Properties of renormalization group transformations

Robert B. Griffiths

Li-Xiang Cen, Z. D. Wang

The property of quantum many-body systems under spatial reflection and the relevant physics of renormalization group (RG) procedure are revealed. By virtue of the matrix product state (MPS) representation, various attributes for translational invariant systems associated with spatial reflection are manifested. We demonstrate subsequently a conservation rule of the conjugative relation for reflectional MPS pairs under RG transformations and illustrate further the property of the fixed points of RG flows. Finally, we show that a similar rule exists with respect to the target states in the density matrix renormalization group algorithm.

M. S. Green

Zhu Zu-song

Gauss mode is allowed to take a continuous spin value,so the study of Gauss mode is of great value to understand better the critical properties of ferromagnets.In this paper,using the renormalization-group transformation technique,the phase transition and critical properties of the modified Gauss model on a type of hierarchical lattice are studied,and its fixed points is K=b33 obtained.In terms of the RG transformation and scaling theory,critical exponents α=0.758 and v=0.414 are obtained.

Гласко Андрей Владленович

Chin‐Kun Hu

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.