperature fixed point, we can derive and have a physical picture of properties of P^(2n) as n→∞, which can also be derived directly from the matrix theory. We also show that the fixed point of the RG transformations can be obtained directly from the eigenvector corresponding to the largest eigenvalue of P. The formulation is applied to the spin-j-Ising model as an illustration.