The thermodynamic properties of spin models on the lattice have been studied by employing phenomenological renormalization group techniques. The spin-1 Blume- Capel model and the spin-1 and spin-3/2 Baxter-Wu model have been treated. The Blume Capel model with spin-1, in its diluted version, on the simple cubic lattice has been studied by using the mean field renormalization group in the pair aproximation. We have obtained the critical temperature as a function of the vacant sites concentration as well a function of the crystal field anisotropy. We consider exchange e superexchange interactions between Fe atoms induzed by Al atoms and applied this model to FeAl alloys to describe its thermodynamics properties. We also studied the spin-1 and 3/2 Baxter-Wu model in a crystal field by employing the finite size renormalization group approach. The eigenvalues of the transfer matrix have been computed in order to get the corresponding phase diagram. Estimates of the firstand second-order transition temperatures have been obtained, as well as multicritical points, critical exponents and central charges, the latter two being obtained for some values of the crystal field. The Gibbs phase rule has been used to check the consistency of the phase diagram of some magnetic models, including those treated in this work. In this case it is necessary to take into account the symmetries carried by the magnetic Hamiltonians.
The thermodynamic properties of spin models on the lattice have been studied by employing phenomenological renormalization group techniques. The spin-1 Blume- Capel model and the spin-1 and spin-3/2 Baxter-Wu model have been treated. The Blume Capel model with spin-1, in its diluted version, on the simple cubic lattice has been studied by using the mean field renormalization group in the pair aproximation. We have obtained the critical temperature as a function of the vacant sites concentration as well a function of the crystal field anisotropy. We consider exchange e superexchange interactions between Fe atoms induzed by Al atoms and applied this model to FeAl alloys to describe its thermodynamics properties. We also studied the spin-1 and 3/2 Baxter-Wu model in a crystal field by employing the finite size renormalization group approach. The eigenvalues of the transfer matrix have been computed in order to get the corresponding phase diagram. Estimates of the firstand second-order transition temperatures have been obtained, as well as multicritical points, critical exponents and central charges, the latter two being obtained for some values of the crystal field. The Gibbs phase rule has been used to check the consistency of the phase diagram of some magnetic models, including those treated in this work. In this case it is necessary to take into account the symmetries carried by the magnetic Hamiltonians.
The Blume-Capel model on a square lattice and the Baxter-Wu model on a triangular lattice have been studied through extensive Monte Carlo simulations using the single spin update according to the Metropolis algorithm. Both models have been treated with spin S=1 and spin S=3/2 in the presence of crystal field interactions. The transition temperature and correlation length critical exponent have been computed, as a function of the crystal field interaction, by employing the recent developed method of the zeros of the energy probability distribution. The important question of the universality class along the second-order transition lines could be resolved by looking at the corresponding universal probability distributions of the energy and magnetizations. Mixing field arguments turned out to be quite important to decide the possible change in the universality class of the models as the crystal field varies. Single histograms techniques have also been utilized in order to obtain simulational data close to the second-order transitions and multicritical points. As expected, the Blume-Capel model has first- and second-order transitions, where a tricritical point occurs for spin S=1, while an isolated double-critical-endpoint is present for spin S=3/2. Although the topology of the phase diagram of the Baxte-Wu model for spin S=1 is similar to that of the Blume-Capel model, with the tricritical point being replaced by a pentacritical point, the Baxter-Wu model has critical exponents that change along the tetracritical line as the crystal field is increased. On the other hand, the Baxter-Wu model with spin S=3/2 has a phase diagram that is completely different from the one obtained for the Blume-Capel, with the presence of a pentacritical point and a tetracritical endpoint. The present data also show that, despite the change of the probability distributions as the crystal field varies, the models with different spin values belong to the same universality class.
The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.
Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.
We have studied the critical, multicritical and first order behavior of the Spin-1 Baxter-Wu model in a crystal field. We have employed the mean field renormalization group approach and extensive Monte Carlo simulations using the single spin flip Metropolis algorithm and hystogram techniques. The global phase diagram is obtained and it is shown that the system belongs to the same universality class as the corresponding spin-1/2 version and presents a multicritical point separating a second order phase transition line from a first order line. The probability distribution of this pentacritical point is also determined.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.
The critical properties of a random mixed-bond Ising model on a cubic lattice are studied so much numerically with analytically. Analytical studies on Mixed-bond Ising model through renormalization group technique predict the existence of reentrant magnetism in certain range of values of the competition parameter α. This phenomenon typically found in systems that present spin-glass phase. In this work, did we use the algorithm cluster of Wolff to simulate the dynamics of the system. Using the technique of the Histogram obtains the thermodynamic amounts of interest and based on the theory finite size scaling found the critical temperatures, the phase diagrams and the critical exponents of the model. Our results were compared with those obtained using a new technique in effective field theory which employs similar probability distribution within the framework of two-site clusters.