Planar Theory Made Plainer

Abstract

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.