A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.
Using a general-order ab initio many-body Green's function method, we numerically illustrate several pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron Feynman propagators. (i) The perturbation expansion of the frequency-dependent self-energy is not convergent at the exact self-energy in many frequency domains. (ii) An odd-perturbation-order self-energy has a qualitatively wrong shape and, as a result, many roots of the corresponding Dyson equation are nonphysical in that the poles may be complex or residues can exceed unity or be negative. (iii) A higher even-order self-energy consists of vertical lines at many frequencies, predicting numerous phantom poles with zero residues. (iv) Infinite partial resummations of diagrams by vertex or edge renormalization tend to exacerbate these pathologies. (v) The nonconvergence is caused by the nonanalyticity of the rational-function form of the exact Green's function at many frequencies, where the radius of convergence of its Taylor expansion is zero. This is consistent with the fact that (vi) Pad\'e approximants (power-series expansions of a rational function) can largely restore the correct shape and poles of the Green's function. Nevertheless, not only does the nonconvergence render higher-order Feynman-Dyson diagrammatic perturbation theory useless for many lower-lying ionization or higher-lying electron-attachment states but it also calls into question the validity of its combined use with the Ans\"atze requiring the knowledge of all poles and residues. Such Ans\"atze include the Galitskii-Migdal identity, the self-consistent Green's function methods, and some models of the algebraic diagrammatic construction.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.
Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.
On the basis of an exact perturbational expression for the interacting one-particle Green function
G corresponding to bosons / fermions in terms of the bare interaction potential
v and permanents / determinants of the non-interacting one-particle Green function
G0, we deduce four recursive perturbation expansions for the self-energy
Σ. With
W denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of
Σ in terms of (i) self-energy diagrams and
(v,G0), (ii)
G-skeleton self-energy diagrams and
(v,G), (iii)
W-skeleton self-energy diagrams and
(W,G0), and (iv)
G- and
W-skeleton self-energy diagrams and
(W,G). For the calculation of
W, we rely on a similar exact perturbational expression for the interacting two-particle Green function
G2 as for
G. From this expression, we deduce four recursive perturbation expansions for the polarization function
P, necessary for the calculation of
W, that are similar to those for
Σ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-
21 fermions on a lattice in arbitrary
d spatial dimensions. In two further appendices, we present methods and short programs for determining the
νth-order diagrams corresponding to the perturbation expansions of
G in terms of
(v,G0), and
Σ in terms of
(v,G0) and
(v,G) on the basis of the cycle decompositions of the elements of the symmetric group
S2ν.
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.
We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O( n 4 ) and O( n 5 ) of molecular size ( n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O( n 5 ) and O( n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.