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Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and -dimensional Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality (or the fractal dimensionality ). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.

A decorated lattice is suggested and the Ising model on it with three kinds of interactions K1, K2, and K3 is studied. Using an equivalent transformation, the square decorated Ising lattice is transformed into a regular square Ising lattice with nearest-neighbor, next-nearest-neighbor, and four-spin interactions, and the critical fixed point is found at K1 = 0.5769, K2 = −0.0671, and K3 = 0.3428, which determines the critical temperature of the system. It is also found that this system and the regular square Ising lattice, and the eight-vertex model belong to the same universality class.

Recently, the physical properties and applications of the magnetic nanotube have attracted a great deal of theoretical and experimental attention. The magnetization and phase transition of spin-1 Blume-Capel model on a cylindrical Ising nanotube with bimodal random crystal fields are investigated by using the effective field theory. Employing numerical calculations, we obtain the phase diagrams and the magnetization, which depend on the temperature and the parameters of random crystal fields. Our obtained results are as follows. (i) Changing the probability (p) and the ratio of the crystal fields (), the bimodal random crystal fields may describe different doped atoms acting on spins. Especially, for p = 0.5, choosing = 0,-1.0,-0.5 and 0.5, the bimodal random crystal fields can respectively degrade four typical distributions of random crystal fields, i. e., the distribution of diluted crystal fields, the distribution of symmetry staggered crystal fields, the distribution of non-symmetry staggered crystal fields, and the distribution of same-direction crystal field. (ii) The system exhibits different magnetic properties and phase transition behaviors in the diluted, staggered and same-direction crystal field. The diluted and staggered crystal fields may reduce the magnetization of the system, resulting in the ground state saturation value of magnetization, which is less than 1, while the same-direction crystal fields cannot result in a similar behavior. (iii) The system shows several phase transition temperatures, i.e., first-order and second-order phase transitions and reentrant phenomenon as the parameters of bimodal random crystal fields change. The tricritical point and reentrant phenomenon do exist for certain values of the probability, the negative crystal field and the ratio of the crystal fields in the system. The relevant experiment is needed to verify the above-mentioned theoretical results.

The dynamics of one-dimensional random quantum Ising model with both nearest-neighbor and next-nearest-neighbor （NNN） interactions is investigated in the high temperature limit by the method of recurrence relations. Spin autocorrelations and the corresponding spectral densities of the system are calculated. Supposing that the exchange couplings （or the transverse fields） satisfy the double-Gaussian distribution, the effects of this distribution on the dynamics of the system is studied. The results show that the dynamics of the system undergoes a crossover from a collective-mode behavior to a central-peak one when the standard deviations σ J （or σ B ）of the random variables are small and there is no crossover when σ J （or σ B ）are large. Meanwhile, the effects of NNN interactions on the dynamics of the system are studied. It is found that the central-peak behavior becomes more obvious and the collective-mode behavior becomes weaker as K i increase, especially when K i > J i /2（ J i and K i are exchange couplings of the NN and NNN interactions, respectively）. However, the effects are small when the NNN interactions are weak （ K i J i /2）.