Critical Dynamics of the Ising Model with Ising Machine

-The dynamical critical exponent z of he two-dimensional Ising model with sequential-flip heat-bath dynamics is numerically estimated to be 2.2, which agrees with that of Metropolis dynamics. The dedicated machine "m-TIS" designed and constructed by the present authors is used for the simulations. The Monte barlo simulation is one of the most powerful methods in studying cooperative phenomena [I, 21. The recent developments of digital computers make large scale simulations possible. There are two approaches to large scale simulations. One is to use vector processors. In the following, we discuss the Monte Carlo simulations of the Ising models. The fastest simulation of the three-dimensional ferromagnetic-Ising model up to now records 0.85 G spin-trials per second on the HITAC S820/80 by Ito and Kanada [3]. The details of the use of vector processors are desribed in reference [3]. The other approach is to design, construct and use a dedicated (or special purpose) machine [4, 51. Recently, the present authors have designed and constructed a special purpose machines for the Ising spin systems named "m-TIS" [5]. This m-TIS works as a subroutine in simulation programs. When the local spin configuration is transfered from the host computer, it returns the next spin configuration. In one operation, 16 spins are simulated. It takes one clock cycle for the simulation of one spin and the presebt clock is 5 MHz. Therefore the maximum speed is 5M spins per second. The present host is EPSON PC-286, whose CPU is 10 MHz Intel 80286, and the data transfer is not fast enough to work the m-TIS at its peak speed. The present system can treat 2.2 M spins for square lattices and 2M spins for cubic lattices. With this machine, we have studied the dynamical critical exponent z of the square lattice Ising model with Metropolis sequential-flip dynamics [6]. The time correlation function of the magnetization CM (7) at temperatures higher than the critical point is defined length. This z is called the dynamical critical exponerit [7, 81. Instead of varing the temperature, the system size is varied at the critical point of the infinite system and the finite-size-scaling analysis is tried. The result was z = 2.132% 0.008. This z is estimated by other authors. R&z and Collins [9] have obtained z = 2.125 rt 0.01 from the high temperature expansion. Mori and Tsuda [lo] estimated z = 2.076 from simulations at temperatures near T,. All these values are consistent with each other, although the details of the dynamics are not the same. We have studied the difference of the dynamical nature between the heat-bath and the Metropolis dynamics. For this purpose, the correlation functions of the sequential flip dynamics with heat-bath algorithm are calculated. Our dynamics is described in the following. The sl?in flip is tried sequentially from the (1, 1)-site to (1, n)site and then from the (2, 1)-site to the (2, n)-site and SO on. The probability of the configuration (5) at Monte Carlo step t , P ( ( 0 ) , t) , is determined by where L is defined by L = L(n,n) 7 . . L(,,l) L(,-I,,) --. L(2.n) -.. L(2,1) L(l,n) ... L(1,l) (4) and