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Journal of Physics Communications

Exact phase diagram for the mixed spin-1/2 and spin-S Ising models on the square lattice

A. Dakhama, Mouhcine Azhari, Noureddine Benayad

Published Jun 11, 2018• 15 citations
Publisher

Abstract

We propose a new approach which permits us to obtain exact results for the mixed spin-1/2 and Spin- S ( S > 1/2) Ising models with single ion anisotropy Δ on the square lattice. We drive an explicit expression for the critical temperature for arbitrary values of S. We determine the exact phase diagrams for different values of S , and we show that there is no tricritical point. For S integer, there is no long range order when the anisotropy exceeds a critical value which is independent of S . Furthermore, the exact Ising transition temperature T C is always recovered, for any values of S, in the limit of Δ → − ∞ . These exact results are based on a conjecture which extend the analyticity of the n-spin-1/2 correlations functions (except at T = T C (Ising)) for any finite number n, to n → ∞ . This is confirmed by our calculations since our results are practically similar to those obtained by Monte Carlo simulations for S = 1, 2 and 3/2.

DOI
10.1088/2399-6528/aacbbe

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We propose a new approach which permits us to obtain exact results for the mixed spin-1/2 and Spin- S ( S > 1/2) Ising models with single ion anisotropy Δ on the square lattice. We drive an explicit expression for the critical temperature for arbitrary values of S. We determine the exact phase diagrams for different values of S , and we show that there is no tricritical point. For S integer, there is no long range order when the anisotropy exceeds a critical value which is independent of S . Furthermore, the exact Ising transition temperature T C is always recovered, for any values of S, in the limit of Δ → − ∞ . These exact results are based on a conjecture which extend the analyticity of the n-spin-1/2 correlations functions (except at T = T C (Ising)) for any finite number n, to n → ∞ . This is confirmed by our calculations since our results are practically similar to those obtained by Monte Carlo simulations for S = 1, 2 and 3/2.
"We used a Monte Carlo simulation to analize the magnetic behavior of Ising model of mixed spins S A i = ß3=2 ; ß1=2 and æ B j = ß 5 = 2 ; ß 3 = 2 ; ß 1 = 2 , on a square lattice. Were studied the possible critical phenomena that may emerge in the region around the multiphase point (D= j J 1 j = ° 3 ;J 2 = j J 1 j = 1) and the dependence of the phase diagrams with the intensities of the anisotropy field of single ion (D= j J 1 j ) and the ferromagnetic coupling of exchange spin S A i (J 2 = j J 1 j). The system displays first order phase transitions in a certain range of the parameters of the Hamiltonian, which depend on D= j J 1 j and j J 2 = j J 1 j . In the plane (D= j J 1 j ;k B T= j J 1 j), the de- crease of j D= j J 1 jj , implies that the critical temperature, T c , increases and the first order transition temperature, T t , decreases. In the plane (J2 = j J 1 j ;k B T= j J 1 j ), T c increases with the increasing of J 2 = j J 1 j , while that T t decreases."
The critical properties of the bond-diluted mixed spin-1/2 and spin-1 transverse Ising system with single-ion anisotropy are investigated by means of the effective field theory with correlations. Particular emphasis is given to the square lattice for which phase diagrams are obtained. If transverse field varies in the certain ranges we find that the tricritical point is obtained for the value of the bond concentration in a restricted region. We also observe that the reentrant phase transition may occur in the present system if single-ion anisotropy parameter is not large and if transverse field is small. On the other hand, for certain values of the system parameters, new induced magnetic ordering can be obtained. We find a number of interesting phenomena that are not predicted by previous literatures. The influence of the transverse field on the behaviours of the reentrant phase transition and induced magnetic ordering is discussed. A detailed description of the phase transition is presented.
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.
The mixed-spin-(1∕2,SB,SC)(1∕2,{S}_{\mathrm{B}},{S}_{\mathrm{C}})(1∕2,SB​,SC​) Ising model on a decorated square lattice with two different kinds of decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ $({S}_{\mathrm{B}}\ensuremath{\ne}{S}_{\mathrm{C}})$ placed on its horizontal and vertical bonds, respectively, is exactly solved by establishing a precise mapping relationship with the corresponding spin-1∕21∕21∕2 Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy acting on both types of decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ is examined, in particular. If decorating spins SB{S}_{\mathrm{B}}SB​ and SC{S}_{\mathrm{C}}SC​ are integer and half-odd-integer, respectively, or if the reverse is the case, the model under investigation displays a very peculiar critical behavior that had bearing on the spontaneously ordered ``quasi-one-dimensional'' spin system, which appears as a result of the single-ion anisotropy strengthening. We have found convincing evidence that this remarkable spontaneous ordering virtually arises even though all integer-valued decorating spins tend toward their ``nonmagnetic'' spin state S=0S=0S=0 and the system becomes disordered only upon further increase of the single-ion anisotropy. The single-ion anisotropy parameter is also at an origin of various temperature dependences of the total magnetization when imposing the pure ferrimagnetic or the mixed ferro-ferrimagnetic character of the spin arrangement.
By Monte Carlo simulations we study critical properties of the mixed spin-1/2 and spin-1 Ising model on a triangular lattice, considering two different ways of the spin-value distributions on the three sublattices: (1/2,1/2,1) and (1/2,1,1). In the former case, we find standard Ising universality class phase transitions between paramagnetic and magnetic phases but no phase transitions between two distinct magnetic phases (±1/2,±1/2,±1) and (±1/2,±1/2,0) at any finite temperature, except for some interesting non-critical anomalies displayed by response functions. On the other hand, the latter case turns out to be a rare (or perhaps the only) example of a two-dimensional mixed spin-1/2 and spin-1 Ising model on a standard lattice that displays tricritical behavior.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.