A family of the diamond-type hierarchical lattices as a kind of fractals is proposed, on which the Ising model is exactly solved. The convergent condition of the free energy per bond in the thermo- dynamic limit is obviously given. We find that the unstable fixed point of the renormalization-group transformation moves toward K=0 (T=οο) from K=οο(T=O) as the number of branches P iS in. creased, and when P=οο(i.e., df=οο) the unstable fixed point K=O exactly, in contrast with that of the Bethe lattice, We also calculate the critical exponent of the correlation length; we find that in the df=2 and 3 cases the exponent is different from that of the regular lattice with d=2 and 3. re- s19ectively, whmh seems to imply that more general criteria for the ClaSSlnCatlOn of UnlVerSalltV should be oroposed. We have also discussed the upper crmcal fractal dlmensmn.

With a special Sierpfnski carpet (SC)，the Ising model is exactly solved by a combinatorial approach and graph technique. The rigorous partition function and free energy are obtained and the phase transi-tion is investigated. We argue that the existence of a phase transition strongly depends on the order of ramification of the SC. Our method is extended to deal with other lattices.

In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest. neighbor and next-nearest-neighbor interactions，for which an approximate solution of the magnetiza-tion per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery，Sarker，and Tobochnik [Phys. Rev, B 24, 5402 (1981)], our observation shows that its upper. bound value is the same as the known value, thus implying that z is independent of the range of the in-teraction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. the time-delayed correlation function 1S also calculated.

pin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspon-dences between Sf and Or and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and△).

In this paper, the single-spin transition dynamics is used to investigate the kinetic Gaussian model in a periodic external field. We first derive the fundamental dynamic equations, and then treat an isotropic d-dimensional hypercubic lattice Gaussian spin system with Fourier's transformation method. We obtain exactly the local magnetization and the equal-time pair-correlation function. The critical characteristics of the dynamical relaxation tq, the complex susceptibility x(v,q), and the dynamical response are discussed. The results show that the time evolution of the dynamical quantities and the dynamical responses of the system strongly depend on the frequency and the wave vector of the external field.