Many-particle Correlation Functions Research Articles

Entanglement properties and ground-state statistics of free bosons

We calculate analytically the entanglement and R\'enyi entropies, the negativity, and the mutual information together with all the density and many-particle correlation functions for free bosons on a lattice in the ground state, for both homogeneous and inhomogeneous systems. We show that all those quantities can be derived from a multinomial form of the reduced density matrix in the configuration space whose diagonal elements dictate the statistics of the particle distribution, while the off-diagonal coherence terms control the quantum fluctuations. We provide by this analysis a unified approach based on a reduced density matrix technique useful to calculate both the entanglement properties and an infinite number of correlation functions.

Quantum Current Algebra Symmetry and Description of Boltzmann Type Kinetic Equations in Statistical Physics

We review a non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions and apply it to description of invariantly reduced Hamiltonian systems of the Boltzmann type kinetic equations, related to naturally imposed constraints on many-particle correlation functions. As an interesting example of deriving Vlasov type kinetic equations, we considered a quantum-mechanical model of spinless particles with delta-type interaction, having applications for describing so called Benney-type hydrodynamical praticle flows. We also review new results on a special class of dynamical systems of Boltzmann–Bogolubov and Boltzmann–Vlasov type on infinite dimensional functional manifolds modeling kinetic processes in many-particle media. Based on algebraic properties of the canonical quantum symmetry current algebra and its functional representations, we succeeded in dual analysis of the infinite Bogolubov hierarchy of many-particle distribution functions and their Hamiltonian structure. Moreover, we proposed a new approach to invariant reduction of the Bogolubov hierarchy on a suitably chosen correlation function constraint and deduction of the related modified Boltzmann–Bogolubov kinetic equations on a finite set of multi-particle distribution functions. There are also presented results of application of devised methods to describing kinetic properties of a many-particle system with an adsorbent surface, in particular, the corresponding kinetic equation for the occupation density distribution function is derived.

On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits discrete stochastic calculus with respect to initial data in form of higher-order Poincare inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we discuss the Lenard–Balescu limit for a spatially homogeneous system away from thermal equilibrium.

Correlations in Nonequilibrium Luttinger Liquid and Singular Fredholm Determinants

We study interaction-induced correlations in Luttinger liquid with multiple Fermi edges. Many-particle correlation functions are expressed in terms of Fredholm determinants det(1+ÂB[over ^]), where A(ε) and B(t) have multiple discontinuities in energy and time spaces. We propose a general asymptotic formula for this class of determinants and provide analytical and numerical support to this conjecture. This allows us to establish nonequilibrium Fermi-edge singularities of many-particle correlation functions. As an example, we calculate a two-particle distribution function characterizing genuinely nonequilibrium quantum correlations between left- and right-moving fermions that have left the interaction region.

Finite-temperature correlations in the one-dimensional trapped and untrapped Bose gases

We calculate the dynamic single-particle and many-particle correlation functions at nonzero temperature in one-dimensional trapped repulsive Bose gases. These correlation functions decay towards zero for increasing distance, and this decay is governed by a scaling exponent that has a universal expression in terms of observed quantities. This expression is valid in the weak-interaction Gross-Pitaevskii as well as in the strong-interaction Girardeau-Tonks limit, but the observed quantities involved depend on the interaction strength. The confining trap introduces a weak center-of-mass dependence in the scaling exponent. Based on previous exact results for a related untrapped Bose system and the results reported here, we also suggest the proper asymptotic form for the density-density correlation function in the trapped Bose gas as considered here.

Theory of spin diffusion in liquid-phase polymer systems

A general theory of spin diffusion in condensed media is constructed by the method of Zwanzig-Mori projection operators using the superpositional approximation to decouple the many-particle correlation functions. The spin diffusion coefficient is expressed in the form D sp=D tr+D f , where D tr is the contribution associated with translational displacements of the molecules and D f is the contribution caused by intermolecular flip-flop processes. The expression for D tr differs from the well-known Kubo-Green formula for the self-diffusion coefficient D sd in that the integrand contains an additional factor P f (t), which is the probability of the molecular spins not participating in intermolecular flip-flop transitions over the time t. A microscopic expression is obtained for D f in the form of a time integral of the intermolecular dipole-dipole dynamic correlation functions. For liquid-phase polymer system with fairly high molecular mass the condition D sp≫D sd is satisfied.

Coherence in the quasiparticle scattering by the vortex lattice in pure type-II superconductors

The effect of quasi-particle (QP) 'scattering' by the vortex lattice on the de-Haas van-Alphen oscillations in a pure type-II superconductor is investigated within mean field,asymptotic perturbation theory. Using a 2D electron gas model it is shown that, due to a strict phase coherence in the many-particle correlation functions, the 'scattering' effect in the asymptotic limit ($\sqrt{E_F/\hbar\omega_c}\gg 1$) is much weaker than what is predicted by the random vortex lattice model proposed by Maki and Stephen, which destroys this coherence . The coherent many particle configuration is a collinear array of many particle coordinates, localized within a spatial region with size of the order of the magnetic length. The amplitude of the magnetization oscillations is sharply damped just below $% H_{c2}$ because of strong $180^{\circ}$ out of phase magnetic oscillations in the superconducting condensation energy ,which tend to cancel the normal electron oscillations. Within the ideal 2D model used it is found, however, that because of the relative smallness of the quartic and higher order terms in the expansion, the oscillations amplitude at lower fields does not really damp to zero, but only reverses sign and remains virtually undamped well below $H_{c2}$. This conclusion may be changed if disorder in the vortex lattice, or vortex lines motion will be taken into account. The reduced QP 'scattering' effect may be responsible for the apparent crossover from a strong damping of the dHvA oscillations just below $H_{c2}$ to a weaker damping at lower fields observed experimentally in several 3D superconductors.

Single- and many-particle correlation functions and uniform-phase bases for strongly correlated systems

The need for suitable many- or infinite-fermion correlation functions to describe strongly correlated systems is discussed, and the question linked to the need for a correlated basis, in which the ground state may be positive definite for certain low-dimensional geometries. In seeking a positive-definite basis a particular trial basis is proposed, based on that for hard-core bosons in pure one-dimensional systems. Single-particle correlations in this basis are evaluated for the case of the ground state of a quasi-1D Hubbard model in the limit of extreme correlation. The model is a strip of the 2D square lattice wrapped around a cylinder, and is related to a ladder geometry with periodic boundary conditions along its edges. This is done using both a novel mean-field theory and exact diagonalization, and the basis is indeed found to be well suited for examining (quasi-) order in the model. The model has a paramagnetic region and a Nagaoka ferromagnetic region. In the numerical calculation the correlation function in the paramagnetic phase has a power-law decay and the charge motion is qualitatively hard-core bosonic. The mean field leads to an example of a BCS-type model with single-particle bosonic long-range order.

Charge-spin separation and the spectral properties of Luttinger liquids

We compute the spectral function rho (q, omega ) of the one-dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between vsigma q and vrho q, where vsigma and vrho are the velocities of spin and charge fluctuations, while spectral weight above vrho q and below -vrho q is generated by the hybridization of the Fermi surface at +or-kF by interactions. There are non-universal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger-liquid picture, which then would suggest very strong long-range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities, but they probe different and complementary aspects of the Fermi-surface interactions.

On many-particle correlation functions of identical bosons in the 1/N expansion

Many-identical-boson correlation functions in the 1/N expansion approach are studied. They are calculated explicitly under the same assumptions used for two- and three-particle correlation functions. Graphical rules are obtained, which outline the group-theoretical structure of the problem and determine without ambiguity the correction term to the totally incoherent limit (N → ∞) for the number of correlated particles; the correction term (N finite) is shown to factorize with respect to the correlation function calculated independently in the limitN → ∞.

The many-particle correlation functions of quasi-one dimensional systems

The Ok F four-particle function describing the correlation of the order parameter modulus squared is shown to be the most divergent correlation function in the presence of the Peierls instability. The corresponding anomaly can manifest itself at high temperatures, e.g. as the unshifted sheet of x-ray scattering patterns. Concerning the 4k F function the influence of commensurability effects is studied. The experimental situation for TTF-TCNQ is discussed.

Simple Ising models still thrive!

Abstract A selection of recent work by various authors concerning spin 1 2 ; Ising modls with pair couplings is reviewed. Topics addressed include: the universality of corrections-to-scaling and nonlinear scaling fields in planar Ising models; aspects of many-particle correlation functions; the testing and verification of scaling hypotheses for local magnetizations in the presence of walls and boundaries, including the singular nature of the critical region scaling functions and the decay of the influence of a far wall; a soluble model of a surface roughening transition; nonuniversal critical effects of one-dimensional perturbations in planar Ising models; the effects of spatially anisotropic further neighbor interactions, in particular ANNNI models in d 2 dimensions and soluble mock ANNNI models in 2 and 3 dimensions.

Marcinkiewicz's theorem and approximation schemes for many-particle correlation functions: Comment on a paper by Rajagopal and Sudarshan

Recently Rajagopal and Sudarshan proposed a generalization of Marcinkiewicz's theorem to quantum systems. From it they conclude that approximation schemes, such that the irreducible equal-time correlation functions vanish for all orders larger than l but are nonvanishing for some l larger than 2, are inconsistent with basic positivity conditions on the density operator. We show that this statement is false for fermion systems. For boson systems the statement is correct and its extension to irreducible correlation functions for unequal times can be made plausible, a fully rigorous proof does not yet seem available, however. The possibility is briefly discussed that an extension of Marcinkiewicz's theorem may hold for fermion systems, provided one confines his attention to equilibrium states of sucyh systems. (auth)