We used rearranged Feenberg perturbation formula which was adopted to describe the states with two quasiparticles of the momenta ħk1→ and ħk2 →. It was noticed that two “diagrams of states” of order N° (N is the number of the particles in the system) in the fourth order perturbation term contained the interaction between particles out of the condensate. The interaction energy described by these diagrams was calculated for the potential V q = V ( -q2 + g2), ), which had the main features of the real potential between two helium atoms, and for antiparallel momenta ħk1→ = ħk2 →. It was shown that the spectrum depended on the density strongly and in particular that the interaction changed its sign and became repulsive with the increase of density. Owing to the form of the interaction diagrams, Bogoliubov summing was possible. After numerical integration we found that this approximation had a significant influence on the spectrum, although qualitative picture of the interaction was unchanged.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in
rs43. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
This thesis is devoted to a rigorous study of interacting quantum gases. The main objects of interest are the closely related concepts of excitation spectrum and quasiparticles. The immediate motivation of this work is to propose a spectral point of view concerning these two concepts. In the first part of this thesis we discuss the concepts of excitation spectrum and quasiparticles. We provide an overview of physical motivations and based on that we propose a spectral and Hamiltonian-based approach towards these terms. Based on that, we formulate definitions and propositions related to these concepts. In the second part we recall the Bogoliubov and Hartree-Fock-Bogoliubov approximations, which in the physics literature are used to obtain the quasiparticle picture. We show how these two approaches fit into a universal scheme which allows us to arrive at a quasiparticle picture in a more general setup. This scheme is based on the minimization of Hamiltonians over the so-called Gaussian states. Its abstract formulation is the content of Beliaev's Theorem. In the last part we present a rigorous result concerning the justification of the Bogoliubov approximation. This justification employs the concept of the mean-field and infinite-volume limit. We show that for a large number of particles, a large volume and a sufficiently high density, the low-lying energy-momentum spectrum of the homogeneous Bose gas is well described by the Bogoliubov approximation. This result, which is formulated in the form of a theorem, can be seen as the main result of this thesis.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the average energy and the momentum distribution through second order in the two-body interaction potential. A phenomenological theory of helium II is proposed. The chief underlying feature of this phenomenological theory, which is shown to have a firm basis in the microscopic theory, is the presence of a double quasiparticle spectrum. Phenomenological expressions for the average energy and other quantities of interest are also proposed. Simple prescriptions are obtained for calculating the quasiparticle energies and the chemical potential from the average energy. Via these prescriptions, the quasiparticle energies and the chemical potential are calculated through second order in the interaction potential at T ≠ 0 °K, and the results are compared with earlier work in the literature.
A recently developed formalism for Helium II is generalized by introducing a 2-body interaction of spheres with diameter depending on the momentum exchanged between two atoms in an interaction process. A larger class of atomic collisions is also admitted. These modifications allow to account for some details of the interatomic potential V He (r) between two 4 He atoms, which were previously disregarded, and to improve the theoretical graphs of Helium II momentum distribution and normal fluid fraction.
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.
We investigate the normal selfenergy and the mass enhancement factor in the Hubbard model on the two-dimensional square lattice. Our purpose in this paper is to evaluate the mass enhancement factor more quantitatively than the conventional third order perturbation theory. We calculate it by expanding perturbatively up to the fourth order with respect to the on-site repulsion
U. We consider the cases that the system is near the half-filling, which are similar situations to high-
Tc cuprates. As results of the calculations, we obtain the large mass enhancement on the Fermi surface by introducing the fourth order terms. This is mainly originated from the fourth order particle-hole and particle-particle diagrams. Although the other fourth order terms have effect of reducing the effective mass, this effect does not cancel out the former mass enhancement completely and there remains still a large mass enhancement effect. In addition, we find that the mass enhancement factor becomes large with increasing the on-site repulsion
U and the density of state (DOS) at the Fermi energy
ρ(0). According to many current reseaches, such large
U and
ρ(0) enhance the effective interaction between quasiparticles, therefore the superconducting transition temperature
Tc increases. On the other hand, the large mass enhancement leads the reduction of the energy scale of quasiparticles, as a result,
Tc is reduced. When we discuss
Tc, we have to estimate these two competitive effects.