Two-particle Green function

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

Quantum field theory provides an unambiguous way to find energy levels of any composite system. They are determined by the positions of the poles of the respective Green functions. This idea was first realized in the form of the Bethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)... 

The amputated two-particle Green function Gt satisfies the equation... 

A new kernel corresponding to the free two-particle propagator in (1.8) may be defined via this amputated two-particle Green function... 

The new bound state equation is constructed for the two-particle Green function defined by the relationship... 

The two-particle Green function G has the same poles as the initial Green function G and satisfies the BS-like equation... 

Figure 2 Exact single-particle spectrum (solid line) for four electrons on a four-site ring compared with the results of a GW approximation (dash-dotted line) and those obtained from an effective two-particle Green-function (dashed line) for U — 1.0.

The zeroth-order wavefunction of an excited state in an extended system cannot be described by a single determinant since the energies of the excited determinants form continuous bands. The states associated with the same band are degenerate since the energy difference between two adjacent states is an infinitesimal number. Thus, to calculate the energy spectrum of excited states, one needs to use degenerate MBPT, two-particle green function theory or a method like EOM-CC [5],... 

It should be mentioned that, in deriving equation (8.14), we neglected all terms in H that contain two particle or two hole operators (the wave function contains also only the operators of one electron-hole pair). If one wants to consider also double excitations (two electron-hole pairs) a two-particle Green function must be used and a diagrammatic technique. Further even at the one electron-hole case one has to introduce so-called vertex corrections. This has been done for the most important diagrams, but a description of this rather complicated formalism lies outside the framework of this book. 

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