Hiroshi Takano

Migdal-Kadanoff renormalization group approach to the spin-1/2 anisotropic Heisenberg model

The spin-1/2 anisotropic Heisenberg model is studied by generalizing the Migdal-Kadanoff renormalization transformations to quantum spin systems. An approximate one-dimensional decimation is employed besides the potential-moving approximation in this generalization. It is shown that these approximations are valid at high temperatures. The results obtained from these approximations suggest that the two-dimensional spin-1/2X-Y model shows the critical behavior similar to that expected for the classicalX-Y and planar models.

Migdal renormalization group approach to quantum spin systems

Abstract The Migdal RG approximation is extended to quantum spin systems such as the Heisenberg and XY-models. This yields the non-existence of phase transition in the two-dimensional Heisenberg model. The phase transition of the two-dimensional XY-model is also studied.

Finite-Size Scaling Approach to the Kinetic Ising Model

The kinetic Ising model on a square lattice is studied by means of the finite· size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical exponent is universal for certain types of transition pr:obabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z~2.2±O.1.

Real-Space Renormalization Group Approach to the Kinetic Ising Model

A systematic real·space dynamic renormalization group method is proposed and it is applied to the kinetic Ising model on a triangular lattice. The coarse grained master equation is constructed by using the memory function formalism and the Markov approximation. The high temperature series renormalization group method proposed by Betts, Cuthiell and Plischke is extended for the present purpose to make a perturbation expansion in a self·consistent way with respect to all orders of interactions. Terms nonlocal in space appear in the coarse grained time-evolution operator in our approach, because of the memory effect accompanied with the coarse graining procedure. It is shown within the second order calculation that these nonlocal terms are irrelevant around the nontrivial fixed point. The values of the static exponent v and dynamic exponent z are given by 1/v=0.99 and z=2.23 from the present second order calculation. These results are quite reasonable.

Growth properties of Eden model with acceleration sites

One plus one dimensional growth of an Eden model with acceleration sites is investigated by simulations, where the acceleration sites which are distributed at random before the process starts become immediately Eden cells if the surface of Eden cluster touches them. The critical concentration of acceleration sites where the growth rate of the average cluster height diverges is found as pc=0.592±0.005 corresponding to the site percolation threshold of the square lattice. The exponent which characterizes this divergence near the percolation threshold have been found as ν=1.33±0.08. An effective roughness exponent α which characterizes the surface morphology is found to belong to the same universality class as the Eden model for p<pc. At the critical concentration, the present system changes to hold a self-similar surface.

Renormalization group approach to the one-dimensional hubbard model

An approximate decimation method is applied to the one-dimensional half-filled Hubbard model. The specific heat, the entropy and the magnitude of local moments are calculated. The results are in good agreement with those obtained by Shiba at high temperatures.

SURFACE GROWTH OF BINARY EDEN MODEL AT PERCOLATION THRESHOLD CONCENTRATION

We extend the Eden model to the binary system of two kinds of sites with different noise reduction parameters n and m(⩾n). The new model called binary Eden model connects the original Eden model (n=m=1) to the percolation model (n=1 and m=∞). From our numerical simulation on a square lattice of a lattice edge width L, the surface width W(L,m) is found to satisfy the scaling relation at the percolation threshold concentration. Moreover, the surface length s(L,m) is also found to obey the same type scaling relation.

Fractal property of Eden growth morphology with acceleration effect

Fundamental growth process of Eden model has been used in many simulational studies of growing surfaces and the roughness exponent αE=12 and growth exponent βE=13[2] are well known. In this study we have added acceleration effect to the Eden model and examined universality class. We have simulated the growing Eden surface for several concentrations of acceleration sites p so as to observe changing morphology with p. As the simulation is limited to a finite lattice, we have to estimate exponents for infinite lattice size by simulating several lattice sizes Lx×Ly. At the conclusion we have estimated roughness and growth exponents for infinite lattice size and obtained αA∞≃αE=12 and βA∞≃βE=13 which are similar to those of normal Eden model for all concentrations of acceleration sites.