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Progress of Theoretical Physics

Perturbation Theory for Degenerate Problems of Many-Fermion Systems

Tohru Morita

Published Mar 1, 1963• 106 citations
Publisher

Abstract

The linked cluster expansion of the eigenvalues and eigenfunctions of a many-fermion system is extended to the case where the eigenvalues of the unperturbed system are degenerate. It is shown that the eigenvalues and eigenfunctions are to be obtained by solving a secular equation in the space of degenerate unperturbed wave functions; the matrix elements between these unperturbed functions take the form of linked clusters with energy denominators which are the differences of unperturbed energies. Some discussions are given on the relation of this paper to the formalism of Bloch and Horowitz, in which the energy denominators are the differences of an unperturbed energy and the true energy which is to be obtained as the solution to the secular equation.

DOI
10.1143/ptp.29.351

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The linked cluster expansion of the eigenvalues and eigenfunctions of a many-fermion system is extended to the case where the eigenvalues of the unperturbed system are degenerate. It is shown that the eigenvalues and eigenfunctions are to be obtained by solving a secular equation in the space of degenerate unperturbed wave functions; the matrix elements between these unperturbed functions take the form of linked clusters with energy denominators which are the differences of unperturbed energies. Some discussions are given on the relation of this paper to the formalism of Bloch and Horowitz, in which the energy denominators are the differences of an unperturbed energy and the true energy which is to be obtained as the solution to the secular equation.
This paper is a study in a possible steady-state perturbation theory for arbitrary excited multiply degenerate states of a normal Fermi system in the statistical limit. An operator technique has been developed to transform the relevant Hamiltonian into an operator that is a function of occupation number operators only. The perturbation theory equations have been proved to be solvable in the space of quasinormal form operators and their formal solutions have been derived. It is shown that the operator series of perturbation theory can be transformed to a linked cluster expansion with the nonphysical powers of volume eliminated. Elimination of nonphysical terms is effected without use of the diagrams technique.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin \textonehalf{}, e.g., 3He^{3}\mathrm{He}3He, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins \textonehalf{} localized at lattice sites.
The solution of the quantum-mechanical eigenvalue problem is discussed for cases when a series of approximate eigenfunctions is known. If these ``unperturbed'' states are divided into two classes, a perturbation formula is derived giving the influence of one class of states on the other in the final solution. The formula contains as special cases: (i) the Schrödinger-Brillouin formula for the eigenvalue of a nondegenerate state, (ii) a new simple formula for treating a class of degenerate states, and (iii) the splitting of the secular equation in cases where the system naturally consists of two independent parts in mutual interaction.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms $\ensuremath{\Delta}E$ are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.