Analysis of Higher Dimensional Systems by use of the Tensor Product Variational Approach

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We have developed a new numerical renormalization group (RG) method for two-dimensional (2D) quantum systems and 3D classical systems, using a variational state represented as a product of local weights. As an example, we employed 2D IRF model, that contains 3 parameters, as a variational state for the square lattice S=1/2 XXZ model, which is one of the representative 2D quantum systems. We obtained a good energy estimate, even though we have only 3 parameters. In this case the variational formulation works better in the anisotroplc limit, the XY mdoel.For the application for 3D Classical systems, we choose 3D lsing model as a reference system, and prepare a variational state that contains 162 variational parameters. In this case the local factor has auxiliary spin variable, that can be interpreted as the renormalized spin. Since there are so many parameters, one has to survey them automaticaliy. For this purpouse we developed a self-consistent equation for the local weight, and improv … More e them vie iterative numerical procedure. As a result, we showed that the phase transition temperature is obtained accurately within the error of 1%.In the above cases, the variational state is uniform. This Is because CTMRG, the numerical RG method we have used, can treat uniform 2D models only. In order to improve this restriction, we considered a usage of the density matrix renormalization group (DMRG) for the system that have non-uniform ground state. As an example, we have started calculations of thermal state hi the ANNNI model, which has been considered to have complex structure in the ordered phase that appears in the Intermediate temperature. At present, we get a partial phase diagram, that suggests the suppression of the area of comensulate phase.As a bi-product of these researches, we unexpectedly get a new usage of CTMRG for the stochastic systems. The system has a kind of speed of light and information can be transferred within the light cone. We find a new targeting scheme for this case, and proposed a new numerical RG method, "the Ught Cone CTMRG method". Depending on the parameter condition the method becomes instable, and to improve this drawback is one of the task in the future studies Less