Usual Mean-field Approximation Research Articles

From entropic to energetic barriers in glassy dynamics: the Barrat–Mézard trap model on sparse networks

Trap models describe glassy dynamics as a stochastic process on a network of configurations representing local energy minima. We study within this class the paradigmatic Barrat–Mézard model, which has Glauber transition rates. Our focus is on the effects of the network connectivity, where we go beyond the usual mean field (fully connected) approximation and consider sparse networks, specifically random regular graphs (RRG). We obtain the spectral density of relaxation rates of the master operator using the cavity method, revealing very rich behaviour as a function of network connectivity c and temperature T. We trace this back to a crossover from initially entropic barriers, resulting from a paucity of downhill directions, to energy barriers that govern the escape from local minima at long times. The insights gained are used to rationalize the relaxation of the energy after a quench from high T, as well as the corresponding correlation and persistence functions.

One-meson-loop NJL model: Effect of collective and noncollective excitations on the quark condensate at finite temperature

We explore the effect of including quantum fluctuations in the two flavor Nambu$-$Jona-Lasinio model at finite temperature. This is accomplished, in a symmetry preserving way, by including collective and noncollective modes in the one-meson-loop gap equation which originate from poles and branch cuts in the complex plane, respectively. The inclusion of a boson cutoff, $\Lambda_b$, is necessary to regularize the meson-loop momenta. This new parameter is used to study the influence of going beyond the usual mean field approximation in the quark condensate. As the temperature increases, chiral symmetry tends to get restored, the collective modes melt and only noncollective modes contribute to the quark condensate. With the inclusion of such modes, the quark condensate at finite temperature has a different behavior than at mean field level that will be explored.

Vacuum effects on the properties of nuclear matter under an external magnetic field

The effects of the Dirac sea of the nucleons are investigated within a covariant model of the hadronic interaction. We extend the usual Mean Field Approximation and present a procedure to deal with divergences which are proportional to polynomials on the magnetic field intensity. For this purpose a nucleon propagator is used which takes accounts of the full effect of the magnetic field as well as the presence of the anomalous magnetic moments of both protons and neutrons. We examine single particle properties and bulk thermodynamical quantities and conclude that within a reasonable range of densities and magnetic intensities the effects found are moderate.

Quantum Landau damping in dipolar Bose-Einstein condensates

We consider Landau damping of elementary excitations in Bose-Einstein condensates (BECs) with dipolar interactions. We discuss quantum and quasi-classical regimes of Landau damping. We use a generalized wave-kinetic description of BECs which, apart from the long range dipolar interactions, also takes into account the quantum fluctuations and the finite energy corrections to short-range interactions. Such a description is therefore more general than the usual mean field approximation. The present wave-kinetic approach is well suited for the study of kinetic effects in BECs, such as those associated with Landau damping, atom trapping and turbulent diffusion. The inclusion of quantum fluctuations and energy corrections change the dispersion relation and the damping rates, leading to possible experimental signatures of these effects. Quantum Landau damping is described with generality, and particular examples of dipole condensates in two and three dimensions are studied. The occurrence of roton-maxon configurations, and their relevance to Landau damping is also considered in detail, as well as the changes introduced by the three different processes, associated with dipolar interactions, quantum fluctuations and finite energy range collisions. The present approach is mainly based on a linear perturbative procedure, but the nonlinear regime of Landau damping, which includes atom trapping and atom diffusion, is also briefly discussed.

Spin-1/2 anisotropic Heisenberg antiferromagnet model with Dzyaloshinskii-Moriya interaction via mean-field approximation

The spin-1/2 anisotropic Heisenberg model with antiferromagnetic exchange interactions in the presence of a longitudinal external magnetic field and a Dzyaloshinskii-Moriya interaction is studied by employing the usual mean-field approximation. The magnetic properties are obtained and it is shown that only second-order phase transitions take place for any values of the theoretical Hamiltonian parameters. Contrary to previous results from effective field theory, no anomalies have been observed at low temperatures. However, some reentrancies still persist in some region of the phase diagram.

Effects of thermal fluctuations on directed self-assembly in cylindrical confinement

We investigate the directed self-assembly (DSA) of cylinder-forming block copolymers inside cylindrical guiding templates. To complement and corroborate our experimental investigations, we use field-theoretic simulations to examine the fluctuation-induced variations in the size and position of the cylindrical microdomain that forms in the middle of the guiding hole. Our study goes beyond the usual mean-field approximation and self-consistent field theory simulations (SCFT) and incorporates the effects of thermal fluctuations in the description of the self-assembly process using complex Langevin (CL) dynamics. In addition to CL simulations, we present an efficient SCFT-based approach that can inform about the positional error of the formed cylinders. In this new scheme, an external chemical-potential field is applied to displace the inner cylinder away from its centered, lowest energy configuration. In both our experimental and modeling efforts, we focus on two wall-wetting conditions: (1) minor-block-attractive sidewalls and bottom substrates and neutral top surfaces and (2) neutral sidewalls, substrates, and top surfaces. For both cases, we explore the properties of the formed cylinders, including fluctuations in the center position and the size of the domain, for various prepattern conditions. Our results indicate robust critical dimensions (CDs) of the DSA cylinders relative to the prepattern CD, with a standard deviation <0.9 nm. Likewise, we find that the DSA cylinders are accurately registered in the center of the guiding hole, with deviations in the hole-in-hole distance on the order of ∼0.7 to 1.4 nm, translating to errors in the hole-to-hole distance of ∼1 to 2 nm.

Influence of surface and bulk layers on the crossover features for interaction parameters of a ferroelectric thin film

The phase diagram of a ferroelectric thin film is studied by using the usual mean-field approximation. The crossover features, from the ferroelectric-dominant phase diagram to the paraelectric-dominant phase diagram, for the physical parameters of the thin film are discussed. Special attention is devoted to the investigation of surface and bulk layers, and how to influence the crossover values of exchange interactions and transverse fields.

Phase diagram study of a dimerized spin-S zig–zag ladder

The phase diagram of a frustrated spin-S zig–zag ladder is studied through different numerical and analytical methods. We show that for arbitrary S, there is a family of Hamiltonians for which a fully-dimerized state is an exact ground state, being the Majumdar–Ghosh point for a particular member of the family. We show that the system presents a transition between a dimerized phase to a Néel-like phase for S = 1/2, and spiral phases can appear for large S. The phase diagram is characterized by means of a generalization of the usual mean field approximation. The novelty in the present implementation is to consider the strongest coupled sites as the unit cell. The gap and the excitation spectrum is analyzed through the random phase approximation. Also, a perturbative treatment to obtain the critical points is discussed. Comparisons of the results with numerical methods like the Density Matrix Renormalization Group are also presented.

Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model.

We study random transitions between two metastable states that appear below a critical temperature in a one-dimensional self-gravitating Brownian gas with a modified Poisson equation experiencing a second order phase transition from a homogeneous phase to an inhomogeneous phase [P. H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103 (2010)]. We numerically solve the N-body Langevin equations and the stochastic Smoluchowski-Poisson system, which takes fluctuations (finite N effects) into account. The system switches back and forth between the two metastable states (bistability) and the particles accumulate successively at the center or at the boundary of the domain. We explicitly show that these random transitions exhibit the phenomenology of the ordinary Kramers problem for a Brownian particle in a double-well potential. The distribution of the residence time is Poissonian and the average lifetime of a metastable state is given by the Arrhenius law; i.e., it is proportional to the exponential of the barrier of free energy ΔF divided by the energy of thermal excitation kBT. Since the free energy is proportional to the number of particles N for a system with long-range interactions, the lifetime of metastable states scales as eN and is considerable for N≫1. As a result, in many applications, metastable states of systems with long-range interactions can be considered as stable states. However, for moderate values of N, or close to a critical point, the lifetime of the metastable states is reduced since the barrier of free energy decreases. In that case, the fluctuations become important and the mean field approximation is no more valid. This is the situation considered in this paper. By an appropriate change of notations, our results also apply to bacterial populations experiencing chemotaxis in biology. Their dynamics can be described by a stochastic Keller-Segel model that takes fluctuations into account and goes beyond the usual mean field approximation.

Bose–Einstein condensate in a double-well potential: Feynman path integral variational approach

A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.

Effects of surface modification on the critical behaviour in multiple-surface-layer ferroelectric thin films

Using the usual mean-field theory approximation, the critical behaviour (i.e. the Curie temperature Tc and the critical surface transverse field Ω sc ) in a multiple-surface-layer ferroelectric thin film is studied on the basis of the spin- transverse Ising model. The dependence of the Curie temperature Tc on the surface transverse field Ω s and the surface layer number Ns are discussed in detail. Meanwhile the dependence of the critical surface transverse field Ω sc on the surface layer number Ns is also examined. The numerical results indicate that the critical behaviour of ferroelectric thin films is obviously affected by modifications of the surface transverse field Ω s and surface layer number Ns.

Dynamics of chemical reacting systems in low dimensional lattices via the Equation Free Method

In this report we show how it is possible to reconstruct the dynamics of a chemical reacting systems on lattice via an appropriate choice of macroscopic variable set. When chemical reactions fire on a low dimensional setting the usual mean field approximation (MF) may fail to accurately describe the proper dynamics of the system. Consequently it is not possible to use the concentrations of different species as sole quantities to describe macroscopically such systems and more involved approximations are needed. Traditional approaches introduces higher order correlation functions as extra macroscopic variables, however such expansions usually require closure procedures and tend to quickly demand high number of terms even for simple systems. It is also possible to describe the system in a microscopic way via a Kinetic Monte Carlo (KMC) algorithm, but this approach requires much longer computation time. In this report we study such a problem from a different perspective using the Equation Free Method (EFM) framework; with such approach it is possible to “bridge” between the macroscopic dynamics (mean field) and the microscopic one (KMC). We analyze different techniques to move between these two descriptions of the system trying to understand what information needs to be preserved at the macroscopic level in order to recover correctly all the features of the microscopic dynamics.

Fulde-Ferrell-Larkin-Ovchinnikov state in the dimensional crossover between one- and three-dimensional lattices

We present a full phase diagram for the one-dimensional (1D) to three-dimensional (3D) crossover of the Fulde-Ferrell--Larkin-Ovchinnikov (FFLO) state in an attractive Hubbard model of 3D-coupled chains in a harmonic trap. We employ real-space dynamical mean-field theory which describes full local quantum fluctuations beyond the usual mean-field and local density approximation. We find strong dimensionality effects on the shell structure undergoing a crossover between distinctive quasi-1D and quasi-3D regimes. We predict an optimal regime for the FFLO state that is considerably extended to intermediate interchain couplings and polarizations, directly realizable with ultracold atomic gases. We find that the 1D-like FFLO feature is vulnerable to thermal fluctuations, while the FFLO state of mixed 1D-3D character can be stabilized at a higher temperature.

Phase diagrams of a ferroelectric superlattice: A Green’s function technique study

A Fermi-type Green’s function method has been used to investigate the phase transition properties of a ferroelectric superlattice with two alternating materials on the basis of the transverse Ising model. By performing a higher-order decoupling to the equations of motion for the Green’s functions, the eigenfrequencies of the infinite ferroelectric superlattice are obtained. Moreover, we discuss the dependence of the phase diagrams on the interface coupling strength, the transverse field, and the thicknesses of two slabs. The comparison between the Green’s function technique and the usual mean-field approximation is illustrated.

Dynamic properties of phase diagram in cylindrical ferroelectric nanotubes

The dynamic properties of phase diagrams in cylindrical ferroelectric nanotubes were investigated by utilizing the effective-field theory with correlations. Three different structure mechanisms are discussed by performing three different interaction functions. Effects of structure factors of the ferroelectric nanotubes are shown in phase diagrams. A linear relation between the crossover value of transverse field in inner-layer of nanotube and the exchange interaction are revealed. Moreover, for the first time, the phase transition characteristics are shown both in 2-D and 3-D phase diagrams by comparing the variations of the ferroelectric region and the morphology vividly. In comparison, results of the usual mean-field approximation are also presented in some phase diagrams, it suggest that the differential operator technique with correlations reduces some extent of the ferroelectric features of the ferroelectric nanotube.

The van Hemmen–Kondo model for disordered cerium systems

The interplay between disorder and strong correlations has been observedexperimentally in disordered cerium alloys such as Ce(Ni, Cu) orCe(Pd, Rh). In the case of Ce(Ni, Cu) alloys with a Cu concentrationx between 0.6 and 0.3, the first studies have shown a smooth transition withdecreasing temperature from a spin glass phase to ferromagnetism; forx smaller than 0.2, a Kondo phase has been observed. The situation is more complicated nowdue to the recent observation of magnetic clusters. The competition between the Kondo effect,the spin glass (SG) and the ferromagnetic (FE) ordering has been extensively studiedtheoretically. The Kondo effect is described by the usual mean-field approximation; we havetreated the SG behavior successively by the Sherrington–Kirkpatrick model, then by theMattis model and finally by the van Hemmen model, which takes both a ferromagnetic partand a site-disorder random part for the intersite exchange interaction. We present here theresults obtained by the van Hemmen–Kondo model: for a large Kondo exchangeJK, a Kondo phase is obtained while, for smallerJK, the succession of an SG phase, a mixed SG–FE one and finally an FE one has beenobtained with decreasing temperature. This model improves the theoretical description ofdisordered Kondo systems by providing a simpler approach for further calculations ofmagnetic clusters and can, therefore, account for recent experimental data on disorderedcerium systems.

Three-site Bose-Hubbard model subject to atom losses: Boson-pair dissipation channel and failure of the mean-field approach

We employ the perturbation series expansion for derivation of the reduced master equations for the three-site Bose-Hubbard model subject to strong atom losses from the central site. The model describes a condensate trapped in a triple-well potential subject to externally controlled removal of atoms. We find that the $\ensuremath{\pi}$-phase state of the coherent superposition between the side wells decays via two dissipation channels, the single-boson channel (similar to the externally applied dissipation) and the boson-pair channel. The quantum derivation is compared to the classical adiabatic elimination within the mean-field approximation. We find that the boson-pair dissipation channel is not captured by the mean-field model, whereas the single-boson channel is described by it. Moreover, there is a matching condition between the zero-point energy bias of the side wells and the nonlinear interaction parameter which separates the regions where either the single-boson or the boson-pair dissipation channel dominate. Our results indicate that the $M$-site Bose-Hubbard models, for $M&gt;2$, subject to atom losses may require an analysis which goes beyond the usual mean-field approximation for correct description of their dissipative features. This is an important result in view of the recent experimental works on the single-site addressability of condensates trapped in optical lattices.

Green's Function Approach to Crossover Properties for Interaction Parameters of an N-Layer Ferroelectric Thin Film

Utilizing the higher order decoupling approximation to the Fermi-type Green’s function, crossover properties of interaction parameters of an n-layer ferroelectric thin film from the ferroelectric-dominant phase diagram (FPD) to the paraelectric-dominant phase diagram (PPD) are investigated on the basis of the transverse Ising model. The curved surfaces for crossover values of interaction parameters of a thin film with certain layers are constructed in the three-dimensional parameter space. Because both the z-component &lt;Sz&gt; (the polarization) and the transverse component &lt;Sx&gt; of the spin are further included in the eigenfrequency, the results are in agreement with that of the effective-field theory with correlations to some extent. It shows that the higher order decoupling approximation diminishes the ferroelectric feature of a ferroelectric thin film compared with the usual mean-field approximation.

A general analytical equation for phase diagrams of an N-layer ferroelectric thin film with two surface layers

Within the framework of effective-field theory with correlations, the phase diagrams of an N-layer ferroelectric thin film with two surface layers are studied by the differential operator technique based on the spin-1/2 transverse Ising model. A general analytical equation for the phase diagram of a ferroelectric thin film with arbitrary layer number as well as exchange interactions and transverse fields is derived, and then the effects of exchange interactions and transverse fields on phase diagrams are discussed for an arbitrary layer number N. Meanwhile, the crossover features, from the ferroelectric-dominant phase diagram (FPD) to the paraelectric-dominant phase diagram (PPD), for various parameters of an N-layer ferroelectric thin film with two surface layers are investigated. As a result, an N-independent common intersection point equation is obtained, and the three-dimensional curved surfaces for the crossover values are constructed. In comparison with the usual mean-field approximation, the differential operator technique with correlations reduces to some extent the ferroelectric features of a ferroelectric thin film.

Ultrafast magnetization dynamics in diluted magnetic semiconductors

We present a dynamical model that successfully explains the observed time evolution of the magnetization in diluted magnetic semiconductor quantum wells after weak laser excitation. Based on the pseudo-fermion formalism and a second-order many-particle expansion of the exact p–d exchange interaction, our approach goes beyond the usual mean-field approximation. It includes both the sub-picosecond demagnetization dynamics and the slower relaxation processes that restore the initial ferromagnetic order in a nanosecond timescale. In agreement with experimental results, our numerical simulations show that, depending on the value of the initial lattice temperature, a subsequent enhancement of the total magnetization may be observed within the timescale of a few hundred picoseconds.