Roger Alan Smith

CROSSING-SYMMETRIC RINGS, LADDERS, AND EXCHANGES

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.

Parquet theory of inhomogeneous boson systems

Abstract The parquet summation for the two-body vertex is used as the framework for a theory of interacting inhomogeneous boson systems. We present a precisely defined set of approximations for the single-particle self-energy and for the two-body vertex which lead to a set of equations which are equivalent to the inhomogeneous variational hypernetted-chain theory for inhomogeneous boson systems. The present calculation makes clear the distinction between the self-energy which is used in constructing the two-body vertex and the self-energy which one would construct with the two-body vertex.

Planar Theory Made Plainer

In recent years, A.D. Jackson, A. Lande, and I have been approaching the many-body problem from the point of view of summing an important set of Feynman diagrams, the planar or parquet diagrams. It had long been recognized that the hypernetted-chain (HNC) variational theory approach and the summation of rings and ladders in perturbation theory were both dealing with diagrams of very similar topological structure. Early work by Sim, Buchler, and Woo [1] showed that the optimized hypernetted-chain (HNC) variational theory for bosons summed all ring and ladder diagrams exactly and in addition generated terms whose structure was that of other legitimate diagrams. Neither the numerical factors nor the diagrams generated this way were identified. Ref. [2] showed that the diagrams generated by the optimized HNC theory were a subset of the parquet diagrams and in general were not generated with the right numerical factors. Roughly speaking, the boson parquet theory sums ring diagrams in which there can be ladders between bubbles and ladder diagrams in which the rungs of the ladders can be chains of bubbles. These two types of diagrams are embedded in each other in a self-consistent way. A first effort was made in this work to generate an approximate sum of parquet diagrams. For the energy of liquid 4He, the results were comparable to those obtained from optimized Jastrow theory; the corresponding liquid-structure function was reasonable except for the behavior at small k.

PARQUET THEORY: THE DIAGRAMS

The summation of parquet diagrams provides an interesting and powerful approach to many-body theory. In this paper, we present several results on the diagrammatic structure of parquet theory. In comparison with previous diagrammatic discussions at this series of workshop[l] or elsewhere [2,3], this paper will derive the final form of the parquet equations (a new result) using simpler methods than the earlier work. The presentation will also be self-contained. As an illustration of the power of the present approach, we also derive the equations which would form the starting point for three-body parquet. Ultimately, we feel that the three-body parquet will be useful in obtaining more accurate results in physical systems, as well as being intimately related to the generation of better vertex approximations following Baym and Kadanoff[4] and Baym[5]. While the discussion here is self-contained, it is presented solely in terms of the diagrammatic structure. There is certainly much more to doing parquet theory than knowing the diagrams, and these other considerations will be discussed in the final section.

Fermion Parquet Equations

The parquet approach to many-body theory focuses on the effective interaction and expresses it in terms of a sum of a large and physically interesting class of Feynman diagrams.

Fermion Parquet: The Approximations

The full parquet approach to many-body theory represents a self-consistent summation of Feynman diagrams for the two-body Green’s function, G 2, in terms of the bare interaction, V, and the one-body Green’s function, G. The parquet equations sum the reducible diagrams for G 2 which are generated from any initial set of irreducible diagrams. The full one-body G which is used in the construction process is obtained from the proper self-energy ∑ and the non-interacting Green’s function with the usual Dyson equation. One level of self-consistency is attained by constructing the ∑ from the G 2. The ∑ is obtained by closing off the G 2 with a V as shown in fig. 1.

Inhomogeneous Parquet Theory

The parquet summation of Feynman diagrams provides a powerful approach for calculating the properties of many-particle systems. For inhomogeneous boson systems, we have showed previously1 that the parquet theory, with suitable approximations, could lead to the inhomogeneous variational theory of Krotscheck, Qian and Kohn2 (KQK). In this paper, we focus more clearly on the nature of the approximations which were implicitly made in the previous work and discuss in more detail the nature of the single-particle self-energy which is a natural part of the parquet theory which is less clearly manifested in the corresponding variational theory. At the same time, we introduce a new notation which enables rather transparent connections between the diagrammatic quantities and equations of the perturbation theory and the corresponding entities of the variational theory.

Inhomogeneous Boson Systems Made Planar

The hypernetted-chain variational theory and the parquet summation of perturbation theory provide diverse insights into the structure of many-body systems. Both methods effectively deal with the short-range correlations induced by the twobody interaction and the long-range correlations characteristic of quantum many-body systems. In the language of perturbation theory, they both perform a reasonably selfconsistent summation of ring diagrams and ladder diagrams. In past work, we have examined very carefully the relationship between the optimized Jastrow hypernettedchain theory without elementary diagrams and local approximations made in the parquet theory. For both central potentials [1] and spin-dependent potentials[2], we have shown that the final expressions for evaluating the distribution functions and energies are similar to the point of being identical. Yet while the static features are described identically, the two approaches lead naturally to rather different ways of computing other properties of the system. In the perturbation theory, the energy dependence enters in a transparent way into the fundamental constructs, and there are natural suggestions for ways to take the results of a parquet calculation of the ground-state properties of the bulk liquid and use them to determine the single-particle self-energies. [3] The perturbation theory also leads naturally to a discussion[4–5] of the Baym-Kadanoff[6–7] conserving vertex theory.

Baym-Kadanoff Theory Made Even Planar

In the last workshop in this series, I presented some results obtained by expressing the Baym-Kadanoff algorithm[1–2] in the language of the parquet theory [3–14].