We study the magnetic properties of a mixed Ising ferrimagnetic system, in which the two interacting sublattices have spins $σ$, $(\pm 1/2)$ and spins $S$, $(\pm 3/2,\pm 1/2)$ in the presence of a random crystal field, with the mean field approach. The obtained results show the existence of some interesting phenomena, such as the appearance of a new ferrimagnetic phase namely the partly ferrimagnetic phase $(m_σ=\frac{-1}{2},m_S=+1)$ and consequently the existence of three topologically different types of phase diagrams. The effect of increasing the exchange interaction parameter $J$, at very low temperature is investigated. The transitions shown in these phase diagrams are in good agreement with those obtained in the ground state case.
Ground-state phase diagram of the mixed spin-$1/2$ and spin-$1$ Ising chain with axial and rhombic zero-field splitting parameters is exactly calculated within the framework of the transfer-matrix method. It is shown that the rhombic zero-field splitting parameter prefers the magnetically ordered phase instead of the disordered phase.
In the vicinity of boundaries the bulk universality class of critical phenomena splits into several boundary universality classes, depending upon whether the tendency to order in the boundary is smaller or larger than in the bulk. For Ising universality class there are five different boundary universality classes: periodic, antiperiodic, free, fixed and mixed (mixture of the last two). In this paper we present the new set of the universal amplitude ratios for the mixed boundary universality class. The results are in perfect agreement with a perturbated conformal field theory scenario proposed …
By utilizing the framework of the effective-field theory with correlation, hysteresis and compensation behaviors of mixed spin-1 and spin-2 hexagonal Ising nanowire system is investigated in detail. The effects of Hamiltonian parameters on hysteresis behaviors are discussed, in detail. The compensation behavior of the system is also studied and according to values of Hamiltonian parameters, the Q-, S-, R- and N- type compensation behaviors is obtained in the system. Finally, the obtaining results are compared with some experimental and theoretical results and found in a qualitatively good agre…
In the many fields in which the Ising model is applied nowadays, the spin variables are often assumed to be of spin-class $\{-1,1\}$ or $\{0,1\}$, even though for any mix of binary real valued spin-classes a proper Ising model distribution exists. Here we show that in their basis all these spin-classes are the same, as a simple expressions exist that allow us to transform the variables from one particular mix of spin-classes to any other combination, without changing the probabilities for the system states as a whole.
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 $(\pm 1/2,\pm 1/2,\pm 1)$ and $(\pm 1/2,\pm 1/2,0)$ at any finite temperature, except for some interesting non-critical anomalies displayed by response functions. On the o…
Phase transitions of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on several decorated planar lattices consisting of interconnected diamonds are investigated within the framework of the generalized decoration-iteration transformation. The main attention is paid to the systematic study of the finite-temperature phase diagrams in dependence on the lattice topology. The critical behaviour of the hybrid quantum-classical Ising-Heisenberg model is compared with the relevant behaviour of its semi-classical Ising analogue. It is shown that both models on diamond-like decorated planar lattices…
We study the dynamic phase transition properties for the mixed spin-(1/2, 1) Ising model on a square lattice under a time-dependent magnetic field by means the effective-field theory (EFT) based on the Glauber dynamics. We present the dynamic phase diagrams in the reduced magnetic field amplitude and reduced temperature plane and find that the phase diagrams exhibit the dynamic tricitical behavior, the multicritical and zero-temperature critical points as well as reentrant behavior. We also investigate the influence of the frequency (w) and observe that for small values of w the mixed phase di…
The mixed spin-(1,1/2) Ising-Heisenberg model on a distorted diamond chain with the spin-1 nodal atoms and the spin-1/2 interstitial atoms is exactly solved by the transfer-matrix method. An influence of the geometric spin frustration and the parallelogram distortion on the ground state, magnetization, susceptibility and specific heat of the mixed-spin Ising-Heisenberg distorted diamond chain are investigated in detail. It is demonstrated that the zero-temperature magnetization curve may involve intermediate plateaus just at zero and one-half of the saturation magnetization. The temperature de…
Phase transitions of the mixed spin-1/2 and spin-S Ising model on a three-dimensional decorated lattice with a layered structure
在一个精确映射到简单自旋1/2伊辛模型的框架下，研究了具有层状磁结构的三维缀饰晶格上自旋1/2和自旋S（S >= 1/2）混合伊辛模型的相变.对于混合自旋的分层伊辛模型，这种映射对应产生了合理的结果，或者通过采用正交晶格上自旋为1/2的伊辛模型的简化解[Z. D. Zhang，Philos.Mag.87（2007）5309-5419]或通过对四重晶格上的相应自旋1/2伊辛模型进行广泛的蒙特卡罗模拟。结果表明，临界行为显着依赖于轴向零场分裂参数，层间和层内相互作用的相对强度。引人注目的自发秩序捕获的“准1D”自旋系统中发现的相互作用参数的限制区域中，零场分裂参数的力量，所有整数值的装饰自旋对他们的“非磁性”的自旋状态。
The entanglement between spin-1/2 interstitial Heisenberg dimers in the mixed spin-(1,1/2) Ising-XXZ double sawtooth ladder is investigated at low temperature. Here, we consider a cyclic four-spin exchange interaction in square plaquette of each block, and investigate the effects of this amazing interaction on the bipartite entanglement between spin-1/2 interstitial dimers. Interestingly, we observe a remarkable difference in concurrence behavior with respect to the cyclic four-spin exchange interaction and magnetic field. Also, the critical points at which the concurrence vanishes are changed…
A mixed spin-1/2 and spin-3/2 Ising model on a decorated square lattice with a nearest- neighbor interaction, next-nearest-neighbor bilinear interaction, three-site four-spin in- teraction and single-ion anisotropy is exactly investigated using a generalized decoration- iteration transformation, Callen-Suzuki identity and differential operator technique. The ground-state and finite-temperature phase boundaries are obtained by identifying all rel- evant phases corresponding to minimum internal or free energy of the system. The thermal dependencies of magnetization, correlation functions, entrop…
The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concern…
The mixed spin 3- spin 3/2 Ising model has been simulated using cooling algorithm on cellular automaton (CA). The simulations have been made in the interval -6<=D<=6 for J=1 for the square lattices with periodic boundary conditions. The ground state phase diagram of the model has different type ferrimagnetic orderings. Through D/J=2 line, compensation points occurs at kT/J=0. The values of the critical exponents ( ν, α, β and γ) are estimated within the framework of the finite-size scaling theory and power law relations for selected D/J values (-2, 0, 1, 2 and 4). The estimated critical expone…
Within the framework of the effective-field theory (EFT) with correlations, the phase diagrams of a cylindrical spin-1 Ising nanowire is investigated, depending on the ratio of the Hamiltonian parameters. We discuss in detail the influence of interfacial coupling, crystal field and transfer field ratio on the phase diagrams, and some characteristic phenomena have been found such as tricritical point and first-order transitions.
Hysteresis characteristics of the crystal field diluted general Spin-S ($S>1$) Blume-Capel model have been studied within the effective field approximation. Particular emphasis has been paid on the large negative valued crystal field and low temperature region and it has been demonstrated for this region that, rising dilution of the crystal field results in decreasing number of windowed hysteresis loops. The evolution of the multiple hysteresis loop with the dilution of the crystal field has been investigated and physical mechanism behind this evolution has been given.
In this paper we present a study of the effects of two different single-ion anisotropies in the phase diagram and in the compensation temperature of the mixed spin-2 and spin-5/2 Ising ferrimagnetic system. We employed the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy. Also we found the Landau expansion of the free energy in the order parameter to describe the phase diagrams. In the plane critical temperature versus single-ion anisotropie the phase diagram displays tricritical behavior. The critical and compensation temperatures increase when the single-ion ani…
Transfer-matrix methods are used, in conjunction with finite-size scaling and conformal invariance concepts, to generate an accurate phase diagram for a two-dimensional square-lattice Ising spin-1/2 magnet, with couplings which are positive along one coordinate axis, and negative along the other, in a uniform external field. Our results indicate that the critical line starts horizontally at the zero-temperature end of the phase boundary, at variance with the reentrant behavior predicted in some earlier studies. Estimates of the thermal scaling exponent are very close to the Ising value $y_T=1$…
We study zero-temperature hysteresis in the random-field Ising model on a Bethe lattice where a fraction $c$ of the sites have coordination number $z=4$ while the remaining fraction $1-c$ have $z=3$. Numerical simulations as well as probabilistic methods are used to show the existence of critical hysteresis for all values of $c > 0$. This extends earlier results for $c=0$ and $c=1$ to the entire range $0 \le c \le 1$, and provides new insight in non-equilibrium critical phenomena.
The dynamic behaviors of a mixed spin (1/2-1) hexagonal Ising nanowire (HIN) with core-shell structure in the presence of a time dependent magnetic field are investigated by using the effective-field theory with correlations based on the Glauber-type stochastic dynamics (DEFT). According to the values of interaction parameters, temperature dependence of the dynamic magnetizations, the hysteresis loop areas and the dynamic correlations are investigated to characterize the nature (first- or second-order) of the dynamic phase transitions (DPTs). Dynamic phase diagrams, including compensation poin…
Mixed-spin Ising model on a decorated Bethe lattice is rigorously solved by combining the decoration-iteration transformation with the method of exact recursion relations. Exact results for critical lines, compensation temperatures, total and sublattice magnetizations are obtained from a precise mapping relationship with the corresponding spin-1/2 Ising model on a simple (undecorated) Bethe lattice. The effect of next-nearest-neighbour interaction and single-ion anisotropy on magnetic properties of the ferrimagnetic model is investigated in particular. It is shown that the total magnetization …
Weak universality, bicritical points and reentrant transitions in the critical behaviour of a mixed spin-1/2 and spin-3/2 Ising model on the union jack (centered square) lattice
Critical and compensation properties of a mixed spin-1 and spin-3/2 Ising ferrimagnet on a square lattice are investigated by standard and histogram Monte Carlo simulations. The critical temperature is studied as a function of a single-ion anisotropy strength. The second order of the phase transition is established by finite-size scaling for the entire boundary. Some previously obtained results, such as a tricritical point, predicted by the mean field theory (MFT) and effective field theory (EFT), or a first-order transition line separating two different ordered phases, obtained by the cluster…
A spin-1 transverse Ising model with longitudinal crystal field in a longitudinal magnetic field is examined by introducing an effective field approximation (IEFT) which includes the correlations between different spins that emerge when expanding the identities. The effects of the crystal field as well as the transverse and longitudinal magnetic fields on the thermal and magnetic properties of the spin system are discussed in detail. The order parameters, Helmholtz free energy and entropy curves are calculated numerically as functions of the temperature and Hamiltonian parameters. A number of …
Ground-state and finite-temperature behaviour of the mixed spin-1/2 and spin-1 Ising-Heisenberg model on decorated planar lattices consisting of inter-connected diamonds is investigated by means of the generalised decoration-iteration mapping transformation. The obtained exact results clearly point out that this model has a rather complex ground state composed of two unusual quantum phases, which is valid regardless of the lattice topology as well as the spatial dimensionality of the investigated system. It is shown that the diamond-like decorated planar lattices with a sufficiently high coord…
The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field ($h$) is switched on, the phase transition is preserved, which turns to first-order for $h/Γ\to \infty$, $Γ$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/Γ>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-sp…
Weak universal critical behaviour of the mixed spin-(1/2, S) Ising model on the union jack (centered square) lattice: integer versus half-odd-integer spin-S case
The mixed spin-(1/2, 1) Ising model on two fully frustrated triangles-in-triangles lattices is exactly solved with the help of the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the equivalent spin-1/2 Ising model on a triangular lattice. It is shown that the mutual interplay between the spin frustration and single-ion anisotropy gives rise to various spontaneously ordered and disordered ground states, which differ mainly in an occurrence probability of the non-magnetic spin state of the integer-valued decorating spins. We have convincingly e…
The ground state, zero-temperature magnetization process, critical behaviour and isothermal entropy change of the mixed-spin Ising model on a decorated triangular lattice in a magnetic field are exactly studied after performing the generalized decoration-iteration mapping transformation. It is shown that both the inverse and conventional magnetocaloric effect can be found near the absolute zero temperature. The former phenomenon can be found in a vicinity of the discontinuous phase transitions and their crossing, while the latter one occurs in some paramagnetic phases due to a spin frustration…
The magnetic properties of the mixed spin-$\frac{1}{2}$ and spin-$\frac{7}{2}$ Ising model with a crystal-field in a longitudinal magnetic field are investigated on the Bethe lattice using exact recursion relations. The ground-state phase diagram is constructed. The temperature-dependent one is displayed in the case of uniform crystal-field on the $(k_{\text{B}}T/|J|, D/|J|)$ plane in the absence of the external constraint for lattice coordination numbers $z = 3, 4, 6$. The order parameters and corresponding response functions as well as the internal energy are calculated and examined in detai…
The mixed spin-(1/2, 3/2) Ising model on a decorated square lattice, which takes into account lattice vibrations of the spin-3/2 decorating magnetic ions at a quantum-mechanical level under the assumption of a perfect lattice rigidity of the spin-1/2 nodal magnetic ions, is examined via an exact mapping correspondence with the effective spin-1/2 Ising model on a square lattice. Although the considered magnetic structure is in principle unfrustrated due to bipartite nature of a decorated square lattice, the model under investigation may display anomalous spin frustration driven by a magnetoelas…
We used mean-field theory based on the Bogoliubov inequality for the Gibbs free energy to examine the magnetic properties of a mixed spin-5/2 and spin-7/2 Blume-Capel ferrimagnetic system. The thermal behaviours of the system magnetization are classified according to the extended Néel nomenclature. The system exhibits compensation phenomena where a complete cancellation of sublattice magnetizations is observed below the critical temperature. Temperature-dependent phase diagrams are constructed for the case of unequal sublattice crystal field interactions. Under appropriate conditions, our calc…
In this work, we have performed Monte Carlo simulations to study phase transitions in a mixed spin-1 and spin-3/2 Ising ferrimagnetic system on the square and cubic lattices and with two different single-ion anisotropies. This lattice is divided in two interpenetrating sublattices with spins $S^A = 1$ (states $\pm1$ and 0) on the sublattice $A$ and $S^B = 3/2$ (states $\pm 3/2$, $\pm 1/2$) on the sublattice $B$. We have used single-ion anisotropies $D_{A}$ and $D_{B}$ acting on the sites of the sublattice $A$ and $B$, receptively. We have determined the phase diagrams of the model in the tempe…
The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic field $h$ via the recursion relations method. A ground-state phase diagram is constructed which may be needful to explore important regions of the temperature phase diagrams of a model. The order-parameters, the corresponding response functions and internal energy are thoroughly investigated in order to typify the nature of the phase transition and to get the corresponding temperatures. So, in the absence of the magnetic field, the temperature phase diagrams are displayed in the case …
The ground state and magnetization process of the mixed spin-(1,1/2) Ising diamond chain is exactly solved by employing the generalized decoration-iteration mapping transformation and the transfer-matrix method. The decoration-iteration transformation is first used in order to establish a rigorous mapping equivalence with the corresponding spin-1 Blume-Emery-Griffiths chain in a non-zero magnetic field, which is subsequently exactly treated within the framework of the transfer-matrix technique. It is shown that the ground-state phase diagram includes just four different ground states and the l…
Exact solution of the mixed-spin Ising model on a decorated square lattice with two different kinds of decorating spins on horizontal and vertical bonds
The mixed spin-1/2 and spin-S Ising model on the Union Jack (centered square) lattice with four different three-spin (triplet) interactions and the uniaxial single-ion anisotropy is exactly solved by establishing a rigorous mapping equivalence with the corresponding zero-field (symmetric) eight-vertex model on a dual square lattice. A rigorous proof of the aforementioned exact mapping equivalence is provided by two independent approaches exploiting either a graph-theoretical or spin representation of the zero-field eight-vertex model. An influence of the interaction anisotropy as well as the u…
Bootstrapping mixed correlators in three dimensional conformal field theories with a $\mathbb Z_2$ global symmetry has previously led to a closed allowed region in ($Δ_σ$, $Δ_ε$) space surrounding the 3D Ising model. We repeat that analysis in two dimensions. By further assuming a gap in the spin-2 sector, we also find a closed allowed region in ($Δ_σ$, $Δ_ε$) space surrounding the 2D Ising model.
Competitive effects of so-called three-site four-spin interactions, single ion anisotropy and bilinear interactions is studied in the mixed spin-1/2 and spin-1 Ising model on a decorated square lattice. Exploring the decoration-iteration transformation, we have obtained exact closed-form expressions for the partition function and other thermodynamic quantities of the model. From these relations, we have numerically determined ground- state and finite-temperature phase diagrams of the system. We have also investigated temperature variations of the correlation functions, internal energy, entropy…
The mixed spin-1/2 and spin-S Ising model on a decorated planar lattice accounting for lattice vibrations of decorating atoms is treated by making use of the canonical coordinate transformation, the decoration-iteration transformation, and the harmonic approximation. It is shown that the magnetoelastic coupling gives rise to an effective single-ion anisotropy and three-site four-spin interaction, which are responsible for the anomalous spin frustration of the decorating spins in virtue of a competition with the equilibrium nearest-neighbor exchange interaction between the nodal and decorating …
With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane $y>0$ with different boundary conditions $a$ and $b$ on the negative and positive $x$ axes. For $ab=-+$ and $f+$, they determined the one and two-point averages of the spin $σ$ and energy $ε$. Here $+$, $-$, and $f$ stand for spin-up, spin-down, and free-spin boundaries, respectively. The case $+-+-+\dots$, where the boundary conditions switch between $+$ and $-$ at arbitrary points, $ζ_1$, $ζ_2$, $\dots$ on the $x$ axis was also analyzed. In this paper the alternating …
Mixed spin-1/2 and spin-1 Ising ferrimagnets on a triangular lattice with sublattices A, B and C are studied for two spin value distributions $(S_{\rm A},S_{\rm B},S_{\rm C})=(1/2,1/2,1)$ and $(1/2,1,1)$ by Monte Carlo simulations. The non-bipartite character of the lattice induces geometrical frustration in both systems, which leads to the critical behavior rather different from their ferromagnetic counterparts. We confirm second-order phase transitions belonging to the standard Ising universality class occurring at higher temperatures, however, in both models these change at tricritical poin…
Nonequilibrium phase transition properties of a mixed Ising ferrimagnetic model consisting of spin-1/2 and spin-3/2 on a square lattice under the existence of a time dependent oscillating magnetic field have been investigated by making use of Monte Carlo simulations with single-spin flip Metropolis algorithm. A complete picture of dynamic phase boundary and magnetization profiles have been illustrated and the conditions of a dynamic compensation behavior have been discussed in detail. According to our simulation results, the considered system does not point out a dynamic compensation behavior,…
