On accuracy of different cluster models used in describing ordering phase transitions in fcc alloys

A general formulation of cluster methods applied to calculations of thermodynamic quantities of alloys in terms of renormalizing fields describing interaction between a cluster and its environment is given. We have shown that the well-known cluster variation method and the cluster field method, which was suggested earlier, are special cases of our approach. These methods have been used in calculations of phase diagrams of fcc alloys with L1₂ and L1₀ ordering transitions with several realistic interaction models. It turns out that, for all these models, the simple tetrahedron version of the cluster field method suggested in this paper describes the phase diagrams almost as accurately as more complicated cluster variation techniques. Possible applications of the tetrahedron version of the cluster field method to inhomogeneous states and kinetics of phase transitions in fcc alloys are discussed. Zh. éksp. Teor. Fiz. 115, 158–179 (January 1999)     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

Using the effective-field theory based on the Glauber-type stochastic dynamics (DEFT), we investigate dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model under an oscillating magnetic field. We presented the dynamic phase diagrams in (T/J, h₀/J), (D/J, T/J) and (K/J, T/J) planes, where T, h₀, D, K and z are the temperature, magnetic field amplitude, crystal–field interaction, biquadratic interaction and the coordination number. The dynamic phase diagrams exhibit several ordered phases, coexistence phase regions and special critical points, as well as re-entrant behavior depending on interaction parameters. We also compare and discuss the results with the results of the same system within the mean-field theory based on the Glauber-type stochastic dynamics and find that some of the dynamic first-order phase lines and special dynamic critical points disappeared in the DEFT calculation.     

Magnetic fluids in an external field

Within mean field approximation we investigate the phase diagrams of magnetic fluids in presence of a magnetic field. In a finite field the magnetic phase transition is absent, but instead a line of first order liquid-liquid transitions ending in a critical point occurs for a magnetic interaction, which is sufficiently strong. Varying the magnetic field these critical points extend from the tricritical point at H=0 to a critical endpoint. For a fluid with Ising spins we calculate the critical lines and several tricritical exponents analytically. For Heisenberg fluids we obtain the phase diagrams from a numerical solution of the mean field equations of state. Received 20 March 1998     

Dynamic Dipole and Quadrupole Phase Transitions in the Kinetic Spin-3/2 Model

The dynamic phase transition has been studied, within a mean-field approach, in the kinetic spin-3/2 Ising model Hamiltonian with arbitrary bilinear and biquadratic pair interactions in the presence of a time dependent oscillating magnetic field by using the Glauber-type stochastic dynamics. The nature (first- or second-order) of the transition is characterized by investigating the behavior of the thermal variation of the dynamic order parameters and as well as by using the Liapunov exponents. The dynamic phase transitions (DPTs) are obtained and the phase diagrams are constructed in the temperature and magnetic field amplitude plane and found nine fundamental types of phase diagrams. Phase diagrams exhibit one, two or three dynamic tricritical points, and besides a disordered (D) and the ferromagnetic-3/2 (F_3/2) phases, six coexistence phase regions, namely F _3/2+ F _1/2, F _3/2+ D, F _3/2+ F _1/2+ FQ, F _3/2+ FQ, F _3/2+ FQ + D and FQ + D, exist in which depending on the biquadratic interaction. PACS number(s): 05.50.+q, 05.70.Fh, 64.60.Ht, 75.10.Hk     

Mixed-Spin Ising Model in an Oscillating Magnetic Field and Compensation Temperature

Godoy et al. (Phys. Rev. B 69, 054428, 2004) presented a study of the magnetic properties of a mixed spin (1/2,1), Ising ferrimagnetic model on a hexagonal lattice without an oscillating magnetic field. They employed dynamic mean-field calculations and Monte Carlo simulations to find the compensation point of the model and to present the phase diagrams. It has been found that the N-type compensation temperature appears only when the intrasublattice interaction between spins in the σ sublattice is ferromagnetic. Moreover, the system only undergoes a second-order phase transition. In this work, we extend the study a dynamic compensation temperature of a mixed spin-1/2 and spin-1 Ising ferrimagnetic system on a hexagonal lattice in the presence of oscillating magnetic field within the framework of dynamic mean-field calculations. We find that the system displays the N-type compensation temperature. We also calculate dynamic phase diagrams in which contain the paramagnetic, ferrimagnetic, nonmagnetic fundamental phases and two different mixed phases, depending on the interaction parameters and oscillating magnetic field. The system also exhibits tricritical and reentrant behaviors.     

Numerical study of the mixed spin-1 and spin-5/2 BEG model on the Bethe lattice

The mixed spin-1 and spin- \frac52\frac{5}{2} ferromagnetic Ising model with bilinear (J) and biquadratic (K) nearest-neighbor exchange interactions and a single-ion potential or crystal-field interaction (D) is studied on the Bethe lattice by means of exact recursion equations. First, the phase diagram of the system at zero temperature is obtained in the (D/Jq, K/Jq) plane, where q denotes the coordination number of the lattice. Second, the sublattice magnetizations as functions of the temperature, the crystal-field and the biquadratic interaction strengths are thoroughly investigated. For q = 3, the resulting phase diagrams show first and second order phase transitions as well as compensation points where the net magnetization of the whole lattice should vanish in the antiferromagnetic version of the model. One interesting feature of the model concerns the presence of tricritical points. Our calculations show that at non-zero temperature, none of the sublattice can order separately. However, under an external magnetic field, some interesting phase diagrams with partially ordered phases arise.     

Phase diagrams in an ultra-thin spin-1 transverse Ising film with bond or site dilution at surfaces

The phase diagrams of two nanoscaled thin films with bond and site dilutions at the surfaces, described by the spin-1 transverse Ising model, are investigated by the use of an effective field theory with correlations. A number of characteristic phenomena have been found in them, which are heavily dependent on the ratios (r = J₁/J and p = Ω_S/Ω, where J is the exchange interaction in the inner layer, J₁ is the exchange interaction between the surface and the next inner layer, Ω_S is the transverse field at the surfaces and Ω is the transverse field in the inner layer). Some of them have exhibited very similar behaviors found in the two spin-1/2 nanoscaled thin films with bond and site dilutions at the surfaces.     

Phase transitions in the mixed Ashkin-Teller model

By using a mean-field approximation (MFA) and Monte-Carlo (MC) simulations, we have studied the effect on the phase diagrams of mixed spins ( and S =1) in the Ashkin-Teller model (ATM) on a hypercubic lattice. By varying the strength describing the four spin interaction and the single ion potential, we have obtained by these two methods quite rich phase diagrams with several multicritical points. This model exhibits a new partially ordered phase which does not exist neither in the spin-1/2 ATM nor in the spin-1 ATM. While MFA yields phase diagrams which are sometimes qualitatively incorrect, accurate results are obtained from MC simulations. From the critical exponents which have been calculated using finite-size scaling ideas, we have shown that all phase transitions are Ising-like except for the paramagnetic-Baxter critical surface on which the critical exponents vary continuously, by varying only the strength of the coupling interaction independently of the value of the single ion potential. Received 5 July 1999 and Received in final form 4 July 2000     

Socio-economical dynamics as a solvable spin system on co-evolving networks

We consider social systems in which agents are not only characterized by their states but also have the freedom to choose their interaction partners to maximize their utility. We map such systems onto an Ising model in which spins are dynamically coupled by links in a dynamical network. In this model there are two dynamical quantitieswhich arrange towards a minimum energy state in the canonical framework:the spins, s_i, and the adjacency matrix elements, c_ij.The model is exactly solvable because microcanonical partition functions reduce to productsof binomial factors as a direct consequence of the c_ij minimizing energy. We solve the system for finite sizes and for the two possible thermodynamic limits and discussthe phase diagrams.     

Islands of stability of the d-wave order parameter in s-wave anisotropic superconductors

In this paper we find and present on diagrams in the coordinates of η=2t₁/t₀ (the ratio of the second and the first nearest neighbor hopping integrals) and n (the carrier concentration) the areas of stability for the superconducting spin-singlet s- and d-wave and the spin-triplet p-wave order parameters hatching out during the phase transition from the normal to the superconducting phase. The diagrams are obtained for an anisotropic two-dimensional superconducting system with a relatively wide partially-filled conduction band. We study a tight-binding model with an attractive nearest neighbor interaction with the amplitude V₁, and the on-site interaction (with the amplitude V₀) taken either as repulsive or attractive. The problem of the coexistence of the s-, p- and d-wave order parameters is addressed and solved for chosen values of the ratio V₀/V₁. A possible island of stability of the d-wave order parameter in the s-wave order parameter environment for a relatively strong on-site interaction is revealed. The triple points, around which the s-, d-, and p-wave order parameters coexist, are localized on diagrams. It is shown that results of the calculations performed for the two-dimensional tight-binding band model are dissimilar with some obtained within the BCS-type approximation.     

Dynamic phase transition in the kinetic Blume--Emery--Griffiths model: Phase diagrams in the temperature and interaction parameters planes

As a continuation of our previously published work, the dynamic phase transitions are studied further, within a mean-field approach, in the kinetic Blume--Emery--Griffiths model in the presence of a time varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different planes, namely in the reduced temperature (T) and biquadratic interaction (k) plane and found eight fundamental types of phase diagrams for various values of reduced crystal-field interaction (d) and magnetic field amplitude (h), and in the (T,?d) plane and obtained six distinct topologies for different values of k and h. Phase diagrams exhibit one or two dynamic tricritical points and a dynamic double critical end point, dynamic triple and quadruple points, and besides disordered and ordered phases, three coexistence phase regions exist in which occurring of these strongly depend on the values of d, k and h.     

Order parameter quantum fluctuations in a two-dimensional system of mesoscopic Josephson junctions

The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n ₀ (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY-model—and approaches the latter as n ₀ increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n ₀<10, while quantum fluctuations in the phase are significant when n ₀<8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY-model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/n ₀ using the method of functional integration. Zh. éksp. Teor. Fiz. 113, 261–277 (January 1998)     

On the accuracy of various statistical methods for describing ordering phase transitions in BCC alloys

The accuracy of various statistical methods for describing B2-or D0₃-type (CuZn or Fe₃Al) ordering phase transitions characteristic of BCC alloys is examined. The temperature-concentration phase diagrams were calculated for several models of alloys discussed in the literature with the use of three methods, namely, the mean-field and pair cluster approximations and the tetrahedron cluster field method developed in this work. The calculation results were compared with each other and with Monte Carlo calculations. The accuracy of various methods was found to sharply depend on the type of interactions in the system, primarily, on the presence of strong competing interactions and short-range correlations that hinder ordering. In the absence of such correlations, in particular, in Fe-Al type alloys with extended interactions, the use of the pair cluster approximation allows the results obtained in the mean-field approximation to be substantially refined without any noticeable complication of calculations. At the same time, for systems with short-range and competing interactions, the use of this approximation could drastically distort the form of phase diagrams, whereas the tetrahedron cluster field method allows the phase diagrams of these systems to be calculated fairly accurately for temperatures and concentrations of interest to physics.     

Nonlocal quark model beyond mean field

The nonlocal chiral quark model is extended beyond mean field using a strict 1/N _c expansion scheme. The nonlocal interaction has the advantage that all diagrams are finite and leads to unique evaluation of the 1/N _c corrections. Parameters of the nonlocal model are refitted making use of the physical values of the pion mass and the weak pion decay constant. The size of the 1/N _c correction to the quark condensate is carefully studied in nonlocal and local Nambu-Jona-Lasinio models. It is found that even the sign of the corrections can be different.     

Phase transition in antiferromagnets with anisotropic exchange interactions and uniaxial anisotropy

Abstract

Following the Bogoliubov variational principle, the equilibrium and stability equations of the free energy for the two sublattice antiferromagnetic system with inter- and intrasublattice exchange interactions and with an external magnetic field are investigated. For the Ising spin system with uniaxial anisotropy, the phase diagrams have been calculated for various values of anisotropy constant d and the ratio of intra- to intersublattice interaction constants γ. It is shown that first-order, as well as second-order transitions, occur for γ > 0, whereas only a second-order transition occurs for γ ≦ 0, irrespective of the sign of d. Furthermore, similar calculations are extended for the anisotropic Heisenberg spin system and quite interesting phase diagrams have been obtained. Next, the effects of the anisotropic exchange interactions on the magnetic ordered states and the magnetizations of the singlet ground state system of spin one and with a uniaxial anisotropy term are investigated in the vicinity of the level crossing field H ? D/gμ _B . A field-induced ordered state without the transverse component of magnetization is shown to appear in a certain range of magnetic field as the spin dimensionality decreases. It has also turned out that the phase transition between this ordered state and the canted antiferromagnetic state ordinarily found for the isotropic singlet ground state system is of first order. Lastly, the stable spin configurations at a temperature of absolute zero for a two-sublattice uniaxial antiferromagnet under an external magnetic field of arbitrary direction are studied. In particular, the effects of a single ionic anisotropy D-term and anisotropy in the exchange interactions on the magnetic phases are investigated. The antiferromagnetic state has turned out to appear only for the external magnetic field along the easy axis of sublattice magnetization, and makes a first-order phase transition to the canted-spin state or the ferromagnetic state. For other field directions, no antiferromagnetic state appears and only a second-order phase transition between the canted-spin and the ferromagnetic states occurs. The critical field as a function of external field direction has been calculated for several D-values.     

Phase diagram of randomly polymerized membrane

Using a replica formalism, a generalization of a recent mean field model corresponding to the observed wrinkling transition in randomly polymerized membranes is presented. In this model we study the effects of global fluctuations of the surface normals to the flat membrane, which can be introduced by a random local field. In absence of these global fluctuations, we show that, the model exhibits both continuous and discontinuous transitions between flat and wrinkled phases, contrary to what has been predicted by Bensimon et al. and Attal et al. Phase diagrams both in replica symmetry and in breaking of replica symmetry in sense of Almeida and Thouless are given. We have also investigated the effects of global fluctuations on the replica symmetry phase diagram. We show that, the wrinkled phase is favored and the flat phase is unstable. For large global fluctuations, the transition between wrinkled and flat phases becomes first order. Received: 3 December 1997 / Revised: 31 March 1998 / Accepted: 3 August 1998     

Comparative analysis of energy-level splitting of Pr3＋ doped in LiYF4 and LiBiF4 crystals： a complete energy matrix calculation

Based on the combination of Racah's group-theoretical consideration with Slater's wavefunction, a 91 × 91 complete energy matrix is established in tetragonal ligand field D_2d for Pr³⁺ ion. Thus, the Stark energy-levels of Pr³⁺ ions doped separately in LiYF₄ and LiBiF₄ crystals are calculated, and our calculations imply that the complete energy matrix method can be used as an effective tool to calculate the energy-levels of the systems doped by rare earth ions. Besides, the influence of Pr³⁺ on energy-level splitting is investigated, and the similarities and the differences between the two doped crystals are demonstrated in detail by comparing their several pairs of curves and crystal field strength quantities. We see that the energy splitting patterns are similar and the crystal field interaction of LiYF₄:Pr³⁺ is stronger than that of LiBiF₄:Pr³⁺.     

Zero and finite temperature phase diagram of the spinless fermion model in infinite dimensions

The phase diagram of the model of spinless fermions with repulsive nearest neighbour interaction is calculated analytically on a hypercubic lattice in infinite dimensions (d → ∞). In spite of its simplicity the model displays a rich phase diagram depending on the doping δ, the interaction U and the temperature T. The system can be in the homogeneous phase (HOM), the nonsegregated AB charge density wave (AB-CDW), the AB phase separation region (PS-AB/HOM; coexistence of AB-CDW and HOM), the incommensurate phase (IP) or the IP phase separation region (PS-AB/IP; coexistence of AB-CDW and IP). We identify three important values of the interaction U_IPL = 0.572 < U_IPH = 1.914 < U_IP/PS = 4.212 which distinguish four intervals of U. These imply four different types of phase diagrams. In all the three phase diagrams with U below U_IP/PS the IP appears. We propose a new general ansatz for the order parameter of this phase. A competition between the IP, the PS-AB/IP and the PS-AB/HOM is found. The relevance of our findings for the phase scenario of the Hubbard model is shown.     

Mean Field Magnetic Phase Diagrams for the Two Dimensional <Emphasis Type="Italic">t</Emphasis> — <Emphasis Type="Italic">t</Emphasis>′ — <Emphasis Type="Italic">U</Emphasis> Hubbard Model

We study the ground state phase diagram of the two dimensional t — t′ — U Hubbard model concentrating on the competition between antiferro-, ferro-, and paramagnetism. It is known that unrestricted Hartree–Fock- and quantum Monte Carlo calculations for this model predict inhomogeneous states in large regions of the parameter space. Standard mean field theory, i.e., Hartree–Fock theory restricted to homogeneous states, fails to produce such inhomogeneous phases. We show that a generalization of the mean field method to the grand canonical ensemble circumvents this problem and predicts inhomogeneous states, represented by mixtures of homogeneous states, in large regions of the parameter space. We present phase diagrams which differ considerably from previous mean field results but are consistent with, and extend, results obtained with more sophisticated methods. PACS: 71.10.Fd, 05.70.Fh, 75.50.Ee