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Canadian Journal of Physics

A continued fraction representation of the mass operator

Dipak K. Saraswati

Published Jan 1, 1976• 1 citation
Publisher

Abstract

We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.

DOI
10.1139/p76-003

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We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.
A recursive method is developed to treat electrons coupled to phonons. It is applied to small systems with $E\ensuremath{\bigotimes}\ensuremath{\beta}$ Jahn-Teller coupling. Two cases are considered, a model with one electron and two orbitals on a single site (related to the Rabi Hamiltonian) and a model with two electrons on two sites. The corresponding Green's functions are represented by rational functions. It is found that the spectra change substantially when one phonon couples to the electron but are relatively robust under an increasing number of phonons.
The connection between the projection operator approach and the continued fraction approach to perturbation theory is investigated. A concise solution to the linear operator equation A mod x>= mod b> is found in terms of the level shift operator using projection operator techniques. The analogous process to the use of projection operators in the continued fraction method of solving the same problem is identified, and a parallel development performed. The connection between the two approaches is thereby established, and continued fraction expressions for the level shift operator obtained. The abstract equation is then specialised to deal with (i) the eigenvalue problem and (ii) the calculation of transition probabilities for quantum mechanical systems described by a time-independent Hamiltonian. Particular attention is paid to the problem of degeneracy and it is shown that the most convenient expressions are found by a hybrid of the two approaches.
We present the continued fraction method (CFM) as a microscopic approximation to the spectral density of the Hubbard model in the correlated metal phase away from half filling. The quantity expanded as a continued fraction is the single-particle Green's function. Leading spectral moments are taken into account through a set of real expansion coefficients, as known from the projection technique. Further stages are added to the continued fraction, with complex coefficients, thus defining a terminator function. This enables us to treat the entire spectral range of the Green's function on equal footing and determine the energy scale of the Fermi liquid quasiparticles by minimizing the total energy. The solution is free of phenomenological parameters and remains well defined in the strong-coupling limit, near the doping-controlled metal-insulator transition. Our results for the density of states agree reasonably with those from several variants of the dynamical mean-field theory. The CFM requires minimal numerical effort and can be generalized in several ways that are interesting for applications to real materials.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.