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Physical Review A

Two-body and three-body parquet theory

Alexander Lande, R. A. Smith

Published Jan 1, 1992• 26 citations
Publisher

Abstract

One of the fundamental approaches to microscopic many-body theory is through the use of perturbation theory. This paper presents a clear derivation of the equations that sum the two-body and three-body reducible diagrams that are generated from some input set of irreducible diagrams (typically the bare interaction) in a crossing-symmetric form. The diagrammatic structure is useful for both bosons and fermions at zero and finite temperature. The essential features of the procedure for the two-body case are the following. (i) All constructions involve only direct diagrams. (ii) Diagrams can be joined together by five different ladder, chain, and vertex correction operations. (iii) The sequences in which these operations are applied satisfy certain associativity relations. (iv) These associativity relations are used to insure that each permitted diagram is generated exactly once. (v) The ladder-and-chain operations are used to construct reducible vertex sums ${\mathrm{\ensuremath{\Gamma}}}_{0}$ which have no external vertex corrections. (vi) The full direct vertex \ensuremath{\Gamma} consists of the vertex ${\mathrm{\ensuremath{\Gamma}}}_{0}$ plus ${\mathrm{\ensuremath{\Gamma}}}_{0}$ dressed on either or both sides by the full direct vertex \ensuremath{\Gamma}. (vii) Finally, the full vertex is obtained by adding the exchange of the direct vertex. The extension to the three-body formalism proceeds in much the same way.

DOI
10.1103/physreva.45.913

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One of the fundamental approaches to microscopic many-body theory is through the use of perturbation theory. This paper presents a clear derivation of the equations that sum the two-body and three-body reducible diagrams that are generated from some input set of irreducible diagrams (typically the bare interaction) in a crossing-symmetric form. The diagrammatic structure is useful for both bosons and fermions at zero and finite temperature. The essential features of the procedure for the two-body case are the following. (i) All constructions involve only direct diagrams. (ii) Diagrams can be joined together by five different ladder, chain, and vertex correction operations. (iii) The sequences in which these operations are applied satisfy certain associativity relations. (iv) These associativity relations are used to insure that each permitted diagram is generated exactly once. (v) The ladder-and-chain operations are used to construct reducible vertex sums \ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​ which have no external vertex corrections. (vi) The full direct vertex \ensuremath{\Gamma} consists of the vertex \ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​ plus \ensuremathΓ0{\mathrm{\ensuremath{\Gamma}}}_{0}\ensuremathΓ0​ dressed on either or both sides by the full direct vertex \ensuremath{\Gamma}. (vii) Finally, the full vertex is obtained by adding the exchange of the direct vertex. The extension to the three-body formalism proceeds in much the same way.
A new approach is used in the description of fermion and boson systems at zero or finite temperature. We generalize the familiar ladder and chaining operations to construct a crossing-symmetric approximation to the two-particle vertex from the bare interaction. The explicit rules for this construction are given in terms of Feynman propagators. The high level of symmetry of the vertex simplifies the theory. In one form, the structure appears as a generalization of parquet diagrams. The energy, self-energy, and vertex satisfy a number of consistency relations.
The Baym-Kadanoff theory of the conserving vertex is discussed within the context of the parquet summation of Feynman diagrams in nonrelativistic many-body theory. Since the parquet language provides a convenient way of characterizing all two-body vertices, the Baym-Kadanoff algorithm for generating a conserving two-body vertex takes a very straightforward form. We present the general theory and then illustrate the results using many of the simple conserving vertices. We then analyze the parquet two-body vertex within the theory, concluding that parquet is not a conserving theory. We show that any conserving theory that generates the direct parquet diagrams must also generate an infinite number of irreducible diagrams and derive integral equations for the minimal set of irreducible diagrams that would be needed in a conserving theory that produced the direct parquet diagrams. We present an algorithm that enables one to determine, for any direct or exchange parquet diagram, the parquet diagram whose contribution to the self-energy, when functionally differentiated, will generate the original parquet diagram. We then show that any set of diagrams for the two-body vertex that includes the bare interaction and satisfies the important properties of being conserving and antisymmetric under exchange of the outgoing particles must necessarily include all diagrams.
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.
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called v8v_8v8​ v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the 3P2^3\textrm{P}_23P2​ 3 P 2 and 3P2^3\textrm{P}_23P2​ 3 P 2 - 3F2^3\textrm{F}_23F2​ 3 F 2 gaps. We also find pairing in 3P0^3\textrm{P}_03P0​ 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
Abstract The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure. Graphic abstract
In many-body theory, the single-particle self-energy can be constructed from the two-body vertex, the single-particle Green's function, and the interaction in a well-known way. Functional differentiation of the self-energy leads back to the two-body vertex. Given a set of two-body vertex diagrams, the process of constructing self-energy diagrams and functionally differentiating them generates a second set of two-body vertex diagrams. If the initial set of diagrams includes the noninteracting two-body Green's function, the two sets will be equal if and only if the initial set includes all diagrams. We discuss this result from the perspective of the parquet theory.