Recent Progress in Many-Body Theories

The following sections are included:

• CONTENTS

• PREFACE

• SERIES EDITORIAL BOARD AND OTHER COMMITTEES

• FOREWORD BY THE EDITORS

The Eugene Feenberg Medal is awarded to Anthony J. Leggett in recognition of his seminal contributions to Many-Body Physics, including the explanation of the re-markable properties of superfluid ³He in the millikelvin regime, important results in Fermi-liquid theory applied to metals, fundamental new insights into macroscopic quantum coherence, elucidation of key aspects of high-temperature superconduc-tivity, and pioneering studies of the implications of Bose-Einstein condensation in atomic systems.

It is a great honor and privilege to have been named as the recipient of the 1999 Eugene Feenberg Memorial Award. Although I never knew Eugene Feenberg personally, I have studied many of his papers and have always had the greatest respect for his work. Receipt of the award named for him puts me in some very distinguished company, and I should like to thank the community for nominating me and the committee for choosing me. I should also like to thank the many colleagues and students who have participated throughout the years in the work for which I am cited…

In this paper I report my solution to MBX Challenge Competition. Namely, the Bertsch, nonparametric model of neutron matter is analyzed and strong indications are found that, in the infinite system limit, the ground state is a Fermi liquid with an effective mass.

I am pleased and honored to be here on the occasion of the year of my 75th birthday at the special session of this conference on many-body theory. The idea for this appears to have come from members of the department of physics at the University of Washington. I was first contacted by my acquaintance and friend from decades ago, Larry Wilets. In deciding what I could briefly say at the end of this session, I decided to summarize the highlights of the years of my work on many-body theory…

In this paper I intend to give a brief survey of the most challenging problems existing now in condensed matter theory. I will try as hard as possible not to follow just my personal taste, but to stick to some reasonably universal organizing principle.

This organizing principle I find in the general structure of physical theory in its relation to experiment. Namely, physical theory operates on three levels…

Although the effects of interactions in solid state systems still remains a widely open subject, some limiting cases such as the three dimensional Fermi liquid or the one-dimensional Luttinger liquid are by now well understood when one is dealing with interacting electrons in periodic crystalline structures. This problem is much more fascinating when periodicity is lacking as it is the case in quasicrystalline structures. Here, we discuss the influence of the interactions in quasicrystals and show, on a controlled one-dimensional model, that they lead to anomalous transport properties, intermediate between those of an interacting electron gas in a periodic and in a disordered potential.

We present a new, asymptotic exact, theory for the U(1) gauge fluctuations on top of the d-wave RVB mean-field theory of the high T_c cuprates. After integrating out the gauge fluctuations exactly we obtain an unconstrained theory of gapless fermion excitations coupling to superconducting phase fluctuations. Several salient features of this theory will be discussed.

We show, using a Bogoliubov-de Gennes (BdG) mean field theory, that the local pairing amplitude Δ(r) becomes highly inhomogeneous with increasing disorder in an s-wave superconductor. The probability distribution P(Δ) is peaked about the BCS value at low disorder, but with increasing disorder, progressively develops into a broad distribution with significant build up of weight near Δ ≈ 0. At high disorder, the system is found to form superconducting “islands” separated by a non-superconducting sea. Surprisingly, a finite energy gap persists into the highly disordered state in spite of many sites having negligible pairing amplitude and is understood in detail within the BdG framework. Once the pairing amplitude becomes inhomogeneous, the role of quantum phase fluctuations becomes crucial in driving a superconductor-insulator transition at a critical disorder. The insulator is unusual as it has a finite gap for all disorder strengths in marked contrast to the Anderson insulator in non-interacting systems. We treat the phase fluctuations within a self consistent harmonic approximation and obtain the superfluid stiffness as a function of disorder, which agrees well with our earlier quantum Monte Carlo studies.

Recent experiments seem to confirm predictions that interactions lead to charge-density wave ground states in higher Landau levels. These new “correlated” ground states of the quantum Hall system manifest themselves for example in a strongly anisotropic resistivity tensor. We give a brief introduction and overview of this new and emerging field.

We calculate numerically the spectrum of disordered electrons in the lowest Landau level at filling factor 1/5 using the self-consistent Hartree-Fock approximation for systems containing up to 400 flux quanta. Special attention is paid to the correct treatment of the q = 0 component of the Coulomb interaction. For sufficiently strong disorder, the system is an insulator at this filling factor. We observe numerically a Coulomb gap in the single-particle density of states (DOS). The DOS agrees quantitatively with the predictions for classical point charges.

Using a mean field theory on the von Neumann lattice, we study compressible anisotropic states around v = l+1/2 in the quantum Hall system. The Hartree-Fock energy of the unidirectional charge density wave (UCDW) are calculated self-consistently. In these states the UCDW seems to be the most plausible state. We show that the UCDW is regarded as a collection of the one-dimensional lattice fermion systems which extend to the uniform direction. The kinetic energy of this one-dimensional system is induced from the Coulomb interaction term and the self-consistent Fermi surface is obtained.

We compute by means of extensive quantum chemistry methods the nearest-neighbour effective exchange and hopping integrals for α′NAV₂0₅. We find that, unlike usually assumed, the system is basically a two-dimensional asymmetric triangular Heisenberg lattice, where the sites represent unpaired electrons delocalised on the VOV rungs.

We apply the renormalization-group (RG) approach to two model systems where the two-dimensional Fermi surface has portions which give rise to the logarithmically singular two-loop self-energy process.

The normal coupled cluster method is implemented to high orders in a systematic approximation scheme, and is shown to give accurate ground- and excited-state properties of anisotropic Heisenberg antiferromagnets and their quantum phase transitions at zero temperature.

Quantum dots are often referred to as artificial atoms: Metallic gates at the surface of a GaAs-GaAlAs heterostructure confine the two-dimensional electron gas at the interface to an area of ≤ μm² size. Because of the tunneling barriers connecting the quantum dot to external leads, the number of electrons on the dot is (almost) integer, and the Coulomb interaction is important and affects many properties of quantum dots: The spacing of Coulomb blockade resonances, the co-tunneling between resonances, and (possible) localization in Fock space. Some theoretical work relating to these topics is reviewed.

We analyze the smallest Dirac eigenvalues by formulating an effective theory for the Dirac spectrum. We find that in a domain where the kinetic term of the effective theory can be ignored, the Dirac eigenvalues are distributed according to a Random Matrix Theory with the global symmetries of the QCD partition function. The kinetic term provides information on the slope of the average spectral density of the Dirac operator. In the second half of this lecture we interpret quenched QCD Dirac spectra (with eigenvalues scattered in the complex plane) in terms of an effective low energy theory.

This article provides an introduction to the ideas behind the multilevel blocking (MLB) approach to the fermion sign problem in path-integral Monte Carlo simulations, and also gives a detailed discussion of MLB results for quantum dots. MLB can turn the exponential severity of the sign problem into an algebraic one, thereby enabling numerically exact studies of otherwise inaccessible systems. Low-temperature simulation results for up to eight strongly correlated electrons in a parabolic 2D quantum dot are presented.

The Kondo effect in a quantum dot is discussed. In the standard Coulomb blockade setting, tunneling between the dot and the leads is weak, the number of electrons in the dot is well-defined and discrete; the Kondo effect may be considered in the framework of the conventional one-level Anderson impurity model. It turns out however, that the Kondo temperature T_K in the case of weak tunneling is extremely low. In the opposite case of almost reflectionless single-mode junctions connecting the dot to the leads, the average charge of the dot is not discrete. Surprisingly, its spin may remain quantized: s = 1/2 or s = 0, depending (periodically) on the gate voltage. Such a “spin-charge separation” occurs because, unlike an Anderson impurity, a quantum dot carries a broad-band, dense spectrum of discrete levels. In the doublet state, the Kondo effect develops with a significantly enhanced T_K. Like in the weak-tunneling regime, the enhanced T_K exhibits strong mesoscopic fluctuations. The statistics of the fluctuations is universal, and related to the Porter-Thomas statistics of the wave function fluctuations.

We present the results of Diffusion Monte Carlo (DMC) calculations based on accurate multiconfiguration wave functions for N electrons (N ≤ 13) confined to a parabolic quantum dot. The density and correlation energies have been computed and compared with the predictions of local spin density approximation theory (LSDA). We also computed the addition energy a a function of the number of electrons in the dot, and compared them with the results of LSDA and Hartree Fock calculations. DMC results show a behavior qualitatively closer to the result of recent capacitance experiments.

We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a “sign” problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.

The formulation and recent applications of the Quantum Monte Carlo diagonal-ization (QMCD) method are reported. The QMCD has been proposed for solving the quantum many-body interacting systems, providing us with energy eigenvalues, transition matrix elements and wave functions. Its application to the nuclear shell model is referred to as the Monte Carlo Shell Model. By the Monte Carlo Shell Model calculations, the level structure of low-lying states can be studied with realistic interactions, providing a useful tool for nuclear spectroscopy. The Monte Carlo Shell Model has been applied to the study of a variety of nuclei, and can be characterized as the importance truncation scheme to the full diagonalization which is infeasible in many cases due to extremely large dimensions.

We discuss recent results on color superconductivity in QCD at large chemical potential.

Recent studies of light p-shell nuclei reveal that so-called ‘realistic’ nuclear interactions, those based on the NN scattering data augmented with plausible models of the three-nucleon interaction, provide a good description of nuclei through A=8. However, significant discrepancies exist, primarily in the energies of neutron rich systems and in the L·S splittings in the spectra of these nuclei. We briefly describe the methods used in these studies, and describe improved models of the three-nucleon interaction.

Finite nuclei such as those found in the chain of even tin isotopes from ¹⁰²Sn to ¹³⁰Sn, exhibit a near constancy of the excitation energy, a constancy which can be related to strong pairing correlations and the near degeneracy in energy of the relevant single-particle orbits. Large shell-model calculations for these isotopes reveal that the major contribution to pairing correlations in the tin isotopes stems from the ¹S₀ partial wave in the nucleon-nucleon interaction. Omitting this partial wave and the ³P₂ wave in the construction of an effective interaction, results in a spectrum which has essentially no correspondence with experiment. These partial wave are also of importance for infinite neutron matter and nuclear matter and give the largest contribution to the pairing interaction and energy gap in neutron star matter.

By combining diffusion Monte Carlo for the spatial degrees of freedom and auxiliary field Monte Carlo to separate the spin-isospin operators, we can solve for the ground state of many-nucleon systems. We use a path constraint to control the fermion sign problem and apply the method to neutron systems interacting with the Argonne two nucleon potential and the Urbana IX three-nucleon potential. We compare our results with fermion hypernetted chain calculations.

Recent developments in neutron star theory and observation are discussed. Based on modern nucleon-nucleon potentials more reliable equations of state for dense nuclear matter have been constructed. Furthermore, phase transitions such as pion, kaon and hyperon condensation, superfluidity and quark matter can occur in cores of neutron stars. Specifically, the nuclear to quark matter phase transition and its mixed phases with intriguing structures are treated. Rotating neutron stars with and without phase transitions are discussed and compared to observed masses, radii and glitches. The observations of possible heavy ∼ 2M⊙ neutron stars in X-ray binaries and quasi-periodic oscillations require relatively stiff equations of state and restrict strong phase transitions to occur at very high nuclear densities only.

For the past 40 years, Brueckner theory has proven to be a most powerful tool to investigate systematically models for nuclear matter. I will give an overview of the work done on nuclear matter theory, starting with the simplest model and proceeding step by step to more sophisticated models by extending the degrees of freedom and including relativity. The final results of a comprehensive hadronic theory of nuclear matter are compared to the predictions by currently fashionable two-nucleon force models. It turns out that a two-nucleon force can, indeed, reproduce those results if the potential is nonlocal, since nonlocality is an inherent quality of the more fundamental field-theoretic approach. This nonlocality is crucial for creating sufficient nuclear binding.

Understanding an important class of experiments requires that light-front dynamics and the related light cone variables k⁺, k_┴ be used. If one uses k⁺ = k⁰ + k³ as a momentum variable the corresponding canonical spatial variable is x^- = x⁰ - x³ and the time variable is x⁰ + x³. This is the light front (LF) approach of Dirac. A relativistic light front formulation of nuclear dynamics is developed and applied to treating infinite nuclear matter in a method which includes the correlations of pairs of nucleons. This is light front Brueckner theory.

We review the latest variational calculations of the ground state properties of doubly closed shell nuclei, from ¹²C to ²⁰⁸Pb, with semirealistic and realistic two-and three-nucleon interactions. The studies are carried on within the framework of the correlated basis function theory and integral equations technique, with state dependent correlations having central and tensor components. We report results for the ground-state energy, one- and two-body densities and static structure functions. For ¹⁶O and ⁴⁰Ca we use modern interactions and find that the accuracy of the method is comparable to that attained in nuclear matter with similar hamiltonians, giving nuclei underbound by ~2 MeV/A. The computed Coulomb sums are in complete agreement with the latest analysis of the experimental data.

Exact expression of the Pauli exclusion operator Q in the nuclear matter calculation is presented. Numerical calculations of the G matrix with the exact Q operator are carried out by employing the Bonn B and C NN potentials. It is observed that the exact treatment of the operator Q brings about non-negligible and attractive contributions to the binding energy.

A theoretically consistent approach to the calculation of the Coulomb energy of nuclei is presented. New contributions to the single-particle energies are taken into account. We show that the interplay between the Coulomb interaction and the strong interaction leads to an upward shift of the proton single-particle levels, affecting the position of the calculated proton drip line. The same contributions are responsible for significant corrections to the mass difference of the mirror nuclei (Nolen-Schiffer anomaly) and to the effective proton mass.

The description of the properties of liquid Helium is a challenge for any microscopic many-body theory. In this context, we study the ground state and the excitation spectrum of one ³He impurity in liquid ⁴He at T = 0 with the aim of illustrating the power of the correlated basis function formalism in describing heavily correlated systems. The strong interatomic interaction and the large density require the theory to be pushed to a high degree of sophistication. A many-body correlation operator containing explicit two- and thre-particle correlation functions is needed to obtain a realistic ground state wave function, whereas a perturbative expansion including up to two phonon correlated states must be enforced to study the impurity excitation energies. The theory describes accurately the experimental spectrum along all the available momentum range. As empirically shown by the experiments, a marked deviation from the quadratic Landau-Pomeranchuck behavior is found and the momentum dependent effective mass of the impurity increases of ~ 50 % at q ~ 1.7 Å⁻¹ with respect to its q = 0 value. Although the main emphasis is given to the correlated basis function theory, we present also comparisons with other methods, as diffusion Monte Carlo, variational Monte Carlo with shadow wave functions and time dependent correlations.

With the standard techniques to deal with the sign problem in a diffusion Monte Carlo (DMC) calculation, i.e., fixed node (FN) and released node (RN), we have re-explored the application of DMC to the study of normal liquid ³He at zero temperature. Our analysis points to the necessity of including optimized backflow correlations to quantitatively improve the FN-DMC results. The theoretical equation of state so obtained fits the experimental data for the energy, the pressure and the sound velocity The upper bounds provided by the FN method are checked on the one hand by the application of RN, and on the other, by the inclusion of explicit three-body correlations in the backflow operator. Both calculations show negligible corrections with respect to the characteristic size of the statistical error.

In two dimensions the microscopic theory, which provides a basis for the naive analogy between a quantized vortex in a superfluid and an electron in an uniform magnetic field, is presented. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with the energy, and scales with the size of the vortex.

We have studied the ground state properties of two-dimensional ³He-⁴He mixtures at zero temperature. ³He atoms with opposite spins form loosely bound dimers in free space and in low concentration mixtures with ⁴He. The binding energy of the dimer ranges from milli-Kelvins near the saturation density to micro-Kelvins at the solidification density. The radius of such a weakly bound dimer is tens of Angstoms. We also calculate the phase diagram of the mixture. The maximum solubility of ³He ≈ 7% is determined by comparing chemical potentials in the pure and mixed fluids. The upper stability limit of the super-saturated mixture is obtained from the second derivative of the enthalpy. It becomes negative at the concentration 10-15 % depending of the pressure, indicating a softening of the concentration-fluctuation mode. We also find an indication of the phase transition from the dimerized to atomic mixture.

Superfluidity of Λ-hyperons admixed in neutron star cores is investigated by a realistic approach and found to be realizable with the critical temperature of 10⁸⁻⁹ K though in a restricted density region. In reference to the results of Λ, the possible occurence of the Σ⁻- and Ξ⁻- superfluidities is pointed out. The pairing effects of hyperons is also shown to be important in hypernuclear matter relevant to hypernuclei. In a treatment of a gap equation, the necessity to use a “bare” hyperon-hyperon interaction, instead of an “effective” one, is stressed.

Starting from a t-J model, we introduce inhomogeneous terms to mimic stripes. We find that if the inhomogeneous terms break the SU(2) spin symmetry the binding between holes is tremendously enhanced in the thermodynamic limit. In any other model (including homogeneous models) the binding in the thermodynamic limit is small or neglible. By including these inhomogeneous terms we can reproduce experimental neutron scattering data. We also discuss the connection of the resulting inhomogeneity-induced superconductivity to recent experimental evidence for a linear relation between magnetic incommensurability and the superconducting transition temperature, as a function of doping.

We have investigated the vortex in chiral superconductors, especially in p-wave case. In chiral superconductors the Cooper pair has orbital angular momentum hence U(1), parity (P) and time reversal symmetry (T) are broken simultaneously. We have found that the vortex has fractional charge and fractional angular momentum which comes from P- and T-violation. The fractionalization of the angular momentum suggests that the vortex could be an anyon which obeys the fractional statistics. We have also pointed out that the electric field is induced near the vortex core and non-trivial electromagnetic phenomena are expected to occur.

Gaseous Bose-Einstein condensates are a macroscopic condensed-matter system which can be understood from a microscopic, atomic basis. We present examples of how the optical tools of atomic physics can be used to probe properties of this system. In particular, we describe how stimulated light scattering can be used to measure the coherence length of a condensate, to measure its excitation spectrum, and to reveal the presence of pair excitations in the many-body condensate wavefunction.

The most salient features of the Bose-Einstein condensation of a magnetically confined alkali vapor is the diluteness of the gas and the extremely weak effective interactions. From a theoretical point of view, the interesting aspect is the potential formulation of the many-body quantum theory for a non-uniform and potentially non-equilibrium system founded entirely on microscopic physics. The crucial postulate is the rapid attenuation of many particle quantum correlations in the dilute system which can be motivated from universal considerations. In principle, it will be possible to provide direct comparison between theory and experiment over all temperature scales with no phenomenological parameters—a challenge facing the theoretical community in the near future. The dilute gas experiments provide an exciting stage on which to build bridges linking the theory of complex and collective phenomena in superconducting and superfluid systems, with the single particle microscopic physics described in quantum optics and laser physics.

The field theory treatments of Bose condensed systems developed forty years ago have received renewed attention within the context of inhomogeneous systems since the realisation of Bose-Einstein condensation in dilute trapped alkali gases in 1995. These experiments allow quantitative tests of the theories used to describe them and hence provide a unique laboratory in which to evaluate and develop these field theoretical methods to an unprecedented degree. Various examples are discussed to illuminate these developments.

A theoretical investigation is carried out of the ground state of a finite assembly of Bose hard spheres enclosed in a spherical cavity. Total and condensate radial densities are computed, and the non-uniform condensate is studied as a function of the number of particles in the cavity. Comparison with mean-field results is made. Possible experimental implications are discussed.

We present all stationary solutions to the nonlinear Schrödinger equation in one dimension for box and periodic boundary conditions. For both repulsive and attractive nonlinearity we find expected and unexpected solutions. Expected solutions are those that are in direct analogy with those of the linear Schrödinger equation under the same boundary conditions. Unexpected solutions are those that have no such analogy. We give a physical interpretation for the unexpected solutions. We discuss the properties of all solution types and briefly relate them to experiments on the dilute-gas Bose-Einstein condensate.

Stationary states of the nonlinear Schrödinger equation (NLSE) found analytically in previous work are extended into 2 and 3 dimensions by the simplest possible ansatz: namely, it is assumed that the direct product of one dimensional solutions for each dimension will yield a stationary state. The solutions considered mimic the dynamics of a repulsive Bose-Einstein condensate (BEC) in a trap of high aspect ratio. This assumption of separability, as established by direct numerical integration of the NLSE via variable step 4th order Runge-Kutta using a pseudo spectral basis, is found to work well for both ground and excited states for box transverse confinement, and for either box or periodic boundary conditions along the longest trap axis. Addition of white noise at t = 0, followed by similar numerical propagation in either 2 or 3 dimensions, is found to lead to instability once the transverse confining dimension are greater than approximately 6 healing lengths. Such instabilites eventually manifest themselves as vortices fathered by the well known snake instability of the NLSE solitons in dimensionalities higher than 1 The dynamics of interacting solitons may become chaotic as the solitons themselves become unstable in the presence of noise.

If the electron-electron repulsion in an atom or molecule were very weak, it could be treated by orbital-based perturbation theory. If this repulsion were very strong, it could be treated in a model of strict correlation. A simple interaction strength interpolation between these two limits, at fixed electron density, can describe the reality that lies between the extremes. By working entirely within a sophisticated density functional approximation, the meta-generalized gradient approximation, we find that the interpolation error is only about 0.1 % for the exchange-correlation energy and about 4 kcal/mole = 0.17 eV for the atomization energy. We also find that real systems probably lie close to the radius of convergence of density functional perturbation theory.

The ground-state properties of superfluid nuclear systems with ¹S₀ pairing are studied within a local energy-density functional (LEDF) approach. A new form of the LEDF is proposed with a volume part which fitsthe Friedman-Pandharipande and Wiringa-Fiks-Fabrocini equation of state at lowand moderate densities and allows an extrapolation to higher densities which preserves causality. For inhomogeneous systems, a surface term is added, with two free parameters, which has a fractional form like a Padé approximantcontaining the square of the density gradient in both the numerator and denominator. In addition to the direct and exchange Coulomb interaction energy, an effective density-dependent Coulomb-nuclear correlation term is included with one more free parameter. A three-parameter fit to the masses and radii of about 100 spherical nuclei has shown that the latter term gives a contribution of the same order of magnitude as the Nolen-Schiffer anomaly in the Coulomb displacement energy. The root-mean-square deviations from experimental masses and radii with the proposed LEDF come out about a factor of two smaller than those obtained with the conventional functionals based on the Skyrme or finite-range Gogny force, or on relativistic mean-field theory. The generalized variational principle is formulated leading to the self-consistent Gor’kov equations which are sovled exactly, with physical boundary conditions both for the bound and scattering states. The method is used to calculate the differential observables such as odd-even mass differences and staggering in charge radii. With a zero-range density-dependent cutoff pairing interaction incorporating a density-gradient term, the evolution of these observables is reproduced reasonably well, including the kinks at magic neutron numbers and the sizes of the associatecd staggering. An extrapolation from the pairing properties of finite nuclei to pairing in infinite nuclear matter is discussed. A “reference” value of the pairing gap Δ_F ≈ 3.3 MeV is found for subsaturated nuclear matter at about 0.65 of the equilibrium density. With the formulated LEDF approach, we study also the dilute limitin both the weak and strong coupling regimes. Within the sum rules approach itis shown that the density-dependent pairing may also induce sizeable staggering and kinks in the evolution of the mean energies of multipole excitations.

The energy surface of the Helium dimer, as a prototype of a van der Waals bond molecule, is investigated within the framework of density functional theory. For the exchange-correlation energy an implicit density functional, depending on the Kohn-Sham orbitals, is applied in which exchange is treated exactly,while correlation is approximated by the lowest order contribution obtained byKohn-Sham perturbation theory. The resulting energy surface is in fair quantitative agreement with highly accurate empirical data over the complete range ofinternuclear separations, demonstrating that the concept of orbital-dependent functionals can provide a seamless description of dispersion forces. As selfconsistent calculations with implicit functionals on the basis of the optimized potential method are rather time-consuming, the correlation part of the exchange-correlation functional is evaluated perturbatively in the Helium dimer calculations. However, we also present an approximate scheme for the evaluation ofthe corresponding correlation potential.

I review recent advances in understanding the physical character of the exchange-correlation (xc) potential in time-dependent density functional theory. I show that in an electron gas of slowly varying density the dynamical part ofthe xc potential is an extremely nonlocal functional of the density. This difficulty can be circumvented by introducing an xc vector potential that is a function of the local current density. The form of this vector potential is entirely specified by symmetry. Its physical effects are analogous to those of the viscous force in the Navier-Stokes equation of classical hydrodynamics. Recentapplications of the xc vector potential to the calculation of infrared absorption linewidths in quantum wells are described.

We calculated the exchange, correlation and total energies of clusters of alkali metals with N = 1 − 150 atoms in the spherical jellium model. The calculations were made using the Kohn-Sham method with exchange and correlation energies evaluated in the meta-generalized gradient approximation (MGGA), proposed by J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, in the generalized gradient approximation (GGA) of J. P. Perdew, K. Burke and M. Ernzerhof, and in the Local Density Approximation (LDA).

We evaluated the relative deviations of MGGA and GGA energies with respect to LDA. Exchange energies of MGGA and GGA are more negative than the LDA ex-change energy and become closer to this as the cluster size increases. On the other hand, the GGA and MGGA correlation energies, which are almost identical, are less negative than LDA. The deviations of GGA and MGGA exchange-correlationenergies with respect to LDA are smaller than those of the exchange and correlation energies separately.

For clusters with 18 and 20 atoms we have compared our jellium results withVariational and Diffusion Monte-Carlo results. Errors of LDA for exchange and correlation tend to cancel so that the total exchange-correlation energy is close to the Monte-Carlo results. Similar cancellations occur with GGA and MGGA. We also examined the validity of the liquid drop model.

The normal coupled cluster method has been used to obtain simple accurate ap-proximations to the ground and first excited state energies of the linear e ⊗ E pseudo-Jahn-Teller Hamiltonian.

The validity of the coupled cluster method is studied within the lattice gauge field theory given by a SU(2) pure glue theory in 2+1 dimensions. Satisfactory convergence is observed for the ground state, but the method is less successful for the prediction of glueballs. We propose to improve the coupled cluster method for excited states by combining it with standard Monte-Carlo techniques which potentially cure the non-hermiticity problems caused by the truncation.

In 1955 Marshall¹ used a variational method to study the isotropic spin-half Heisenberg antiferromagnet (HAF) specified by the Hamiltonian

where the sum on 〈i,j〉 counts each nearest-neighbourpair once and once only, and J is positive. Furthermore, the lattice was assumed to be bipartite (i.e., the lattice can be divided into two sublattices (A,B) such that all nearest neighbours of a site on onesublattice lie on the other and vice versa). The total number of atoms was N and the spatial dimensionality was not restricted. In the course of this paper it was proven that the exact ground-state wave function can be written as where {|I〉} are the usual basis states in the Ising representation. The coefficients {c_I} were proven to have the property that, c_I = (- 1)^{ϕ (I)}a_I where the coefficients {a_I} are all positive real numbers or zero, and ϕ(I) is the eigenvalue , with respect to the corresponding eigenstate |I〉, of the operator, ϕ = n_A which counts all of the spin-half ‘up’ states on the A-sublattice. Lieb and Mattis² proved that the ground state of the HAF was a singlet, following the work of Marshal¹and Lieb et al.³

We derive an exact probabilistic representation for the evolution of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions in terms of an associated stochastic dynamics of a collection of Poisson processes.

The short time behavior of a disturbed system is influenced by off-shell motion and characterized by the reduced density matrix possessing high energetictails. After this short time regime the time evolution is controlled by small gradients. This leads to a nonlocal Boltzmann equation for the quasiparticle distribution and a functional relating the latter one to the reduced density matrix. The nonlocalities are presented as time and space shifts arising from gradient expansion and are leading to virial corrections in the thermodynamical limit.

We explore a nonlinear field model to describe the interplay between the ability of excitons to be Bose-condensed and their interaction with other modes of a crystal. We apply our consideration to the long-living para-excitons in Cu₂O. Taking into account the exciton-phonon interaction and introducing a coherent phonon part of the moving condensate, we derive quasi-stationary equations for the exciton-phonon condensate. These equations can support localized solutions, and we discuss the conditions for the moving inhomogeneous condensate to appear in the crystal. The stability conditions of the moving condensate are analyzed by use of Landau arguments, and Landau critical parameters appear in the theory. Finally, we apply our model to describe the recently observed interference and strong nonlinear interaction between two coherent exciton-phonon packets in Cu₂0.

We derive a fourth-order diffusion Monte Carlo algorithm for solving quantum many-body problems. The method uses a factorization of the imaginary time propagator in terms of the usual local energy E and Langevin operators L as well as an additional pseudo-potential consisting of the double commutator [E_L, [L,E_L]]. A new factorization of the propagator of the Fokker-Planck equation enables us to implement the Langevin algorithm to the necessary fourth order. We achieve this by the addition of correction terms to the drift steps and the use of a position-dependent Gaussian random walk. We show that in the case of bulk liquid helium the systematic step size errors are indeed fourth order over a wide range of step sizes.

A generalization of symmetrized density matrices in combination with the technique of generating functions allows to calculate the partition function and the static correlation functions of identical particles in a parabolic confining well with harmonic two-body interactions.

The following sections are included:

• AUTHOR INDEX

• SUBJECT INDEX