Equations for subsystems

The exp(S) equations of Coester and Kümmel are rederived from the Schrödinger equations for n-body subsystems which are specified by the amplitudes of n fermions being at certain positions in r-space and the rest moving in shell model states. This derivation needs no formal exp(S) ansatz and allows us to give physical significance to a certain method of truncating the equations. Approximations for systems with short- and long-range forces are distinguished and the application to the electron gas is briefly discussed. It is found that the mutual occurrence of long- and short-range difficulties makes it hard to truncate the equations, and a crude but simple approximation for this case is proposed. In relation to Brueckner-Bethe theory, additional self-consistency is found to be the most important difference. Its importance is stressed and discussed in relation to the concepts of a particle potential and occupation probabilities. The equations are finally studied for a solvable model which simulates long-range forces. It is seen that for this example the truncated equations are highly efficient in getting near to the exact ground state as long as the system shows no phase-transition-like phenomena.     

Variational pair theory of the attractive and extended Hubbard model ind-dimensions

We present a variational approach for treating the Hubbard Hamiltonian in one, two and three dimensions. It is based on 2M-fermion wavefunctions which are allowed to form correlated spin-singlet pairs. Expressions for the ground state energy and correlation functions are derived in terms of general pair coefficient functions. The presented approach offers a convenient starting point for improved variational treatments that allow to include different specific types of pair correlations. We present first applications to the attractive and to the extended Hubbard model using a very simple ansatz for the pair coefficient functions. The ground state energy, chemical potential, order parameter, momentum distribution as well as spin-spin and density-density correlation functions follow from a system of coupled nonlinear equations that has to be solved selfconsistently. All quantities are given for arbitrary band-filling in one, two and three dimensions. Our results are compared with those of other approximations and for the one-dimensional case with the exact results of Krivnov and Ovchinnikov.     

The renormalized variational theory: Its structure,interpretation, and relation to other methods

We apply the Ritz variational principle to a renormalized form of the Iwamoto-Yamada cluster expansion, restricting our discussion to infinite systems. The structure of the resulting theory is governed by the renormalization which keeps track of the normalization denominator in the expectation value. The single-particle potential for hole states (u_i) is introduced as a Lagrange multiplier in the variational principle, and the self-consistent choice of u_i guarantees that the renormalization factors are determined correctly. The importance of the renormalization is illustrated by a discussion of the two-body approximation to our theory. The general formalism is evaluated in more detail for the representation Ψ_T = exp(S)Φ of the trial wave function. Very fundamental considerations show that the theory is especially adapted to that choice of Ψ_T. In addition, if we use that choice of Ψ_T the self-consistent single-particle energies are directly related to experiment, and the theory is almost identical to renormalized Brueckner theory. Thus we are able to clarify many aspects of the latter. We also discuss the relation to the theory of Coester and Kümmel.     

二维费密液体理论(Ⅱ)——3He朗道参数的计算

本文采用相关基波函数(CBF)方法研究了二维³He液体的性质。基于集团展开利用配分函数的变分原理,将这一方法推广到有限温度情形。从Lennard-Jones势出发,在二级微扰近似下数值计算了二维³He液体的基态能、准粒子谱和朗道参数。 关键词：     

Lowest order constrained variational method applied to liquid $^\mathsf{3}$He

The energy of normal liquid ³He is obtained using the lowest order constrained variational (LOCV) method. In order to test the convergence of the cluster expansion series, the three-body cluster energy is calculated, with the LOCV correlation functions, and by imposing the normalization constraint on the two-body distribution function which includes three-body cluster correlations (LOCVE). It is shown that the normalization constraint plays an important role in keeping the higher cluster terms small. The resulting LOCVE calculation for the ground state energy of liquid ³He is compared with the available experimental data and the prediction from different theoretical techniques.Received: 26 May 2003, Published online: 30 January 2004PACS: 61.20.Gy Theory and models of liquid structure - 61.20.Ne Structure of simple liquids - 67.55.-s Normal phase of liquid ³He     

量子隧道态-声子耦合系统的变分法研究

本文采用变分法研究了量子隧道态-声子耦合系统的基态能量。发现:系统的基态可以用位移态及位移-压缩态这两种变分态描述,且随着耦合强度的增加,稳定的基态将由位移态向位移-压缩态转变。本文给出这种转变的相图。 关键词：     

An ab initio calculation of the vibrational energies and transition moments of HSOH

We report new ab initio potential energy and dipole moment surfaces for the electronic ground state of HSOH, calculated by the CCSD(T) method (coupled cluster theory with single and double substitutions and a perturbative treatment of connected triple excitations) with augmented correlation-consistent basis sets up to quadruple-zeta quality, aug-cc-pV(Q+d)Z. The energy range covered extends up to 20 000 cm⁻¹ above equilibrium. Parameterized analytical functions have been fitted through the ab initio points. Based on the analytical potential energy and dipole moment surfaces obtained, vibrational term values and transition moments have been calculated by means of the variational program TROVE. The theoretical term values for the fundamental levels ν_SH (SH-stretch) and ν_OH (OH-stretch), the intensity ratio of the corresponding fundamental bands, and the torsional splitting in the vibrational ground state are in good agreement with experiment. This is evidence for the high quality of the potential energy surface. The theoretical results underline the importance of vibrational averaging, and they allow us to explain extensive perturbations recently found experimentally in the SH-stretch fundamental band of HSOH.     

Two-component Fermi systems: II. Superfluid coupled cluster theory

In this second paper of a series the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi superfluids using a Bardeen-Cooper-Schrieffer (BCS) ground state as a zeroth-order approximation. We concentrate on developing the formalism necessary for carrying out eventual numerical calculations on realistic superconducting systems. We do this by generalising the one-component formalism in an appropriate manner and by using the results in the first paper of this series, where we studied two-component Fermi fluids. We stress the previous successes of the CCM, both from the point of view of analytic and numerical results, and we further indicate its potential for studying superconductivity. We restrict ourselves here to a so-called ring plus single particle energy (RING+SPE) approximation for general potentials and show how it can be formulated as a set of four coupled, bilinear integral equations for the cluster-integrated amplitudes. These latter amplitudes are themselves derived from the four-point functions of the system which provide a measure of the two-particle/two-hole component in the true ground-state wavefunction with respect to the BCS model state. We indicate how to obtain possible analytic solutions.     

Extremal Kähler metrics and Bach–Merkulov equations

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach–Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein–Maxwell equations. Inspired by the work of C. LeBrun on Einstein–Maxwell equations on compact Kähler surfaces, we give a variational characterization of solutions to Bach–Merkulov equations as critical points of the Weyl functional. We also show that extremal Kähler metrics are solutions to these equations, although, contrary to the Einstein–Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces.     

Correlated ground state of the Hubbard model

The variational many-body approach or, more generally, the method of correlated basis functions initiated for a quantitative analysis of strongly interacting quantum fluids may be adapted with minor modifications for exploring the properties of lattice models. This is demonstrated by performing an explicit analysis of the paramagnetic ground state of the Hubbard model. In a first step of the approximation scheme we represent the correlated state by a spin-dependent wave function of Jastrow-type. We analyze in detail the associated density-matrix elements and set up the corresponding Fermi hypernetted-chain equations which determine the irreducible constituents of these quantities. The solutions are discussed and constructed by iteration in terms of cluster approximants. Specializing the input data and the formal results provides a Fermi hypernetted-chain analysis of the correlations induced by a ground state wave function of the Gutzwiller form.     

Bound polaron in a quantum pseudodot under Rashba effect

In the present work, the influence of Rashba effect on bound polaron in a quantum pseudodot is studied. Using the Lee–Low–Pines unitary transformation method and the Pekar type variational procedure, we have derived an expression for the bound polaron ground state energy. The ground state energy as functions of the wave vector, the electron–phonon coupling strength, and quantum confinement size is obtained by considering different Coulomb bound potentials. It is found that (i) the ground state energy is decreased with raising the Coulomb bound potential, the electron–phonon coupling strength, and quantum confinement size. (ii) The ground state energy increases when the wave vector is increasing. (iii) The ground state energy splits into two branches (spin-up and spin-down) due to the Rashba effect.     

Recent progress in simulation of the ground state of many Boson systems

We present two recent advances in the simulation of ⁴He in the condensed phase at zero temperature. Within the variational theory of strongly interacting bosons we have studied a cluster of ⁴He atoms with one alkali ion K⁺. For the wave function we have used a new shadow wave function (SWF) in which the coupling between one ⁴He atom and its shadow variable depends on its distance from the ion. This substantially improves the energy. The first shell around the ion contains 14 atoms which are spatially ordered. However the atoms of the first shell are not completely localized and frequent exchanges with atoms which are further from the ion take place. This also suggests that at least for the ion K⁺ the atoms of the first shell participate in the superfluidity. We have also introduced a new extension of the path integral ground state (PIGS) method which is able to compute exact ground state expectation values without extrapolations and with a SWF as the trial variational wave function to project on the ground state. This is an important extension which opens up the possibility of studying disorder phenomena in the solid phase by an exact method at zero temperature. We have applied this technique to compute the energy of formation of a vacancy at different densities in the solid phase of ⁴He. This computation confirms the variational result that a vacancy is a delocalized defect in the low density helium solid.     

Structure of the superfluid ground state of an atomic Fermi gas near the Feshbach resonance

The ground state of an atomic Fermi gas near the Feshbach resonance for a negative scattering length is investigated using the variational method. The structure of the superfluid state is formed by two coherently coupled subsystems, viz., the quasimolecular subsystem in a closed channel and the subsystem of atomic pairs in an open channel. The derived system of equations makes it possible to describe the properties of the ground state for arbitrary values of the parameters (in particular, to find the gap in the single-particle Fermi excitation spectrum and the speed of sound characterizing the branch of collective Bose excitations).     

Local approach study of electron correlation effects in the Hubbard model

A variational method which corrects the one-electron ground state locally is applied to study electron correlation effects in the Hubbard model. A two-site cluster approximation is used to calculate the ground state energy functional. For the linear chain and for number of electrons per site up to 0.6, the approximation yields more than 80% of the exact results of Lieb and Wu for the ground state energy. Results for higher dimensional systems are also presented.     

Variational model for one-dimensional quantum magnets

A new variational technique for investigation of the ground state and correlation functions in 1D quantum magnets is proposed. A spin Hamiltonian is reduced to a fermionic representation by the Jordan–Wigner transformation. The ground state is described by a new non-local trial wave function, and the total energy is calculated in an analytic form as a function of two variational parameters. This approach is demonstrated with an example of the XXZ-chain of spin-1/2 under a staggered magnetic field. Generalizations and applications of the variational technique for low-dimensional magnetic systems are discussed.     

Comparison between analytical and numerical methods as applied to calculations of excited atomic states

This article presents a comparison between two approaches for implementing a variational method when calculating excited states of atoms, namely a numerical approach in which the equations arising from the requirement of an extremum of the variational functional (the Hartree—Fock equations) are solved, and an analytical approach in which the energy functional expressed in terms of analytical test functions is minimized. Both approaches are analyzed from the point of view of the approximations used to ensure that the conditions are satisfied for the complete wave function of the excited state being sought to be orthogonal to all wave functions of lower-lying energy states having the same symmetry. The well-known ATOM package is used for numerically solving the Hartree—Fock equations and the MINMAX package is used for the analytical variational calculations. It is shown that the analytical approach based on the minimax method possesses greater possibilities for taking account of relaxation effects. A comparison is made between single-electron wave functions, the matrix elements, and the energies of dipole transitions for a number of excited states of the Ne atom, as calculated using both approaches. State Pedagogical University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–128, July, 1998.     

Correlated squeezed-state approach for the ground state of strongly correlated electrons coupled to local Holstein phonons

Summary In the present work we have applied the correlated squeezed-state approach to investigate the ground state of the extended Hubbard model which is coupled to local Holstein phonons. Our study begins with decoupling the electron and phonon subsystems approximately by introducing a variational correlated squeezed-state ansatz for the phonons. Then assuming the renormalized intersite electron correlation of the effective electronic Hamiltonian to be attractive and the renormalized on-site correlation repulsive, we have applied the generalized Hartree-Fock approximation to obtain the ground state of the system, which is a superconducting state with intersite pairing. With optimal values of the variational parameters the correlated squeezed-state approach will by construction yield a ground-state energy lower than those obtained in previous studies. This means that our variational ansatz is more stable as the ground state of the system. Furthermore, our variational study shows that in the correlated squeezed state the polaronic reduction effect of phonons is much more alleviated, and thus the mass enhancement inherent to the polaron effect is noticeably weakened. This weakening of the reduction effect should, in turn, significantly affect other physical properties of the system.     

Influence of quantum transitions in the continuum on ionization of atoms in strong fields

The tight-binding method is used to analyze the ionization of a hydrogenlike atom by an intense monochromatic laser field. The orthogonal and normalized basis in which the solution of the time-dependent Schrödinger equation is expanded contains unperturbed wave functions of the discrete spectrum and generalized Coulomb wave functions of the continuum. In the solution of the coupled equations we make use of the fact that the bound-free and free-free transitions are efficient in different regions of complex time. Simplified equations are constructed and investigated. Results of calculations for ionization of a hydrogen atom from its ground state and of the energy distribution of the electrons in strong and superstrong linearly polarized fields are presented. It is shown that in this case the ground state decays completely, and free-free transitions play a defining role in the dynamics of the process. It is established that the total probability of population of the upper Rydberg states abutting the continuum does not exceed 0.05. The range of applicability of the approach is discussed. A comparison with numerical results obtained by other authors is given.     

Ground state of a Uranium impurity in a metal: Inclusion of Self-Energy terms

A previous investigation of the system ground state for the mixed-valent Uranium ion fluctuating between two magnetic configurations in a free-electron sea is extended by including the Self-Energy terms (by Self-Energy terms we refer to additional terms in the wave function that contain electron-hole excitations of the Fermi sea) in the ground state variational functions. It is found that the presence of one electron-hole pair excitation in the calculation has risen the singlet to magnetic state energy separation as compared to the zero-order case.