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Surface Review and Letters

CRITICAL BEHAVIOR OF THE SEMI-INFINITE MIXED SPINS (7/2, 1/2) MODEL

Hasnae Saadi, El Mostafa Jalal, A. El Antari, N. Hachem, M. Madani, M. El Bouziani

Published Jun 14, 2023• 5 citations
Publisher

Abstract

By the use of the Migdal–Kadanoff renormalization group technique and the mean field approximation, we have explored the critical behavior of the semi-infinite mixed spin-7/2 and spin-1/2 Blume–Capel model. As a function of the computation ratios (bulk-surface) R and Y, different phase diagrams in the bulk and on the surface are classified and determined in the (surface anisotropy, temperature) plane. We have found four types of phase diagrams characterized by ordinary, extraordinary, surface, and special phase transitions. The derivative of the free energy and the behavior of the bulk and surface magnetizations are plotted at very low temperatures proving the existence of first-order transitions for both the surface and bulk. We have also presented the related fixed points and the critical exponents manifesting several classes of universality at the surface. Otherwise, a comparison was made between the two methods as well as with previous studies.

DOI
10.1142/s0218625x23500713

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By the use of the Migdal–Kadanoff renormalization group technique and the mean field approximation, we have explored the critical behavior of the semi-infinite mixed spin-7/2 and spin-1/2 Blume–Capel model. As a function of the computation ratios (bulk-surface) R and Y, different phase diagrams in the bulk and on the surface are classified and determined in the (surface anisotropy, temperature) plane. We have found four types of phase diagrams characterized by ordinary, extraordinary, surface, and special phase transitions. The derivative of the free energy and the behavior of the bulk and surface magnetizations are plotted at very low temperatures proving the existence of first-order transitions for both the surface and bulk. We have also presented the related fixed points and the critical exponents manifesting several classes of universality at the surface. Otherwise, a comparison was made between the two methods as well as with previous studies.
We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R1 and R2 of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.
As a function of the ratio R of bulk and surface interactions and the ratio D of bulk and surface crystal field we have determined, within the framework both of the mean-field approximation and by renormalization-group techniques, various types of phase diagram of a three-dimensional semi-infinite ferromagnetic spin-3/2 Blume-Capel model. We found that there exist four main types of phase diagram showing a variety of phase transitions associated with the surface and unusual multicritical topologies, including ordinary, extraordinary, surface and special phase transitions. Moreover, in the ( R , D ) plane, we determined the domains in which the system exhibits a particular type of phase diagram.
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.
Abstract The Migdal–Kadanoff renormalization group technique is applied to study a three-dimensional semi-infinite Ising system consisting of spin-1/2 sites in the bulk and spin-1 sites at the surface. According to the competition ratio between bulk and surface exchange interactions ( R = J B / J S ), two types of surface phase diagrams have been identified in the (surface crystal field, temperature) plane. These diagrams exhibit a rich variety of phase transition phenomena: surface, ordinary, extraordinary, and special second-order transitions, first-order transitions at the surface, as well as tricritical and multicritical points. Additionally, we list in a table the fixed points that govern the observed phases and phase transitions.
Using a renormalization-group transformation in position space based on the Migdal-Kadanoff recursion relations, we investigate the three-dimensional semi-infinite cubic spin-32\frac{3}{2}23​ Blume-Emery-Griffiths model with single-site uniaxial anisotropy and nearest-neighbor pair interactions, both bilinear and biquadratic. The fixed-point topologies and corresponding phase diagrams predicted for this model are presented. Attention is focused on the renormalization-group description of the surface critical behaviors, in comparison with the mean-field approximation results. Three main classes of system phase diagrams are determined, featuring a rich variety of phase transitions associated with the surface with type being according to the values of bulk and surface interaction parameters. Surface critical, tricritical, bicritical, and multicritical points are also found.
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.