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Journal of Mathematical Physics

Vertex function and crossing symmetry

W. D. Heiss

Published Apr 1, 1980• 4 citations
Publisher

Abstract

An integral equation is proposed for the vertex function occurring in the quantum-mechanical many-body problem, such that the s channel and the t channel are treated simultaneously on the same level. The equation thus provides an intricate blending of the two channels. Its solution is given in closed form, and although this is complicated, it allows the physically interesting features to be investigated numerically.

DOI
10.1063/1.524465

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An integral equation is proposed for the vertex function occurring in the quantum-mechanical many-body problem, such that the s channel and the t channel are treated simultaneously on the same level. The equation thus provides an intricate blending of the two channels. Its solution is given in closed form, and although this is complicated, it allows the physically interesting features to be investigated numerically.
We consider the four-boson and 3 + 1 fermionic problems with a model Hamiltonian which encapsulates the mechanism of the Feshbach resonance involving the coherent coupling of two atoms in the open channel and a molecule in the closed channel. The model includes also the pair-wise direct interaction between atoms in the open channel and in the bosonic case, the direct molecule–molecule interaction in the closed channel. Interactions are modeled by separable potentials which makes it possible to reduce the four-body problem to the study of a single integral equation. We take advantage of the rotational symmetry and parity invariance of the Hamiltonian to reduce the general eigenvalue equation in each angular momentum sector to an integral equation for functions of three real variables only. A first application of this formalism in the zero-range limit is given elsewhere [Y. Castin, C. Mora, L. Pricoupenko, Phys. Rev. Lett. 105 (2010) 223201].
We address the problem of finding self-consistent parquet equations for two-body correlated Greens functions that arise out of a cumulant expansion. The general theory as developed for non-relativistic electron systems reproduces the conventional two-body parquet diagrammatic technique for the vertex functions. As an application, the crossing-invariant linearized basic parquet equations are treated for the correlated two-particle Greens function. It is pointed out that besides the Cooperon there are two additional prominent crossing-symmetric two-particle channels in many-particle systems with pair interaction.
We address the problem of finding self-consistent parquet equations for two-body correlated Greens functions that arise out of a cumulant expansion. The general theory as developed for non-relativistic electron systems reproduces the conventional two-body parquet diagrammatic technique for the vertex functions. As an application, the crossing-invariant linearized basic parquet equations are treated for the correlated two-particle Greens function. It is pointed out that besides the Cooperon there are two additional prominent crossing-symmetric two-particle channels in many-particle systems with pair interaction.