The Hydrodynamic Limit for Local Mean-Field Dynamics with Unbounded Spins

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean-field interactions. We prove convergence of the empirical measure to the solution of a McKean–Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of Gärtner, Comets and others for bounded spins or strict mean-field interactions.     

Hall Effect in a Doped Mott Insulator: DMFT Approximation

JETP Letters - In the framework of dynamical mean-field theory, we analyze the Hall effect in a doped Mott insulator as a parent cuprate superconductor. We consider the partial filling (hole...     

Bistability of Feshbach resonance in optical cavity

We consider Feshbach resonance in an optical cavity where photons interact with atoms and molecules dispersively. From mean-field theory we obtain multiple fixed-point solutions, which is strongly related to the phenomenon of bistability. Adiabatic evolutions demonstrate hysteretic behaviors by varying pump-cavity detuning from opposite directions. We also use the quantum model to check mean-field results which match perfectly. The analysis here may enrich the study of particle-photon interaction systems.     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

Kondo proximity effect: how does a metal penetrate into a Mott insulator?

We consider a heterostructure of a metal and a paramagnetic Mott insulator using an adaptation of dynamical mean-field theory to describe inhomogeneous systems. The metal can penetrate into the insulator via the Kondo effect. We investigate the scaling properties of the metal-insulator interface close to the critical point of the Mott insulator. At criticality, the quasiparticle weight decays as 1/x;{2} with distance x from the metal within our mean-field theory. Our numerical results (using the numerical renormalization group as an impurity solver) show that the prefactor of this power law is extremely small.     

Random Tilings of High Symmetry: I. Mean-Field Theory

We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly.     

Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.     

The Genealogical Tree of a Chromosome

We consider a simple neutral model to describe the genealogy of chromosomes by taking into account the effects of both recombination and coalescence. Seen as a statistical physics problem, the model looks like an inverse problem: A number of properties such as pair or three-point correlations can be computed easily, but the prediction of global properties, in particular the average number of ancestors, remains difficult. In the absence of exact solutions, these global properties can nevertheless be estimated by the usual approximations: series expansions, Monte Carlo simulations, mean-field theory. Simulations exhibit also non-self-averaging properties similar to those of mean-field spin glasses.     

Critical temperature of a trapped Bose gas: comparison of theory and experiment

We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature Tc of a harmonically confined Bose gas as reported in Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data. To unequivocally observe beyond mean-field effects, however, the experimental precision must either improve by an order of magnitude, or consider more strongly interacting systems. This is the first application of a classical field method to make quantitative comparison with experiment.     

On the Time Dependence of the Rate of Convergence Towards Hartree Dynamics for Interacting Bosons

Journal of Statistical Physics - We consider interacting N-Bosons in three dimensions. It is known that the difference between the many-body Schrödinger evolution in the mean-field regime and...     

Jordan-Wigner approach to the frustrated spin one-half XXZ chain

The Jordan-Wigner transformation is applied to study the ground state properties and dimerization transition in the J₁-J₂ XXZ chain. We consider different solutions of the mean-field approximation for the transformed Hamiltonian. Ground state energy and the static structure factor are compared with complementary exact diagonalization and good agreement is found near the limit of the Majumdar-Ghosh model. Furthermore, the ground state phase diagram is discussed within the mean-field theory. In particular, we show that an incommensurate ground state is absent for large J₂ in a fully self-consistent mean-field analysis.     

Sandpile on scale-free networks

We investigate the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model on scale-free (SF) networks, where the threshold height of each node is distributed heterogeneously, given as its own degree. We find that the avalanche size distribution follows a power law with an exponent tau. Applying the theory of the multiplicative branching process, we obtain the exponent tau and the dynamic exponent z as a function of the degree exponent gamma of SF networks as tau=gamma divided by (gamma-1) and z=(gamma-1) divided by (gamma-2) in the range 23, with a logarithmic correction at gamma=3. The analytic solution supports our numerical simulation results. We also consider the case of a uniform threshold, finding that the two exponents reduce to the mean-field ones.     

Mean field kinetic theory of a classical electron gas in a periodic potential. II. Qualitative analysis of the mean-field solution in one dimension

A qualitative analysis is made of the static and dynamic behavior of a one-dimensional classical electron gas in a periodic potential in the framework of a mean-field kinetic theory. The mean-field equations have been formally solved elsewhere in terms of the trajectories of one electron in the mean-field equilibrium potential, which determines the local electronic density. Taking advantage of the relative simplicity of the mean-field expressions in one dimension, we study the effects of the temperature upon the local electronic density, the static structure factor, and the spectrum of the fluctuations in the long-wavelength limit. At high temperatures, the system tends to behave like a homogeneous electron gas; however, the collective plasmon mode at zero wavenumber is damped and shifted below the plasma frequency. At low temperatures, the system behaves as an ensemble of independent electrons strongly localized in the neighborhood of the fixed ions that create the periodic potential; the plasmon mode then vanishes. We consider the physical relevance of these predictions. They turn out to be quite reasonable, despite the failure of meanfield theory to predict the phase of the model.     

Weakly interacting Bose-Einstein condensates under rotation: mean-field versus exact solutions

We consider a weakly interacting, harmonically trapped Bose-Einstein condensed gas under rotation and investigate the connection between the energies obtained from mean-field calculations and from exact diagonalizations in a subspace of degenerate states. From the latter we derive an approximation scheme valid in the thermodynamic limit of many particles. Mean-field results are shown to emerge as the correct leading-order approximation to exact calculations in the same subspace.     

Semi-Classical Dynamics in Quantum Spin Systems

We consider two limiting regimes, the large-spin and the mean-field limit, for the dynamical evolution of quantum spin systems. We prove that, in these limits, the time evolution of a class of quantum spin systems is determined by a corresponding Hamiltonian dynamics of classical spins. This result can be viewed as a Egorov-type theorem. We extend our results to the thermodynamic limit of lattice spin systems and continuum domains of infinite size, and we study the time evolution of coherent spin states in these limiting regimes.     

Magnetic field dependence of the spin—Peierls transition

We consider the effect of a magnetic field on the spin—Peierls transition within mean-field theory. Using a mathematical analogy to the regular Peierls transition and existing results, we derive formulas for the magnetic field dependence of the transition temperature and of the distortion periodicity.     

Coexistence of superconductivity and ferromagnetism in ferromagnetic metals

We address the question of coexistence of superconductivity and ferromagnetism. Using a field theoretical approach we study a one-fermion effective model of a ferromagnetic superconductor in which the quasiparticles responsible for the ferromagnetism form the Cooper pairs as well. For the first time we solve self-consistently the mean-field equations for the superconducting gap and the spontaneous magnetization. We discuss the physical features which are different in this model and the standard BCS model and consider their experimental consequences.     

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

Sudden onset of log-periodicity and superdiffusion in non-Markovian random walks with amnestically induced persistence: exact results

We consider strongly interacting boson-boson mixtures on one-dimensional lattices and, by adopting a qualitative mean-field approach, investigate their quantum phases as the interspecies repulsion is increased. In particular, we analyze the low-energy quantum emulsion metastable states occurring at large values of the interspecies interaction, which are expected to prevent the system from reaching its true ground state. We argue a significant decrease in the visibility of the time-of-flight images in the case of these spontaneously disordered states.