Hydrodynamic correlation functions of hard-sphere fluids at short times

The short-time behavior of the coherent intermediate scattering function for a fluid of hard-sphere particles is calculated exactly through ordert ⁴, and the other hydrodynamic correlation functions are calculated exactly through ordert ². It is shown that for all of the correlation functions considered the Enskog theory gives a fair approximation. Also, the initial time behavior of various Green-Kubo integrands is studied. For the shear-viscosity integrand it is found that at densityn³=0.837 the prediction of the Enskog theory is 32% too low. The initial value of the bulk viscosity integrand is nonzero, in contrast to the Enskog result. The initial value of the thermal conductivity integrand at high densities is predicted well by Enskog theory.     

On the ground state of the half-filled Hubbard model

We calculate the ground state of the half-filled Hubbard model and its energy by starting from a spindensity wave approximation and improving it by incorporating transverse spin fluctuations. The calculations are done by employing a projection method. The quality of the proposed approximation is particularly high for intermediate and large Coulomb repulsionU, where it exceeds considerably e.g. that of the Gutzwiller projected spin-density wave state. To ordert ²/U (wheret is the hopping matrix element), our approximation is shown to be equivalent to a recent Coupled Cluster calculation for the Heisenberg antiferromagnet. Finally we show how to ordert ²/U the linear spin-wave approximation for the Heisenberg antiferromagnet may be obtained.     

Asymptotic time behavior of correlation functions. III. Local equilibrium and mode-coupling theory

The decay to equilibrium is discussed from a general point of view based on the assumed rapid approach to local equilibrium for well-chosen initial states. The assumption is applied to the problem of time correlation functions and it is shown that the mode-coupling formula describes the asymptotics of the so-called projected wavenumber-dependent correlation functions. The local equilibrium assumption thus provides a general basis for thet ^–3/2 behavior of correlation functions derived in previous papers in this series, as well as for the infinite series of correction termst ^–(2–Pn (n 2), withPn=2^–n, and for the corresponding series of corrections of orderk ^3–Pn (n1) to Navier-Stokes hydrodynamics.     

Exact short time dynamics for steeply repulsive potentials

The autocorrelation functions for the force on a particle, the velocity of a particle and the transverse momentum flux are studied for the power law potential v(r)=ε(σr)^ν (soft spheres). The latter two correlation functions characterize the Green–Kubo expressions for the self-diffusion coefficient and shear viscosity. The short-time dynamics is calculated exactly as a function of ν. The dynamics is characterized by a universal scaling function S(τ), where τ=t/τ_ν and τ_ν is the mean time to traverse the core of the potential divided by ν. In the limit of asymptotically large ν this scaling function leads to delta function in time contributions in the correlation functions for the force and momentum flux. It is shown that this singular limit agrees with the special Green–Kubo representation for hard-sphere transport coefficients. The domain of the scaling law is investigated by comparison with recent results from molecular dynamics simulation for this potential.     

A block spin construction of ondelettes. Part I: Lemarié functions

Using block spin assignments, we construct anL ²-orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions. These functions have exponential localization, and for even values of a construction parameterM one can make them classC ^M–1 with vanishing moments up to orderM inclusive. Such a basis has an important application to phase cell cluster expansions in quantum field theory.Supported in part by the National Science Foundation under Grant No. DMS 8603795 and by the Mathematical Sciences InstituteOn leave from the Mathematics Department, Texas A & M University, College Station, Texas 77843 USA     

Density expansion of the radial distribution and bridge functions of the hard sphere fluid

The bridge function and the background correlation function (and consequently the radial distribution function) of the pure hard sphere fluid are expanded up to the sixth power in density. The calculations are based on the Ree–Hoover representation of the diagrams and Monte Carlo integration. The coefficients as functions of the particle–particle separation are fitted to splines taking into account discontinuities in higher derivatives up to the term of the order of (r- const)⁵.     

Muon charge exchange and muonium spin exchange in gases

The spin dynamics of the positive muon undergoing cyclic charge exchange (μ⁺⇌Mu) while it is slowing down is compared with that in the case of repeated muonium spin exchange after thermalization. The expectation value of the spin polarization at timet aftern spin exchange collisions (att ₁,t ₂, ...,t _n ) are calculated explicitly from time dependent wave functions, and the quantity is averaged over the statistical distribution of the times of collisions and over all possible numbers of collisions betweent=0 and the observation timet. This result is complementary and equivalent to the conventional density matrix formalism, but offers an insight into the roles of spin flip and spin non-flip processes. The neutral fraction during slowing down is also discussed.     

Density-matrix renormalization study of the Hubbard model [4]on a Bethe lattice

The half-filled Hubbard model on the Bethe lattice with coordination number z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy per site E, average local magnetization , its fluctuations and various spin correlation functions are determined as a function of the Coulomb interaction strength U/t. The local magnetic moments increase monotonically with increasing Coulomb repulsion U/t showing antiferromagnetic order between nearest neighbors []. At large U/t, is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins [ for ]. shows a maximum for U/t=2.4-2.9 that results from the interplay between the usual increase of with increasing U/t and the formation of important permanent moments at large U/t. While NN sites show antiferromagnetic spin correlations that increase with increasing Coulomb repulsion, the next NN sites are very weakly correlated over the whole range of U/t. The DMRG results are discussed and compared with tight-binding calculations for U=0, independent DMRG studies for the Heisenberg model and simple first-order perturbation estimates. Received 8 February 1999 and Received in final form 14 June 1999     

Polarization Phenomena in Elastic Electron-Proton Scattering

A unified treatment is given by using the scattering matrix in the Pauli spin spaces and polarization density matrices. Various quantities such as polarization P, asymmetry A, spin correlation parameters C_αβ, double polarization asymmetry A_αβ, polarization transfer parameters D_t, A_t, R_t and depolarization and spin rotation parameters D, A, R are expressed in terms of six coefficients which appear in the scattering matrix. Some explicit expressions as well as numerical results for these quantities are also given in the one photon exchange approximation.     

Fluctuations in coagulating systems

Time-dependent fluctuations in a system of coagulating particles are studied, using the master equation for the probability distributionsP(m,t) for the occupation numbersm={m _k} (k=1,2,...) of thek-cluster states. Van Kampen's-expansion is used to determine the deterministic (order ⁰) and fluctuating part (order ^–1/2) of the solution. We calculate the time-dependent behavior of the fluctuations in the cluster size distribution. The model under consideration is of special interest since it exhibits a phase transition (gelation). For monodisperse initial states we give explicit expressions for the probability distribution of the fluctuations and for the equal-time and two-time correlation functions also near the phase transition. For general initial conditions we study the fluctuations (1) for large cluster sizes, (2) in the scaling limit (near the critical point), and (3) for large times. Our results show that the deterministic approach to coagulation processes (Smoluchowski theory) is invalid very close to the gelpointt _c and at large times (tt _M), where the distance from the gelpoint and the timet _M depend upon the size of the system.     

Reacting polymers with highly correlated initial conditions

We propose and theoretically study an experiment designed to measure short-time polymer reaction kinetics in melts or dilute solutions. The photolysis of groups centrally located along chain backbones, one group per chain, creates pairs of spatially highly correlated macroradicals. We calculate time-dependent rate coefficients κ(t) governing their first-order recombination kinetics, which are novel on account of the far-from-equilibrium initial conditions. In dilute solutions (good solvents) reaction kinetics are intrinsically weak, despite the highly reactive radical groups involved. This leads to a generalised mean-field kinetics in which the rate of radical density decay - ∼S(t), where S(t) ∼t ^{- (1 + g/3)} is the equilibrium return probability for 2 reactive groups, given initial contact. Here g≈ 0.27 is the correlation hole exponent for self-avoiding chain ends. For times beyond the longest coil relaxation time τ, - ∼S(t) remains true, but center of gravity coil diffusion takes over with rms displacement of reactive groups x(t) ∼t ^1/2 and S(t) ∼ 1/x ³(t). At the shortest times ( t 10^-6s), recombination is inhibited due to spin selection rules and we find ∼tS(t). In melts, kinetics are intrinsically diffusion-controlled, leading to entirely different rate laws. During the regime limited by spin selection rules, the density of radicals decays linearly, n(0) - n(t) ∼t. At longer times the standard result - ∼d ³(t)/d (for randomly distributed ends) is replaced by ∼d2x ³(t)/d ² for these correlated initial conditions. The long-time behavior, t > τ, has the same scaling form in time as for dilute solutions. Received 18 May 2000     

Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

The Hamiltonian system formed by a Klein-Gordon vector field and a particle in ℝ³ is considered. The initial data of the system are given by a random function, with finite mean energy density, which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are assumed to be translation invariant. The distribution μ _t of the solution at time t ∈ ℝ is studied. The main result is the convergence of μ _t to a Gaussian measure as t → ∞, where μ_∞ is translation invariant.     

Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the square lattice

The instability of the fully polarized ferromagnetic ground state (Nagaoka state) of the Hubbard model on the square lattice is investigated. We use single spin flip variational wave functions including majority spin correlation effects and calculate spin flip energies in the thermodynamic limit. With very local wave functions and with moderate numbers of variational parameters we reproduce the best known estimate for the critical hole density δ_cr = 0.29 and we obtain an estimate of U_cr = 63 t for the critical coupling which is considerably better than the best estimate of U_cr = 42 t previously known. The simplicity of our wave functions makes the physical origin of the various aspects of the instability particularly transparent.     

Electronic properties of metals at low temperatures

A many body theory of an electron gas is developed to find the internal and correlation energies at low but finite temperatures. The contribution from the first order exchange, second order (regular and anomalous) exchange, and ring diagrams are treated. The Fermi momentum and the correlation energy are determined as functions of the density by two different methods, one being based on iteration and the other a direct solution of the number density relation. It was found that the iterative solutions which are correct to ordere ² ore ⁴ become negative forr _s of order 5 while the direct solutions do not, indicating the invalidity of the former. Hence, the correlation energy evaluated to the same orders by iteration will not be satisfactory in the same range. The highest order iterative solution which includes terms of ordere ⁶ does not show such a breakdown. These terms which give the contribution of orderr _s to the correlation energy are therefore important and tend to reduce the magnitude of the correlation energy. The corresponding curve is indeed close to that determined by the direct method for smallr _s but a significant deviation takes place at largerr _s . The Coulomb interaction seems less effective at higher temperatures. The internal energy is also determined as a function of density and temperature.     

Chebyshev polynomials and Fourier transform of <Emphasis Type="Italic">SU</Emphasis>(2) irreducible representation character as spin tomographic star-product kernel

Spin tomographic symbols of qudit states and spin observables are studied. Spin observables are associated with the functions on a manifold whose points are labeled by the spin projections and sphere S ² coordinates. The star-product kernel for such functions is obtained in an explicit form and connected with the Fourier transform of characters of the SU(2) irreducible representation. The kernels are shown to be in close relation to the Chebyshev polynomials. Using specific properties of these polynomials, we establish the recurrence relation between the kernels for different spins. Employing the explicit form of the star-product kernel, a sum rule for Clebsch–Gordan and Racah coefficients is derived. Explicit formulas are obtained for the dual tomographic star-product kernel as well as for intertwining kernels which relate spin tomographic symbols and dual tomographic symbols.     

High-temperature dynamics of the anisotropic spin-1/2 square lattice Heisenberg model studied by moment methods

The spin dynamics of the anisotropic spin-1/2 nearest-neighbour Heisenberg model (XYZ model) on a plane square lattice is studied at infinite temperature. Low-order coefficients of the short-time expansions for spin-spin correlation functions are calculated. The necessary commutator algebra may be performed by a computer. The series obtained for the spin correlation function are the longest ones available up to now. The series coefficients are used to construct rigorous upper and lower bounds to autocorrelation functions and near-neighbour correlation functions.     

Structure of spin polarons in the t-t′ -t″ -Jz model

We calculate the Green function in the t-t '-t ”-J^z model and analyze the deformation of the quantum Néel state in the presence of a moving hole. Solving the problem in a self-consistent Born approximation and using Reiter's wave function we have found various spin correlation functions. We show that the different sign of hopping elements between the hole- and electron-doped system of high- cuprates is responsible for the sharp difference of the polaron structure between the two systems with antiferromagnetism stabilized in the electron-doped case by carriers moving mainly on one sublattice. Received 11 January 2000     

Spin-wave renormalization by mobile holes in a two-dimensional quantum antiferromagnet

The coupling of antiferromagnetic spin excitations and propagating holes has been studied theoretically on a square lattice in order to investigate the dependence of antiferromagnetic order on hole doping, being of relevance, e.g., for the Cu–3 d⁹ system in antiferromagnetic CuO₂-planes of high-T_c superconductors. An effective Hamiltonian has been used, which results from a 2D Hubbard model (hopping integral t) with holes and with strong on-site Coulomb repulsion U. Bare antiferromagnetic excitations and holes with energies of the same order of magnitude t²/U are interacting via a coupling term being proportional to t and allowing holes to hop by emitting and absorbing spinwaves. In terms of a self-consistent one-loop approximation the renormalization of the spectral function both of holes and antiferromagnetic spin excitations are calculated.     

Flow equations and the strong-coupling expansion for the Hubbard model

Applying the method of continuous unitary transformations to a class of Hubbard models, we reexamine the derivation of thet/U expansion for the strong-coupling case. The flow equations for the coupling parameters of the higher order effective interactions can be solved exactly, resulting in a systematic expansion of the Hamiltonian in powers oft/U, valid for any lattice in arbitrary dimension and for general band filling. The expansion ensures a correct treatment of the operator products generated by the transformation, and only involves the explicit recursive calculation of numerical coefficients. This scheme provides a unifying framework to study the strong-coupling expansion for the Hubbard model, which clarifies and circumvents several difficulties inherent to earlier approaches. Our results are compared with those of other methods, and it is shown that the freedom in the choice of the unitary transformation that eliminates interactions between different Hubbard bands can affect the effective Hamiltonian only at ordert ³/U² or higher.     

The muon capture in <Superscript>16</Superscript>O: the angular and polarization correlations

Longitudinal polarization of the daughter nuclei ¹⁶N which arises in μ⁻ capture on ¹⁶O as a function of the recoil angle, together with the angular distribution and the alignment of the recoil nucleus are calculated. The neutrinos born escape mainly along the muon spin. The polarization is found to vary from zero (recoil momentum counter to the muon spin direction) up to 50% (along the muon spin direction). The results can be applied to the experimental tests of T conservation, to the analysis of the projects of constructing the powerful mono-energetic neutrino sources, to the experimental study of the pseudo-scalar form factor and the K-electron capture, and to other spin-polarization correlation experiments.