Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/r^d+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.     

Velocity correlation functions for finite one-dimensional systems

Certain systems consisting of a one-dimensional gas of a finite number of point particles interacting with a “hard-core” potential are considered.We use the technique developed by Jepsen to calculate exactly the velocity correlation functions for these systems. We discover that after a slow decay for times of the order of the relaxation time, there is a “fast” decay to the equilibrium value on a macroscopic time scale characterized by L/v_TH (L is the length of the container and v_TH the thermal velocity).The dependence of the velocity correlation functions on the initial position of the specified particle is also considered. In particular, the behaviour approaching the boundary of the container is analyzed. These considerations are generalized to systems of higher spatial dimension.     

Analyticity for one-dimensional systems with long-range superstable interactions

We consider unbounded spin systems and classical continuous particle systems in one dimension. We assume that the interaction is described by a superstable two-body potential with a decay at large distances at least asr ^?2(lnr)^?(2+ε), ε > 0. We prove the analyticity of the free energy and of the correlations as functions of the interaction parameters. This is done by using a “renormalization group technique” to transform the original model into another, physically equivalent, model which is in the high-temperature (small-coupling) region.     

Numerical test of approximations in the hybrid precompound decay model

We review inconsistencies in expressions which are used for exciton distribution functions in precompound decay models utilizing the Ericson type of partial state density. We then compare results of one formulation, the hybrid model, with and without corrected distribution functions for the second hierarchy of configurations. Results are shown for⁹³Nb (n, n′) and (n, p) reactions at 14.7, 45 and 90 MeV incident nucleon energies. The corrected distribution functions change hybrid model predictions by less than 10% for angle integrated spectra, but certain cases are discussed for which the improved treatment should be used due to larger errors from the use of the Ericson expression.     

Quantum and Classical Correlations in the Solid-State NMR Free Induction Decay

The free induction decay (FID) of the transverse magnetization in a dipolar-coupled rigid lattice is a fundamental problem in magnetic resonance and in the theory of many-body systems. As it was shown earlier the FID shapes for the systems of classical magnetic moments and for quantum nuclear spin ones coincide if there are many nearly equivalent nearest neighbors n in a solid lattice. In this paper, we reduce a multispin density matrix of above system to a two-spin matrix. Then we obtain analytic expressions for the mutual information and the quantum and classical parts of correlations at the arbitrary spin quantum number S, in the high-temperature approximation. The time dependence of these functions is expressed via the derivative of the FID shape. To extract classical correlations for S > 1/2 we provide generalized POVM measurement (positive-operator-valued measure) using the basis of spin coherent states. We show that in every pair of spins the portion of quantum correlations changes from 1/2 to 1/(S + 1) when S is growing up, and quantum properties disappear completely only if S → ∞.     

Study ofpf-shell nuclei with a symmetry conserving generator coordinate method

Bands based on the 0⁺ ground state and the first excited 0⁺ pairing vibrational state of⁴⁸Ti,⁵²Cr and⁵⁶Fe are studied with the generator coordinate method. The generating wave functions for each value of the angular momentumJ are angular momentum and particle number projected selfconsistent Hartree-Fock-Bogoliubov states where the constrained amount of pairing correlations serves as the generator coordinate. The interaction is given by reaction matrix elements derived from the Hamada-Johnston force. The basis includes the four lowest oscillator shells. The excitation energies of the pairing vibrational states can be reproduced fairly well by the present choice of the generating wave functions, whereas the ground band is not much improved compared to projected Hartree-Bogoliubov calculations. We find that the strength of the pairing correlations in the 0⁺ and 2⁺ states of the ground state and the pairing vibrational bands can be related to data of two-particle transfer reactions. The angular momentum dependence of the pairing correlations and of the moments of inertia are studied. The results show that for a strongly paired ground state the ground state band and the pairing vibrational band intersect. This may produce in the yrast band the anomaly of the moment of inertia known from rare earth nuclei.     

Short-time vibrational dynamics of metaphosphate glasses

In this paper we present the picosecond vibrational dynamics of a series of binary metaphosphate glasses, namely Na₂O–P₂O₅, MO–P₂O₅ (M=Ba, Sr, Ca, Mg) and Al₂O₃–3P₂O₅ by means of Raman spectroscopy. We studied the vibrational dephasing and vibrational frequency modulation by calculating time correlation functions of vibrational relaxation by fits in the frequency domain. The fitting method used enables one to model the real line profiles intermediate between Lorentzian and Gaussian by an analytical function, which has an analytical counterpart in the time domain. The symmetric stretching modes ν_s(PO₂^?) and ν_s(P–O–P) of the PO₂^? entity of PØ₂O₂^? units and of P–O–P bridges in metaphosphate arrangements have been investigated by Raman spectroscopy and we used them as probes of the dynamics of these glasses. The vibrational time correlation functions of both modes studied are rather adequately interpreted within the assumption of exponential modulation function in the context of Kubo–Rothschield theory and indicate that the system experiences an intermediate dynamical regime that gets only slower with an increase in the ionic radius of the cation-modifier. We found that the vibrational correlation functions of all glasses studied comply with the Rothschild approach assuming that the environmental modulation is described by a stretched exponential decay. The evolution of the dispersion parameter α with increasing ionic radius of the cation indicates the deviation from the model simple liquid indicating the reduction of the coherence decay in the perturbation potential as a result of local short lived aggregates. The results are discussed in the framework of the current phenomenological status of the field.     

Spinon and <Emphasis Type="Italic">η</Emphasis>-spinon correlation functions of the Hubbard chain

We calculate real-space static correlation functions of spin and charge degrees of freedom of the one-dimensional Hubbard model that are described by operators related to singly occupied sites with spin up or spin down (spinons) and unoccupied or doubly occupied sites (η-spinons). The spatial decay of their correlation functions is determined using density matrix renormalization group results. The nature and spatial extent of the correlations between two sites on the Hubbard chain is studied using the eigenstates and eigenvalues of the two-site reduced density matrix. The results show that the spinon-spinon correlation functions decay algebraically and the η-spinon correlation functions decay exponentially, both in the half-filling and metallic phases. The results provide evidence that these degrees of freedom are organized in boundstates in the interacting system.     

Multiscale Formalism for Correlation Functions of Fermions. Infrared Analysis of the Tridimensional Gross–Neveu Model

We present a multiscale formalism for fermionic systems (with a smooth UV cutoff ) establishing a trivial link between the correlation functions and the effective potential flow, and study the k-point truncated functions of the tridimensional Gross–Neveu model. A new efficient method is used to bound these correlation functions and show polynomial tree decay for long distances. We are guided by a block lattice mechanism with a property of orthogonality between terms in different scales, which leads to simple formulas for the correlations.     

A Gibbs-Like Measure for Single-Time, Multi-Scale Energy Transfer in Stochastic Signals and Shell Model of Turbulence

A Gibbs-like approach for simultaneous multi-scale correlation functions in random, time-dependent, multiplicative processes for the turbulent energy cascade is investigated. We study the optimal log-normal Gibbs-like distribution able to describe the subtle effects induced by non-trivial time dependency on both single-scale (structure functions) and multi-scale correlation functions. We provide analytical expression for the general multi-scale correlation functions in terms of the two-point correlations between multipliers and we show that the log-normal distribution is already accurate enough to reproduce quantitatively many of the observed behavior. The main result is that non-trivial time effects renormalize the Gibbs-like effective potential necessary to describe single-time statistics. We also present a generalization of this approach to more general, non log-normal, potentials. In the latter case one obtains a formal expansion of both structure functions and multi-scale correlations in terms of cumulants of all orders.     

Pairing vibrational and isospin rotational states in a particle number and isospin projected generator coordinate method

Pairing vibrational and isospin rotational states are described in different approximations based on particle number and isospin projected, proton-proton, neutron-neutron and proton-neutron pairing wave functions and on the generator coordinate method (GCM). The investigations are performed in models for which an exact group theoretical solution exists. It turns out that a particle number and isospin projection is essential to yield a good approximation to the ground state or isospin yrast state energies. For strong pairing correlations (pairing force constant equal to the single-particle level distance) isospin cranking (-ωT_x) yields with particle number projected pairing wave function also good agreement with the exact energies. GCM wave functions generated by particle number and isospin projected BCS functions with different amounts of pairing correlations yield for the lowest T = 0 and T = 2 states energies which are practically indistinguishable from the exact solutions. But even the second and third lowest energies of charge-symmetric states are still very reliable. Thus we conclude that also in realistic cases isospin rotational and pairing vibrational states may be described in the framework of the GCM method with isospin and particle number projected generating wave functions.     

Poor decay of correlations in inhomogeneous fluids and solids and their relevance for the physics of phase boundaries

We study the poor decay of correlations for equilibrium states of inhomogeneous fluids and solids, in the regimes of both classical and quantum statistical mechanics. Our main observation is the usefulness of the statistical mechanical expression of thestress tensor and its long-range correlations with the particle density. From this we are able to infer a very slow decay of correlations for the various molecular distribution functions under discussion. The derived results are of relevance both for completely inhomogeneous systems such as quasicrystals or granular structures and for the slightly more regular cases of, e.g., phase separating layers in fluids and solids, ideal crystals, etc. As one of the byproducts we prove the nonexistence of planequantum interfaces in two dimensions (thus extending earlier results of Requardt to the quantum regime). The results hold for arbitrary potentials of not too long range.     

The interaction energy of three molecules united in a nanocluster by long-range attraction forces with account for Coulomb repulsion

The interaction energy of three neutral molecules that form a nanocluster is studied. It is assumed that one molecule (M₀) has a dipole moment, while the other two (M₁ and M₂) are nonpolar. The molecule interaction energy in such a nanocluster is determined by the sum of dispersion interaction energies of each pair of molecules and the sum of inductive energies of the molecules. Analytical expressions for these energies as functions of the distance between the centers of mass of the molecules have been obtained. A method for the determination of damping functions which takes the contribution of repulsive forces into account has been developed. Analytical expressions for the molecule interaction energies for a two-molecule cluster in an external field of the third molecule have been obtained. A nanocluster consisting of a molecule of polar isomer pentene C₅H₁₀ and a nonpolar molecule of polycyclic aromatic hydrocarbon pyrene C₁₆H₁₀ in the external electrostatic field of another pyrene molecule is considered. The calculation showed that the interaction energy of the two-molecule nanocluster increases by a factor of 1.5 if this cluster is in the field of the induced dipole moment of an external pyrene molecule.     

Circulating-current phase in the three-band model for two-leg CuO ladders

We calculate circulating-current (CC), charge-density-wave, and d-wave-like pairing (d-SC) correlation functions in the three-band Hubbard model for two-leg CuO ladders using the density-matrix renormalization group method and detect a dominant fluctuation in a wide range of parameter values and hole-doping rates. We find that, for model parameters leading to a realistic ground state in the undoped ladder, the CC fluctuations decay faster than the d-SC correlations at least up to a hole doping of 10%. It means that no phase with CC order or dominant CC fluctuations occur at low doping.     

Correlation functions of a time-continuous dissipative system with a strange attractor

We determine the exact decay of time correlation functions of a continuous-time chaotic system. In contrast to discrete-time chaotic systems where these correlations decay as a rule exponentially fast we find in our continuous-time system long-time tails well known from many-particle systems.     

Diffusion-limited annihilation in inhomogeneous environments

We study diffusion-limited (on-site) pair annihilation A + A → 0 and (on-site) fusion A + A → A which we show to be equivalent for arbitrary space-dependent diffusion and reaction rates. For one-dimensional lattices with nearest neighbour hopping we find that in the limit of infinite reaction rate the time-dependent n-point density correlations for many-particle initial states are determined by the correlation functions of a dual diffusion-limited annihilation process with at most 2n particles initially. Furthermore, by reformulating general properties of annihilating random walks in one dimension in terms of fermionic anticommutation relations we derive an exact representation for these correlation functions in terms of conditional probabilities for a single particle performing a random walk with dual hopping rates. This allows for the exact and explicit calculation of a wide range of universal and non-universal types of behaviour for the decay of the density and density correlations.     

Quantum corrections to liquid correlations

Sudarshan's semi-classical treatment of correlation functions is applied to the study of quantum corrections to the Van Hove function. In its generalised form, it enables one to choose the best correlation function for a given potential. Higher-order correlations are also sketched briefly and the details are similar to those considered by Oppenheim and Bloom.     

Testing EPR locality using B-mesons

We study the possibility of testing local realistic theory (LRT) based on the Bell inequality for the correlations in the decay modes of entangled K or B-mesons. It is shown that such a test is possible for a restricted class of LRT, despite the passive nature of decay events and/or the non-unitary treatment of the correlations which invalidate the test for general LRT. Unfortunately, the present setup of the KEKB (Belle) experiment, where the coherence of entangled B-mesons has been confirmed recently, does not admit such a test due to the indeterminacy in the separate decay times of the entangled pairs.