The Potts antiferromagnet situated on the fractal family of diamond-type hierarchical lattices is investigated The exact analysis of our findings reveals that the algebraically ordered behavior pre- dicted by Berker and Kadanoff can exist on such lattices with L=odd, where L is the structure pa- rameter of the hierarchical lattices; however, similar behavior to that of the antiferromagnetic Potts model on a decorated square lattice is exhibited for systems with L=even. We find that the different constructions that have the same ffactal dimensionality can have a different cutoff value q 0, in contrast with the translationally invariant system. We also give the ground-state entropy at q=qo in a closed form for the systems with L=even.

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.

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.

We have studied the kinetic phase transition of a three-state-vector Potts system on a triangular lattice in contact with a heat bath and subject to an external energy source. The system evolves via the stochas-tic dynamics composed of Glauber and Kawasaki dynamics simulating interaction with the heat bath and energy source, respectively. The pair approximation IS used. It 1S found that there IS a paramagnetic-ferromagnetic transition at small energy flow, and the transition line is obtained. Also, when the energy is large enough, the system always will be in the paramagnetic state.

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.