Improved Mean Field Approximation Research Articles

Thermodynamic states of the mixed spin 1/2 and spin 1 hexagonal nanotube system obtained from a eighteen-site cluster within an improved mean field approximation

The mean field approximation results in the mixed spin 1/2 Ising model and spin 1 Blume–Capel model, in the hexagonal nanotube two layers system, are obtained from the Bogoliubov inequality. The Gibbs free energy, magnetization and critical frontiers are obtained. Besides the stable branches of the order parameters, we obtain the metastable and unstable parts of these curves and also find phase transitions of the metastable and unstable branches of the order parameters. The classification of the stable, metastable and unstable states is made by comparing the free energy values of these states. An uncommon increase in the magnetization, already observed by Canko et al., is described by the presence of an unstable state with order parameter nonzero.

Thermodynamic States of the Mixed Spin 1/2 and Spin 1 Hexagonal Nanowire System Obtained from a Seven-Site Cluster Within an Improved Mean Field Approximation

The mean field approximation results in the mixedspin 1/2 Ising model and spin 1 Blume-Capel model, in the hexagonal nanowire system, are obtained from the Bogoliubov inequality. The Gibbs free energy, magnetization, and critical frontiers are obtained. Besides the stable branches of the order parameters, we obtain the metastable and unstable parts of these curves and also find phase transitions of the metastable branches of the order parameters. The classification of the stable, metastable, and unstable states is made by comparing the free energy values of these states.

Generalized Potts-Models and their Relevance for Gauge Theories

We study the Polyakov loop dynamics originating from finite-temperature Yang- Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and antiferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents and at the continuous transition between symmetric and antiferromagnetic phases are the same as for the 3-state spin Potts model. Symmetry constraints and strong coupling expansion for the effective action describing the Polyakov loop dynamics of gauge theories lead to effective field theories with rich phase struc- tures. The fields are the fundamental characters of the gauge group with the fundamental domain as target space. The center symmetry of pure gauge theory remains a symmetry of the effective models. If one further freezes the Polyakov loop to the center Z of the gauge group one obtains the well known vector Potts spin-models, sometimes called clock models. Hence we call the effective theories for the Polyakov loop dynamics generalized Z-Potts models. We review our recent results on generalized Z3-Potts models (1). These results were obtained with the help of an improved mean field approximation and Monte Carlo simulations. The mean field approximation turns out to be much better than expected. Probably this is due to the existence of tricritical points in the effective theories. There exist four distinct phases and transitions of first and second order. The critical exponents and at the second order transition from the symmetric to antiferromagnetic phase for the generalized Potts model are the same as for the

Higher Order Terms in the Improved Mean Field Approximation for the IIB Matrix Model and the Emergence of Four-Dimensional Space-Time

The spontaneous breakdown of SO(10) symmetry in the IIB matrix model has been studied using the improved mean field approximation (IMFA). In this report, the eighth-order contribution to the improved perturbative series is obtained, which involves the evaluation of 20410 planar two-particle irreducible vacuum diagrams. We consider SO(d)-preserving configurations as an ansatz (d = 4,7). The development of a plateau, representing the condition of self-consistency, is seen in both ansatz. A large ratio of the space-time extent of the d-dimensional part to that of the remaining (10 - d)-dimensional part is obtained for the SO(4) ansatz evaluated at representative points of the plateau. This is interpreted as the emergence of four-dimensional space-time in the IIB matrix model.

Stability of 4-Dimensional Space-Time from the IIB Matrix Model via the Improved Mean Field Approximation

We investigate the origin of our four-dimensional space-time by considering dynamical aspects of the IIB matrix model using the improved mean field approximation. Previous works have focused on the specific choices of configurations as ansatz which preserve SO(d) rotational symmetry. In this report, an extended ansatz is proposed and examined up to a third-order approximation which includes both the SO(4) ansatz and the SO(7) ansatz in their respective limits. From the solutions of the self-consistency condition represented by the extrerna of the free energy of the system, it is found that some of the solutions found in the SO (4) or SO (7) ansatz disappear in the extended ansatz. This implies that the extension of ansatz can be used to distinguish stable solutions from unstable solutions. It is also found that there is a non-trivial accumulation of extrema including the SO(4)-preserving solution, which may lead to the formation of a plateau.

An improved mean field approximation on the worldsheet for planar φ3 theory

We present an improved version of our earlier work on summing the planar graphs in φ 3 field theory. The present treatment is also based on our world sheet formalism and the mean field approximation, but it makes use of no further approximations. We derive a set of equations between the expectation values of the world sheet fields, and we investigate them in certain limits. We show that the equations can give rise to (metastable) string forming solutions.

Improved Perturbation Theory and Four-Dimensional Space-Time in the IIB Matrix Model

We have analyzed the IIB matrix model on the basis of the improved mean field approximation (IMFA) and have obtained evidence suggesting that the four-dimensional space-time appears as its most stable vacuum.This method is a systematic way to obtain an improved perturbation series and was first applied to the IIB matrix model by Nishimura and Sugino. In a previous paper, we reformulated this method and proposed a criterion for the convergence of the improved series, that is, the appearance of a “plateau”. In this paper, we carry out higher-order calculations, and find that our improved free energy tends to have a plateau, which shows that IMFA works well in the IIB matrix model.

Mean field approximation of IIB matrix model and emergence of four-dimensional space–time

For the purpose of analyzing non-perturbative dynamics of string theory, Nishimura and Sugino have applied an improved mean field approximation (IMFA) to IIB matrix model. We have extracted the essence of the IMFA and obtained a general scheme, the improved Taylor expansion, that can be applied to a wide class of series which is not necessarily convergent. This approximation scheme with the help of the 2PI free energy enables us to perform higher order calculations. We have shown that the value of the free energy is stable at higher orders, which supports the validity of the approximation. Moreover, the ratio between the extent of “our” space–time and that of the internal space is found to increase rapidly as we take the higher orders into account. Our results suggest that the four-dimensional space–time emerges spontaneously in IIB matrix model.

An Improved Mean Field Approximation

AbstractAn upper bound for the free energy may generate an approximation which preserves short order range effects even when a sum of single spins is used as the auxiliary Hamiltonian. Unlike the usual mean field approximation it may distinguish the cubic lattice from the planar triangular lattice. Its application to the Ising chain recovers some exact results.

Ultrasound in spin glasses

The change of the sound velocity δv(ω,T) and the damping of sound waves γ(ω,T) in spin glasses are calculated in the frame-work of an Ising model with a random distribution of exchange interactions. The calculation is based on linearized equations of motion for the spins and on an improved mean field approximation which includes the Onsager reaction field. Near to the freezing temperatureTf and at high temperatures δv(ω,T) and ωγ(ω,T) turn out to be approximately proportional to the real and the imaginary parts of the dynamical susceptibility. For the special case of infinite range interactions atT=Tf one has δv(ω, Tf) ∞ (ω τ)1/2 and γ(ω, Tf) ∞ (τ/ω)1/2 where τ is the relaxation time of independent spins. However, already slightly aboveTf the frequency dependence of both quantities becomes rather small for RKKY spin glasses. At high temperatures both, δv(ω,T) and γ(ω,T) vary asT−1.