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Journal of Mathematical Physics

Some Analytic Properties of Green Functions and Self-Energy Parts for a System of Interacting Bosons

W. E. Parry, Roy E. Turner

Published Apr 1, 1963• 0 citations
Publisher

Abstract

Some analytic properties of the single-particle Green functions and proper self-energy parts for a system of interacting bosons at nonzero temperatures are derived. In particular, it is shown how to obtain the analytical continuations of simple functions of the proper self-energy parts, which can be obtained from perturbation theory at an infinite set of points along the imaginary energy axis, to the whole of the complex energy plane. This is useful in determining the poles of the Green functions.

DOI
10.1063/1.1703987

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Some analytic properties of the single-particle Green functions and proper self-energy parts for a system of interacting bosons at nonzero temperatures are derived. In particular, it is shown how to obtain the analytical continuations of simple functions of the proper self-energy parts, which can be obtained from perturbation theory at an infinite set of points along the imaginary energy axis, to the whole of the complex energy plane. This is useful in determining the poles of the Green functions.
A simple model of noninteracting electrons with a separable one-body potential is used to discuss the possible pole structure of single-particle Green's functions for fermions on unphysical sheets in the complex frequency plane as a function of the system parameters. The poles in the exact Green's function can cross the imaginary axis, in contrast to recent claims that such a behavior is unphysical. As the Green's function of the model has the same functional form as an approximate Green's function for coupled Luttinger liquids, no definite conclusions concerning the concept of ``confined coherence'' can be drawn from the locations of the poles of this Green's function.
Certain general properties of single-particle propagators for a system of interacting fermions are derived. In addition, the properties of the proper self-energy part ${G}_{k}(\ensuremath{\zeta})$ which were used in previous work on the ground-state energy and on the Fermi surface are established. In particular, the fact that to all orders of perturbation theory in the interaction, $\mathrm{Im} {G}_{k}(x\ensuremath{-}i{0}^{+})$ behaves like ${C}_{k}{(x\ensuremath{-}\ensuremath{\mu})}^{2}$ (Ck>0{C}_{k}>0Ck​>0) for xxx very near $\ensuremath{\mu}$, is proved.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of NNN particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1N+1N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of NNN particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
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.