An Improved Heterogeneous Mean-Field Theory for the Ising Model on Complex Networks

Heterogeneous mean-field theory is commonly used methodology to study dynamical processes on complex networks,such as epidemic spreading and phase transitions in spin models.In this paper,we propose an improved heterogeneous mean-field theory for studying the Ising model on complex networks.Our method shows a more accurate prediction in the critical temperature of the Ising model than the previous heterogeneous mean-field theory.The theoretical results are validated by extensive Monte Carlo simulations in various types of networks.     

Dual symmetries and universality of the critical behavior of a disordered ising ferromagnet

The critical behavior of a d-dimensional random Ising ferromagnet described in the critical region by the Ginzburg-Landau model with a random temperature has been considered. The critical temperature dependences of thermodynamic quantities, the equation of state, and the spin correlation function at the Curie point have been calculated by the renormalization group method. The most profound questions of the theory of impurity ferromagnets are the problem of universality and the role of the Griffiths singularities.     

Broad histogram Monte Carlo

We propose a new Monte Carlo technique in which the degeneracy of energy states is obtained with a Markovian process analogous to that of Metropolis used currently in canonical simulations. The obtained histograms are much broader than those of the canonical histogram technique studied by Ferrenberg and Swendsen. Thus we can reliably reconstruct thermodynamic functions over a much larger temperature scale also away from the critical point. We show for the two-dimensional Ising model how our new method reproduces exact results more accurately and using less computer time than the conventional histogram method. We also show data in three dimensions for the Ising ferromagnet and the Edwards Anderson spin glass. Received: 8 August 1997 / Revised: 11 August 1997 / Accepted: 30 October 1997     

Two analytic approximations for thed+1 ising model

Summary The Hamiltonian Ising model ind+1 dimensions is studied by means of two new approximation methods which exploit the geometry of the model. Explicit results for the critical temperature and observables of the theory are obtained. The authors of this paper have agreed to not receive the proofs for correction.     

The Ising model on random lattices in arbitrary dimensions

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.     

Polyakov loop and spin correlators on finite lattices A study beyond the mass gap

We derive an analytic expression for point-to-point correlation functions of the Polyakov loop based on the transfer matrix formalism. For the 21) Ising model we show that the results deduced from point-point spin correlators are coinciding with those from zero momentum correlators. We investigate the contributions from eigenvalues of the transfer matrix beyond the mass gap and discuss the limitations and possibilities of such an analysis. The finite size behaviour of the obtained 21) Ising model matrix elements is examined. The point-to-point correlator formula is then applied to Polyakov loop data in finite temperature SU(2) gauge theory. The leading matrix element shows all expected scaling properties. Just above the critical point we find a Debye screening mass μ_D/T ≈ 4 , independent of the volume.     

The transverse ferromagnet spin-1 Ising model of alternating magnetic superlattice

Within the framework of the effective field theory with a probability distribution technique that accounts for the single-site spin correlations, we examine the critical behavior of the transverse ferromagnetic spin-1 Ising model of an alternating magnetic superlattice. The critical temperature of the alternating magnetic superlattice has been studied as a function of the interlayer and intralayer exchange interactions and the strength of the transverse field and the thickness of the finite superlattice. Received 12 January 2000 and Received in final form 14 September 2000     

Two-Site Effective Field Approximation for Mixed Spin Ising Model

A simple and powerful method (two-site effective field approximation) for mixed spin Ising model was presented. Our result about transition temperature of mixed Ising spin system is much better than that by making use of other approximate methods.     

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

The fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.     

Deconfinement transition and dimensional cross-over in the 3D gauge Ising model

We present a high-precision Monte Carlo study of the finite-temperature gauge theory in 2 + 1 dimensions. The duality with the 3D Ising spin model allows us to use powerful cluster algorithms for the simulations. For temporal extensions of up to N_t = 16 we obtain the inverse critical temperature with a statistical accuracy comparable with the most accurate results for the bulk phase transition of the 3D Ising model. We discuss the predictions of T.W. Capehart and M.E. Fisher for the dimensional cross-over from 2 to 3 dimensions. Our precise data for the critical exponents and critical amplitudes confirm the Svetitsky-Yaffe conjecture. We find deviations from Olesen's prediction for the critical temperature of about 20%.     

Magnetic properties of 2D nano-islands II: Ising spin model with out-of-plane magnetic field

An Ising effective field theory model is presented to calculate the magnetic properties of 2D nano-islands on a nonmagnetic substrate, subject to an externally out-of-plane applied magnetic field. The system Hamiltonian contains nearest neighbor exchange interactions, single-atom magnetic anisotropies, and the Zeeman term. The calculations yield, in particular, the single site spin correlations, the magnetizations, and the isothermal susceptibilities, for the core and periphery domains of the nano-island. The choice of a spin S=1 for the atoms of the system permits the analysis of local spin fluctuations via the single site spin correlations. We investigate in this respect the effects due to the different magnetocrystalline anisotropies and reduced dimensionalities, for the core and periphery domains, and in particular the critical influence of the applied magnetic field. Detailed theoretical results are presented for the square and hexagonal lattice symmetries, with numerical applications for the 2D monolayer Co nano-islands on a Pt substrate. It is shown that the remarkable differences between the magnetic properties of the core and periphery domains in zero field are washed out when an out-of-plane field is applied. The applied field also provokes critical discontinuities for the spin correlations and magnetization reversals, for the core and periphery domains, which are especially evident for the hexagonal lattice nano-island in the range of fields of interest. The discontinuities and magnetization reversals occur over elementary temperature widths, and shift to lower temperatures with increasing field. The field-dependant isothermal susceptibilities show new features very different from those for the susceptibilities in zero field. The present Ising model does not show any blocking temperature transition to superparamagnetism.     

Diagrammatic decompositions on lattice for correlation functions of Ising spin models by iterative method

In the present paper the iterative method enables us to calculate correlation functions of Ising spin models by two approximate ways. The equations resulting from two models A and B, based on different physical considerations concerning correlations in surroundings of the reference spin, are solved for one, two, and three-dimensional Ising models with nearest neighbour interactions. Except of anomalies occuring in low-dimensional systems the model A leads to the critical pointA _c=0.191 for the three-dimensional cubic Ising model.     

Spin compensation temperatures induced by longitudinal fields in a mixed spin-3/2 and spin-5/2 Ising ferrimagnet

I studied the ferrimagnetic Ising model with nearest neighbour interactions for a square lattice and simple cubic one, using mean field theory. The free energy of a mixed spin Ising ferrimagnetic model was calculated from a mean field approximation of the Hamiltonian. By minimizing the free energy, I obtained the equilibrium magnetizations and the compensation temperatures. Clear indications of the single-ion anisotropies on the compensation points of the mixed spin-3/2 and spin-5/2 ferrimagnetic lattices are found. Some interesting behaviors of these systems are obtained depending not only on the values of magnetic anisotropies for both sublattice sites but also on the lattice structure. The longitudinal magnetic fields dependence of the spin compensation temperature is the main focus of research. The possibility of many compensation temperatures is indicated.     

Application of the real space dynamic renormalization group method to the one-dimensional kinetic ising model

We apply the recently developed real space dynamic renormalization group method to the one-dimensional kinetic Ising model. We show how one can develop block spin methods that lead to recursion relations for the space and time dependent correlation functions that correspond to the observables for this system. We point out the importance of carefully choosing the appropriate parameters governing the behavior of individual blocks of spins and the necessity of worrying about the high temperature properties of the temperature recursion relations if one is to obtain the proper exponential decay of correlation functions at large distances away from the critical point at zero temperature. We systematically investigate the accuracy of our approximate recursion relations for various correlation functions by checking them against the known exact results. Our simple methods work surprisingly well over a wide range of temperatures, wavenumbers and frequencies.     

Slowing down in the three-dimensional three-state Potts glass with nearest neighbor exchange ± J: A Monte Carlo study

,Static and dynamic properties of the Potts model on the simple cubic lattice with nearest neighbor -interaction are obtained from Monte Carlo simulations in a temperature range where full thermal equilibrium still can be achieved (). For a lattice size L = 16, in this range finite size effects are still negligible, but the data for the spin glass susceptibility agree with previous extrapolations based on finite size scaling of very small lattices. While the static properties are compatible with a zero temperature transition, they certainly do not prove it. Unlike the Ising spin glass, the decay of the time-dependent order parameter is compatible with a simple Kohlrausch function, , while a power law prefactor cannot be distinguished. The Kohlrausch exponent y ( T ) decreases from at [0pt] to at [0pt] however. The relaxation time is compatible with the exponential divergence postulated by McMillan for spin glasses at their lower critical dimension, but the exponent that can be extracted still differs significantly from the theoretical value, . Thus the present results support the conclusion that the Potts spin glass in d = 3 dimensions differs qualitatively from the Ising spin glass. Received: 8 October 1997 / Accepted: 27 November 1997     

A MONTE CARLO RENORMALIZATION GROUP STUDY OF A 2-DIMENSIONAL TRIANGULAR LATTICE

The method developed by D.J.E.Callaway is applied to Ising model on a two-dimensional triangular lattice. A fixed point and critical exponent are found. The results are consistent with one of the exact theories very well. Obviously this method show superiority to that obtained by some other approximate methods. The method is also applied to Z₂ gauge theory on a 2-dimensional triangular lattice, no fixed points are found, in agreement with other methods.     

Ising Models with Four Spin Interaction at Criticality

We consider two bidimensional Ising models coupled by an interaction quartic in the spins, like in the spin representation of the Eight vertex or the Ashkin-Teller model. By Renormalization Group methods we write a convergent perturbative expansion for the specific heat and for the energy-energy correlation up to the critical temperature. A form of nonuniversality is proved, in the sense that the critical behaviour is described in terms of critical indices which are non trivial functions of the coupling. The logarithmic singularity of the specific heat of the Ising model is removed or changed in a power law (with a non universal critical index) depending on the sign of the interaction.     

Some rigorous results on majority rule renormalization group transformations near the critical point

We consider the majority rule renormalization group transformation with two-by-two blocks for the Ising model on a two-dimensional square lattice. For three particular choices of the block spin configuration we prove that the model conditioned on the block spin configuration remains in the high-temperature phase even when the temperature is slightly below the critical temperature of the ordinary Ising model with no conditioning. We take as the definition of the infinite-volume limit an equation introduced in earlier work by the author. We use a computer to find an approximate solution of this equation and verify a condition which implies the existence of an exact solution.     

Determination of the orientational order parameter of the homologous series of 4-cyanophenyl 4-alkylbenzoate (n.CN) by different methods

The orientational order parameters of a homologous series of 4-cyanophenyl 4-alkylbenzoates have been determined at different temperatures from (13)C-NMR, x-ray diffraction, optical birefringence, high resolution density and diamagnetic susceptibility anisotropy measurements. To determine the temperature dependence of the orientational order parameter from the (13)C chemical shift values, the two-dimensional polarization inversion spin exchange at the magic angle (PISEMA) method was also used for the measurement of (13)C-(1)H dipolar couplings at different sites in the oriented phase. The nematic order parameters determined from each of these methods have been compared. Apart from a slight shift in their values their trends with temperature are very similar. The differences among the results obtained by these five different methods have been discussed. From the high resolution density data, the values of the critical exponents near the T(N-I) transition are found to lie between the Ising model and tricritical behaviour.     

Block spins in the edge of an Ising ferromagnetic half-plane

The characteristic function of a block spin in the face of an Ising ferromagnetic half-plane is obtained in closed form. The distribution function for the block spin converges to a Gaussian at the critical temperature, but the normalization of the block is modified.Partially supported by NRC grant A9344.