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The longitudinal-random-field mixed Ising model consisting of arbitrary spin values has been studied by the use of an effective field theory with correlations (EFT). The phase diagrams of systems with mixed spins: σ = 1/2, S = 1; σ = 1/2, S = 3/2 are plotted. Not only the discontinuity at T = 0 K, is found when both longitudinal fields are trimodal distributed, but also the tricritical behavior is observed in these phase diagrams between the bimodal and trimodal distributions of longitudinal fields, which is different from the single-spin one. The appearance of tricritical point is independent of the coordination number and spin values.

The transverse spin-2 Ising ferromagnetic model with a longitudinal crystal field is studied within the mean-field theory. The phase diagrams and magnetization curves are obtained by diagonalizing the Hamiltonian Hi of the Ising system numerically, and the first order-order phase transitions, the first order-disorder phase transitions, and the second-order phase transitions are discussed in details. Reentrant phenomena occur when the value of the transverse field is not zero and the reentrant diagram is given.

The transverse spin-2 Ising ferromagnetic model with a longitudinal crystal-field is studied within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The ground-state phase diagram and the tricritical point are obtained in the transverse field Ω/zJ-longitudinal crystal D/zJ field plane. We find that there are the first order-order phase transitions in a very small range of D/zJ besides the usual first order-disorder phase transitions and the second order-disorder phase transitions.

The magnetic properties of the mixed ferro-ferrimagnetic compounds with (A(x)B(1-x))(y)C-z where A(z) B-z and C are three different magnetic ions and form three different sublattices, are studied by using the standard mean-field theory. The phase diagram which is related to experimental work of molecule-based ferro-ferrimagnet ((NixMn1-xII)-Mn-II)(1.5)[Cr-III(CN)(6)].zH(2)O is obtained. The magnetization curves(z) internal energy(z) and specific heat of the same mixed (A(x)B(1-x))(y)C system are also investigated.

The phase diagrams of a spin-2 transverse Ising model with a random field on honeycomb, square, and simple-cubic lattices, respectively, are investigated within the framework of an effective-field theory with correlations. We find the behavior of the tricritical point and the reentrant phenomenon for the system with any coordination number z, when the applied random field is bimodal. The behavior of the tricritical point is also examined as a function of applied transverse field. The reentrant phenomenon comes from the competition between the transverse field and the random field.

A theoretical study of a spin-3/2 Ising model in a longitudinal random field with crystal field is studied by using of the effective-field theory with correlations. The phase diagrams and the behavior of the tricritical point are investigated numerically for the honeycomb lattice when the random field is bimodal. In particular, the specific heat and the internal energy are examined in detail for the system with a crystal-field constant in the critical region where the ground-state configuration may change from the spin-3/2 state to the spin-1/2 state. We find many interesting phenomena in the system.

Within the framework of the effective-field theory with correlations, we study the ferromagnetic spin-2 random-field Ising model (RFIM) in the presence of a crystal field on honeycomb (z = 3), square (z = 4) and simple cubic (z = 6) lattices. The effects of the crystal field and the longitudinal random held on the phase diagrams are investigated. Some characteristic features of the phase diagrams, such as the tricritical phenomena, reentrant phenomena and existence of two tricritical points, are found.

The critical behaviors of a mixed spin- and spin-sB Ising system with a transverse crystal field are studied by use of the effective-field theory with correlations. The effect of the transverse crystal field on transition temperatures is investigated numerically for the honeycomb (z = 3) and square (z = 4) lattices. The results show that there is no tricritical point for the system.

The properties of the ground state in the spin-2 transverse Ising model with the presence of a crystal field are studied by using the effective-field theory with correlations. The longitudinal and transverse magnetizations, the phase diagram and the internal energy in the ground state are given numerically for a honeycomb lattice (z = 3).

The spin-1/2 quantum antiferromagnet in three-dimensional space can be described by Heisenberg model:two nearest-neighbor spins with strong interaction in z direction consistitute a dimer, every dimer can be considered as a lattice site in the tetragonal lattice. In z direction, the exchange interaction between two nearest-neighbor dimers is λ'J, and in xy plane it is 2 λJ. In bond-operator representation, within the mean-field decoupling approximation, we have calculated the phase diagram for order-disorder phase transition about λ and λ' at 0 K, obtained the critical value, λc=0.292, within which the disordered phase is stable in two-dimensional case (λ'=0), and found that the disordered phase is always stable for λ' < 1 in one-dimensional case(λ=0). For the disordered phase, the physical quantities, such as energy gap, ground-state energy, two-point correlation functions and correlation lengths in xy plane and in z direction, have been calculated as functions of parameters λ and λ'.

Based on the single-site electron correlation, the local approach has been used to investigate the ferromagnetic phase in the Hubbard-Hirsch model. Using the R =0,α approximation in second-order calculation, the electron Correlation effect on the phase diagram has been studied Carefully. Comparing with those of the R=0 approximation, the electron correlation effect becomes weaker due to taking into aceount the nondiagonal elements of density matrix.