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arXiv:cond-mat/0210679 · Journal of Physics Condensed Matter

Two-particle renormalizations in many-fermion perturbation theory: the importance of the Ward identity

V. Janiš

Published May 16, 2003• 5 citations
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Abstract

We analyse two-particle renormalizations within the many-fermion perturbation expansion. We show that present diagrammatic theories suffer from a lack of direct diagrammatic control over the physical two-particle functions. To rectify this, we introduce and prove a Ward identity enabling an explicit construction of the self-energy from a given two-particle irreducible vertex. Approximations constructed in this way are causal, obey conservation laws, and offer explicit diagrammatic control of singularities in dynamical two-particle functions.

DOI
10.1088/0953-8984/15/21/102

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We analyse two-particle renormalizations within the many-fermion perturbation expansion. We show that present diagrammatic theories suffer from a lack of direct diagrammatic control over the physical two-particle functions. To rectify this, we introduce and prove a Ward identity enabling an explicit construction of the self-energy from a given two-particle irreducible vertex. Approximations constructed in this way are causal, obey conservation laws, and offer explicit diagrammatic control of singularities in dynamical two-particle functions.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
We consider the fulfillment of conservation laws and Ward identities in the one- and two-loop functional renormalization group approach. It is shown that in a one-particle irreducible scheme of this approach Ward identities are fulfilled only with the accuracy of the neglected two-loop terms $O({V}_{\ensuremath{\Lambda}}^{3})$ at one-loop order, and with the accuracy $O({V}_{\ensuremath{\Lambda}}^{4})$ at two-loop order (${V}_{\ensuremath{\Lambda}}$ is the effective interaction vertex at scale $\ensuremath{\Lambda}$). The one-particle self-consistent version of the two-loop renormalization group equations which leads to smaller errors in Ward identities due to the absence of the terms with nonoverlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.
We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $\mathrm{\ensuremath{\Phi}}$ derivability for the self-energy $\mathrm{\ensuremath{\Sigma}}$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function GGG. We call the corresponding approximations for $\mathrm{\ensuremath{\Sigma}}$ partially $\mathrm{\ensuremath{\Phi}}$ derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of $\mathrm{\ensuremath{\Phi}}$ consistently with GGG. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\mathrm{\ensuremath{\Phi}}$ derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive-semidefinite. The latter property can be ensured for a subset of partially $\mathrm{\ensuremath{\Phi}}$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.
Parquet diagrams sum self-consistently Feynman graphs for the vertex function with all two-particle multiple scatterings. We show how the complete parquet equations for the Hubbard-like models can be integrated to a generating functional from which all thermodynamic quantities are derived via (functional) derivatives. An explicit Luttinger-Ward functional Φ[G;Λ,K;U]Φ[G;Λ,{\cal K};U]Φ[G;Λ,K;U] is constructed containing only the renormalized one-particle, GGG, irreducible, ΛΛΛ, and reducible, K{\cal K}K, two-particle propagators as independent variational functions. The parquet approximation is proven to be a thermodynamically consistent, ΦΦΦ-derivable theory obeying the necessary conservation laws.
Using functional methods and the exact renormalization group we derive Ward identities for the Anderson impurity model. In particular, we present a non-perturbative proof of the Yamada–Yosida identities relating certain coefficients in the low-energy expansion of the self-energy to the thermodynamic particle number and spin susceptibilities of the impurity. Our proof underlines the relation of the Yamada–Yosida identities to the U (1) × U (1) symmetry associated with the particle number and spin conservation in a magnetic field.
Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In Part I of this two-part series, we illustrate how the combinatorial properties of Feynman diagrams in MBPT can be studied in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature) and provide a self-contained explanation of Feynman diagrams in this setting. We prove the combinatorial validity of the bold diagrammatic expansion, with methods generalizable to several types of field theories and interactions. Our treatment simplifies the presentation and numerical study of known techniques in MBPT such as the self-consistent Hartree-Fock approximation (HF), the second-order Green's function approximation (GF2), and the GW approximation. The bold diagrams are closely related to the Luttinger-Ward (LW) formalism, which was proposed in 1960 but whose analytic properties have not been rigorously established. The analytical study of the LW formalism in the setting of Gibbs measures will be the topic of Part II.