The procedure to get ground-state energy and excitation spectrum by looking at the normal modes for the simple excitations of a many-body system is extended and applied to a system with singular interaction. It is shown that the singular two-body interaction can be consistently replaced by the so-called reaction matrix in the equations of motion and in the expressions for the energies of ground and excited states.
The procedure to get ground-state energy and excitation spectrum by looking at the normal modes for the simple excitations of a many-body system is extended and applied to a system with singular interaction. It is shown that the singular two-body interaction can be consistently replaced by the so-called reaction matrix in the equations of motion and in the expressions for the energies of ground and excited states.
A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
With the aid of the displacement method, the wave function of interacting bosons is represented as a product of the ground state function and the function describing the excitations in the system. The equations for these functions are solved in the collective variables representation by means of a special perturbation method free of divergences. The expressions obtained for the wave functions, the energy of the ground state, the energy spectrum and the damping of collective excitations are investigated. (auth)
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where n is the density and f0 is the scattering length for the assumed two-body interaction between the bosons. The coefficients a and b are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient b is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
The most of the experimental measurable quantities in many-body electron system, such as the optical absorption, luminescent spectra and exciton effect, are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory, i.e. the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However, the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years, nevertheless, several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation, the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron's self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory, in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package, the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper, the many-body theories for the excitation process, its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions, the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.
Calculational procedures are developed for the ground-state energy of a many-fermion system. The entire theory is developed from the many-body Schr\"odinger equation via perturbation theory. The linked cluster expansion is derived by elementary methods. The analytic singularities of the energy function are analyzed, and procedures designed to avoid or overcome them. Results for the calculation of the energy in the presence of a purely repulsive two-body force are reviewed, and the situation is found to be reasonably satisfactory, though good results are restricted to weak potentials when the system is not dilute. Results for an attractive potential surrounding a strongly repulsive core are reviewed, and procedures discribed for calculations in this case. A sample calculation with an assessment of the probable errors is reviewed.