Degenerate approach to the mean field Bose-Hubbard Hamiltonian

A degenerate variant of mean field perturbation theory for the on-site Bose-HubbardHamiltonian is presented. We split the perturbation into two terms and perform exactdiagonalization in the two-dimensional subspace corresponding to the degenerate states.The final relations for the second order ground state energy and first order wave functiondo not contain singularities at integer values of the chemical potentials. The resultingequation for the phase boundary between superfluid and Mott states coincides with theprediction from the conventional mean field perturbation approach.     

Metastability and the Ising model

We discuss a recent theorem which establishes a precise connection between (i) the approximate degeneracy of the zero eigenvalue for the generator of the Glauber dynamics of the Ising model in a small nonzero field and below the critical temperature, (ii) the existence of a partition of the configuration space into a normal region and a metastable region. This enables us to demonstrate that the recent approach to metastability of Davies and Martin may be viewed as a simple (although in some ways fairly crude) approximation to the conventional approach. We also obtain what appear to be the first results concerning the stability of metastable states under small perturbations.     

Dynamic mean field in the Hubbard model

In the framework of the one-impurity problem formulated for the nondegenerate Hubbard Hamiltonian, general expressions are derived for all its principal correlation functions, as well as its retarded and advanced thermodynamic (Matsubara) Green functions. These results are analyzed for the case of zero temperature.     

Metastability and phase transitions in field theory

λφ⁴ theory is investigated at finite temperature in the presence of an external current, j. Metastable situations are studied that may arise as the sign of j is reversed and they are related to the phase transition that takes place as j→0, β→β_c. A semiclassical approximation to the theory is used, which corresponds, in statistical mechanics, to the “droplet-bubble”.     

Metastability in the two-dimensional ising model

Metastability in the Ising model is studied in two ways. In a dynamical Monte Carlo model, metastable magnetization and lifetime are measured for various magnetic fields and low temperatures. Following up a proposed relation between analytic continuation of transfer matrix eigenvalues and metastability, transfer matrix eigenvalues are studied. We examine the extent to which these approaches agree. The Monte Carlo data also provide quantitative support for the critical droplet model for decay.     

Hyperon--hyperon interaction in relativistic mean field model

By using the new experimental data of \Lambda\Lambda potential, this paper has performed a full calculation for strange hadronic matter with different strangeness contents as well as its consequences on the global properties of neutron star matter in relativistic mean field model. It finds that the new weak hyperon--hyperon interaction makes the equations of state much stiffer than the result of the previous strong hyperon--hyperon interaction, and even stiffer than the result without consideration of hyperon--hyperon interaction. This new hyperon--hyperon interaction results in a maximum mass of 1.75M_{\odot} (where M_{\odot} stands for the mass of the Sun), about 0.2--0.5M_{\odot} larger than the previous prediction with the presence of hyperons. After examining carefully the onset densities of kaon condensation it finds that this new weak version of hyperon--hyperon interaction favours the occurrence of kaons in comparison with the strong one.     

Mechanism of desynchronization in the finite-dimensional Kuramoto model

We study how a decrease of the coupling strength causes a desynchronization in the Kuramoto model of N globally coupled phase oscillators. We show that, if the natural frequencies are distributed uniformly or close to that, the synchronized state can robustly split into any number of phase clusters with different average frequencies, even culminating in complete desynchronization. In the simplest case of N=3 phase oscillators, the course of the splitting is controlled by a Cherry flow. The general N-dimensional desynchronization mechanism is numerically illustrated for N=5.     

Saddle point mean field calculation in the Hubbard model

Using functional techniques, the Hubbard model consisting of self-coupled lattice fermions is transformed into a bosonic functional integral. In this form it is treated by a mean field approximation in its modern version as a systematic expansion around a saddle point. The method is generalized to the extended Hubbard model.     

Nonlocal PNJL model beyond mean field

A nonlocal chiral quark model is consistently extended beyond mean field using a strict 1/N _c expansion scheme. It is found that the 1/N _c corrections lead to a lowering of the temperature of the chiral phase transition in comparison with the mean-field result. On the other hand, near the phase transition the 1/N _c expansion breaks down and a nonperturbative scheme for the inclusion of mesonic correlations is needed in order to describe the phase transition region.     

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.     

Exploiting the Hamiltonian structure of a neural field model

We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model’s behaviour. The lack of robustness of the model to small perturbations in parameters is surprising, and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the time-independent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves. By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.     

Metastability and analyticity in a dropletlike model

We consider an urn model closely related to the Fisher-Felderhof droplet model for the purpose of studying the relation between metastability and analytic continuation. For this model both the statics and dynamics can be solved and we confirm the relation between the metastable decay rate and the imaginary part of the analytically continued free energy (actually, pressure, in this model). We also find that eigenvalue degeneracy, an old theme for static aspects of phase transitions, appears in the dynamics as well. When approaching the phase transition from the stable side it is a degeneracy in the eigenvalues of the linear operator appearing in the master equation that causes the system to lock into a particular phase.Supported in part by the US-Israel Binational Science Foundation and the Technion Fund for the Encouragement of Research.     

Metastability in the Potts model on the Cayley tree

Metastability in the ferromagneticp-state Potts model defined on the Cayley tree is discussed. It is shown that the sign of the boundary fieldH _s determines the order of the transition as well as the stability of the low-temperature phase. Lowering the temperature withH _s >0, a system withp<2 (p>2) will display a second (first)-order transition to a metastable (stable) phase. ForH _s >0 a second (first)-order transition to a metastable (stable) phase occurs ifp>2 (p<2). In this case the system also has a residual entropy which is negative forp<2.     

Self-consistent mean field approximation and application in three-flavor NJL model

In this study, we apply a self-consistent mean field approximation of the three-flavor Nambu–Jona-Lasinio(NJL) model and compare it with the two-flavor NJL model. The self-consistent mean field approximation introduces a new parameter, α, that cannot be fixed in advance by the mean field approach itself. Due to the lack of experimental data, the parameter, α, is undetermined. Hence, it is regarded as a free parameter and its influence on the chiral phase transition of strong interaction matter is studied based on this self-consistent mean field approximation. α affects numerous properties of the chiral phase transitions, such as the position of the phase transition point and the order of phase transition. Additionally, increasing α will decrease the number densities of different quarks and increase the chemical potential at which the number density of the strange quark is non-zero. Finally, we observed that α affects the equation of state(EOS) of the quark matter, and the sound velocity can be calculated to determine the stiffness of the EOS, which provides a good basis for studying the neutron star mass-radius relationship.     

The mean field theory of the three-dimensional ANNNI model

The mean field equations of the simple cubic or tetragonal ANNNI model are studied on finite lattices. Structure combination branching processes are found which allow us to considerably refine previous mean field calculations on the model.     

Aging and non-linear glassy dynamics in a mean field model

The mean field approach of glassy dynamics successfully describes systems which are out-of-equilibrium in their low temperature phase. In some cases an aging behaviour is found, with no stationary regime ever reached. In the presence of dissipative forces however, the dynamics is indeed stationary, but still out-of-equilibrium, as inferred by a significant violation of the fluctuation dissipation theorem. The mean field dynamics of a particle in a random but short-range correlated environment, offers the opportunity of observing both the aging and driven stationary regimes. Using a geometrical approach previously introduced by the author, we study here the relation between these two situations, in the pure relaxational limit, i.e. the zero temperature case. In the stationary regime, the velocity (v)-force (F) characteristics is a power law v∼F ⁴, while the characteristic times scale like powers of v, in agreement with an early proposal by Horner. The cross-over between the aging, linear-response regime and the non-linear stationary regime is smooth, and we propose a parametrization of the correlation functions valid in both cases, by means of an “effective time”. We conclude that aging and non-linear response are dual manifestations of a single out-of-equilibrium state, which might be a generic situation. Received 7 May 2000 and Received in final form 22 August 2000     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.