Two-body Density Matrices Research Articles

Multiconfigurational time-dependent Hartree method for mixtures consisting of two types of identical particles

We specify the formally exact multiconfigurational time-dependent Hartree method originally developed for systems of distinguishable degrees of freedom to mixtures consisting of two types of identical particles. All three cases, Fermi-Fermi, Bose-Bose, and Bose-Fermi mixtures, are treated on an equal footing making explicit use of the reduced one- and two-body density matrices of the mixture. The theory naturally contains as specific cases the versions of the multiconfigurational time-dependent Hartree method for single-species fermions and bosons. Explicit and compact equations of motion are derived and their properties and usage are briefly discussed.

Unified view on multiconfigurational time propagation for systems consisting of identical particles

We show that the successful and formally exact multiconfigurational time-dependent Hartree method (MCTDH) takes on a unified and compact form when specified for systems of identical particles (MCTDHF for fermions MCTDHB for bosons). In particular the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the system's many-particle wave function. We point out that this appealing representation of the equations of motion opens up further possibilities for approximate propagation schemes.

The one-body and two-body density matrices of finite nuclei with an appropriate treatment of the center-of-mass motion⋆

The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wavefunctions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators a^+ and a for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set {a^+} times other exponential of the set {a}. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of Tassie-Barker factors, and point out other model-independent results. The intrinsic wavefunction of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the 4He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wavefunction. We find that the TI introduces important effects in the MDs.

Relation between Density-Matrix Theory and Pairing Theory

The time-dependent density-matrix theory (TDDM) gives a correlated ground state as a stationary solution of the time-dependent equations for one-body and two-body density matrices. The small amplitude limit of TDDM (STDDM) is a version of extended RPA theories which include the effects of ground state correlations. It is shown that the solutions of the Hartree-Fock Bogoliubov theory and the quasi-particle RPA satisfy the TDDM and STDDM equations, respectively, when only pairing-type correlations are taken into account in TDDM and STDDM.

The Multiconfiguration Equations for Atoms and Molecules: Charge Quantization and Existence of Solutions

We prove the existence of ground-state solutions for the multiconfiguration self-consistent field equations for atoms and molecules whenever the total nuclear charge Z exceeds N−1, where N is the number of electrons. Moreover, we show that for arbitrary values of Z and N the scattering charge, i.e., the asymptotic amount of charge lost by an energy-minimizing sequence, is integer-quantized. Our analysis applies to the MC equations of arbitrary rank. As special cases we recover, in a new and unified way, the existence theorems of Zhislin [Zh60] for the N-body Schrodinger equation (infinite rank MC) and of Lieb & Simon [LS77] for the Hartree-Fock equations (rank-N MC). Our approach is a direct study of an invariant, orbital-free formulation in N-body space of the underlying variational principle. Proofs involve (i) the geometric N-body localization methods for the linear Schrodinger equation first introduced by Enss [En77] (and developed in [Sim77, Sig82]), which can be adapted to become powerful tools in nonlinear many-body theory as well, (ii) weak convergence methods from the theory of nonlinear partial differential equations, (iii) careful analysis of the structure of the one- and two-body density matrices of the bound and scattering fragments delivered by geometric localization, which allows us to overcome the fact that the rank of the fragments is not reduced by localization.

High-momentum dynamic structure function of liquid3He−4Hemixtures: A microscopic approach

The high-momentum dynamic structure function of liquid ${}^{3}\mathrm{He}{\ensuremath{-}}^{4}\mathrm{He}$ mixtures has been studied introducing final-state effects. Corrections to the impulse approximation have been included using a generalized Gersch-Rodriguez theory that properly takes into account the Fermi statistics of ${}^{3}\mathrm{He}$ atoms. The microscopic inputs, as the momentum distributions and the two-body density matrices, correspond to a variational (fermi)-hypernetted chain calculation. The agreement with experimental data obtained at $q=23.1{\mathrm{\AA{}}}^{\ensuremath{-}1}$ is not completely satisfactory, the comparison being difficult due to inconsistencies present in the scattering measurements. The significant differences between the experimental determinations and the theoretical results for the ${}^{4}\mathrm{He}$ condensate fraction and the ${}^{3}\mathrm{He}$ kinetic energy still remain unsolved.

Final State Effects in the high q response of 3He– 4He mixtures

A modified Gersch–Rodriguez formalism describing the leading Final State Effects in the high momentum transfer response of low concentration 3He– 4He mixtures is presented and discussed. The leading corrections to the Impulse Approximation are expressed in terms of the interatomic potentials and the semidiagonal two-body density matrices of both the mixture and its boson–boson approximation, in which the 3He atoms are replaced by bosons of the same mass and at the same partial density. Numerical calculations of the Final State Effects functions of 3He and 4He are finally

Double Giant Quadrupole Resonance in Time-Dependent Density-Matrix Theory

state of the isoscalar giant quadrupole resonance in 16 0. It is found that the peak energy of the double phonon state is nearly twice the energy of the single phonon state calculated in the random phase approximation and that its width is slightly larger than that of the single-phonon state. The double phonon states of giant resonances have been the subject of a num­ ber of recent experimental 1) - 3) and theoretical 4) - 10) investigations, and some mi­ croscopic calculations have been performed based on various time-dependent the­ ories. 8) - 10) In this note we propose a new time-dependent approach based on the time-dependent density-matrix theory (TDDM).l1) TDDM is an extended version of the time-dependent Hartree-Fock theory (TDHF), and its equations of motion determine the time evolution of both one-body and two-body density matrices. We demonstrate that, using appropriate initial conditions for the two-body density ma­ trix, we can calculate the strength functions of the double-phonon states in a manner similar to that used to calculate the strength functions of single phonon states in TDHF and TDDM.ll) The new method is applied to the double phonon state of the isoscalar quadrupole resonance in 16 0. We first briefly discuss the equations of motion in TDDM and then present the method for calculating the strength function of the double giant quadrupole resonance (DGQR). The formulation of TDDM is based on a truncation of the BBGKY hierarchy 12) of reduced density matrices, in which genuine correlated parts of a three-body density matrix and higher reduced density matrices are neglected. 13) TDDM thus determines the time evolution of a one-body density matrix p and a two­ body density matrix P2. The equations of motion in TDDM consist of the following three coupled equations for the single-particle wavefunctions 'ljJQ, the occupation matrix nQQI (the expansion coefficients of p) and the correlation matrix CQ(3QI(31 (the expansion coefficients of C2, where C2 is a correlated part of P2): 11)

Coherent and incoherent dynamic structure functions of the free Fermi gas

A detailed calculation of the coherent and incoherent dynamic structure functions of the free Fermi gas, starting from their expressions in terms of the one- and semi-diagonal two-body density matrices, is derived and discussed. Their behavior and evolution with the momentum transfer is analyzed, and particular attention is devoted to the contributions that both functions present at negative energies. Finally, an analysis of the energy weighted sum rules satisfied by both responses is also performed. Despite the simplicity of the model, some of the conclusions can be extended to realistic systems.

Stationary Solution of a Time Dependent Density Matrix Formalism

A stationary solution of a time-dependent density-matrix formalism, which is an extension of the time-dependent Hartree-Fock theory to include the effects of two-body correlations, is obtained for the Lipkin model hamiltonian, using an adiabatic treatment of the two-body interaction. It is found that the obtained result is a reasonable approximation for the exact solution of the model. We have recently reported realistic calculations for the damping of giant reso­ nances1>-3> and for low-energy heavy-ion collisions, 4 > based on an extended version of the time-dependent Hartree-Fock theory. The ETDHF theory called the time­ dependent density-matrix theory (TDDM) determines the time evolution of both one-body and two-body density matrices and allows us to consistently treat the effects of the nuclear mean field and two-body correlations. Although the obtained results are quite encouraging, these numerical calculations contain an inconsistency in the treatment of the ground state: the Hartree-Fock (HF) ground state, which clearly is not a stationary solution of TDDM, is used as the initial ground state. In our previous paper 5 > we tried to find a correlated stationary solution of TDDM using the stationary limit of the TDDM equations and the constraints on reduced density matrices derived from general relations among them. The obtained results for the Lipkin model, 6 > however, were not quite satisfactory and the problem of finding the ·reasonable stationary solutions of TDDM has not been settled yet. In this paper we demonstrate that a better stationary solutions of TDDM can be obtained through an adiabatic treatment of two-body interactions. The TDDM equations have been derived from the truncation scheme, proposed by Wang and Cassing,7), 8 > of the well-known BBGKY hierarchy 9 > for reduced density

An elementary Hubbard Hamiltonian and the representability problem

Two finite size Hubbard models in two dimensions are considered at half filling: a 2×2 square lattice and a 4×4 square lattice, both with periodic boundary conditions. In the elementary case of the 2×2 lattice it is shown that the eigenvalues of the Hamiltonian corresponding to a given simmetry can be found applying the variational approach directly to the one and two-body density matrices. In the case of a 4×4 lattice the allowed region of the relevant one-body and two-body density matrices is numerically determined and the ground state energy is computed for several values of the interaction.

Non-energy-weighted sum rule in the response function method: One- and two-body density matrices

Two expressions for the non-energy-weighted sum rules are studied in detail. One of them contains the information of one- and two-body density matrices of the ground state of a nucleus. The other summarizes the elastic and inelastic processes which determine the nuclear response to an external probe. These processes are well described by the response function method. In this paper, we establish the relation of these two expressions; namely, we show which processes of the response have the information of the one- and two-body density matrices. To make the physical points clear, we use the Coulomb sum rule as an example.

Unitary group approach to reduced density matrices

A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.

The structure of quantum fluids

We outline the principal features of Bose and Fermi fluids that are revealed in particle scattering experiments at high momentum transfer. In this regime, the dynamic structure function is determined by the dominant influence of correlations which are embodied in the static one- and two-body density matrices characterizing a strongly correlated system. We analyze the general structure of these fundamental quantities and of the associated momentum distributions that enter as input quantities for determining the dynamical response. We discuss their physical interpretation and their interrelationships. We further describe the main features of advanced many-body methods, which begin on a nonperturbative basis. They permit a formal and numerical evaluation of various quantities that characterize the structure of the density matrices and therewith of quantum fluids and solids.

Extended coupled-cluster method. III. Zero-temperature hydrodynamics of a condensed Bose fluid.

The extended coupled-cluster method (ECCM) of quantum many-body theory, which has been studied and developed in earlier papers in this series, is now applied to condensed Bose systems. The formalism is seen to provide a concise and convenient description of quite general nonstatic states of such systems with arbitrary spatial inhomogeneity. The entire ECCM description is based on the equations of motion for the set of linked-cluster amplitudes which, we have shown, completely characterize an arbitrary quantum system with a Schr\"odinger dynamics. Since all such amplitudes obey the cluster property, and hence may be regarded as a set of quasilocal, many-body, classical order parameters, the formalism is in principle perfectly capable of describing phase transitions and states of topological excitation or deformation and broken symmetry. At the lowest (mean-field) level of truncation, the formalism degenerates to the well-known Gross-Pitaevskii description of the condensate wave function or one-body order parameter. The treatment is developed in a fully gauge-invariant fashion, and is thereby shown to provide a complete hydrodynamical description, valid in the zero-temperature limit. In particular, by studying the off-diagonal one- and two-body density matrices in terms of the basic ECCM amplitudes, we derive balance (or local conservation) equations for such local observables as the number density, momentum density, and energy density. These are shown to be obeyed not only by the exact untruncated formalism but also by most practical truncation schemes in the ECCM configuration space.

Selfconsistent diabatic approach to dissipative collective nuclear motion

Within a selfconsistent description for the one- and two-body density matrices collective variables are introduced via scaling diabatic states. Equations of collective motion coupled to a collision integral for the single-particle occupation probabilities are derived from the randomness of the two-body interaction matrix elements and from an additional time smoothing procedure. For a linear approximation to the time-dependence of the single-particle energies the collision term conserves energy all by itself, i.e. the time-smoothed time derivative of the correlation energy vanishes.

Selfconsistent diabatic approach to dissipative collective nuclear motion

Within a selfconsistent description for the one- and two-body density matrices collective variables are introduced via scaling diabatic states. Equations of collective motion coupled to a collision integral for the single-particle occupation probabilities are derived from the randomness of the two-body interaction matrix elements and from an additional time smoothing procedure. For a linear approximation to the time-dependence of the single-particle energies the collision term conserves energy all by itself, i.e. the time-smoothed time derivative of the correlation energy vanishes.

Fluctuation effect and the spin-glass transition temperatures in the random-bond Ising models in the hexagonal, triangular, BCC and other lattices-unified description

The cluster variational free energies of the random-bond Ising model are given in terms of effective fields and of effective interactions. The stationarity of the free energy gives the reducibilities between the one-body, two-body and cluster density matrices. From these reducibilities the uniform, staggered and spin-glass susceptibilities are obtained. A simple version of the above treatment is also presented which is called the cactus approximation. The phase diagrams of the binary (and ternary) mixtures of the ferromagnetic, antiferromagnetic (and non-magnetic) bonds in the triangular, hexagonal, and BCC lattices are obtained. The phase diagram of the triangular lattice is qualitatively similar to the FCC lattice and those of the hexagonal and BCC lattice to the square lattice. In particular the bond-diluted antiferromagnet in the triangular lattice is shown to have a spin-glass state.

Correlated BCS theory

A variational theory of the BCS pair-condensed state of strongly interacting Fermi systems is presented. The short-ranged correlations among the particles are described by a correlation operator which induces state-independent correlations of the Jastrow type. The two-body distribution function and the one- and two-body density matrices are expanded to all orders of the dynamical correlations and of the BCS amplitudes, and are expressed in terms of irreducible quantities. A Fermi hypernetted chain (FHNC) technique is derived to sum up all the terms of the corresponding series.

Long-range order in bose fluids

We are interested in the evaluation of the one- and two-body density matrices for extended boson matter. A trial function Ψ = Π i<j A⨍(r ij) of Bijl-Dingle-Jastrow form provides a reasonable first approximation to the ground state wave function of A bosons. The corresponding one-body density matrix has a structure n( r) = ρn c exp [− Q( r)] in the thermodynamic limit and shows off-diagonal long range order, n( r)→- ρn c > 0 as r→ ∞. The condensate fraction is given by n c = exp Q(0). A similar structure is found for the two-body density matrix, n( r 1, r 2, r' 1, r' 2) = χ( r) χ( r') exp[− R (r 1, r 2, r' 1,r' 2)] where r ≡ ¦r 1 − r 2¦, r' ≡ ¦r' 1 − r' 2¦ . The behavior over macroscopic distances is characterized by off-diagonal long range order of pairing type, n( r 1, r 2, r' 1, r' 2) → χ( r) χ( r') as ¦ r 1 − r' 1¦ → ∞ while r, r' < ∞ , in close analogy to the pairing phenomenon in Fermi superfluids. The pairing function χ( r) is related to the one-body density matrix by χ(r) = ⨍(r)n(r). The basic quantities Q( r) and Q(0) are analysed (i) in terms of a standard factor cluster expansion, (ii) in terms of a highly summed cluster expansion in the compact portions of the successive distribution functions g( r 12), g 3( r 1, r 2, r 3), …, (iii) within the hypernetted chain (HNC) framework. The HNC equations for Q( r) are solved analytically for uniform Bose fluids described by ¦g(r) − 1¦ ⪅ K ⪡ 1. We find in the uniform limit Q( r) = −(2 π) −3(4 ρ) −1 ∫ [ S( k) − 1] 2 S −1( k) exp (i k · r)d k where S( k) is the liquid structure function. The HNC formalism for quantities Q( r) and Q(0) is employed in a numerical study of the long-range order in (i) the charged boson system, (ii) a model of neutron matter, (iii) liquid 4He.