By Ramponi A.J., Caird J.

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**Example text**

2 the phase diagram of a three dimensional fermion system with attractive interaction between the particles from weak to strong coupling is shown. For weak coupling the system is a Fermi liquid with single fermions as the particles, while - in I the on crossover from BCS - _ . - - = for strong coupling it is a Bose liquid of bound fermion pairs. These two are separated by a region of dissociation shown as the shaded area Fig. 2, where single fermions and bound pairs coexist. By thermody- regimes in 3.

In short-hand notation the Bethe-Salpeter equation reads F = F, - FIX-V (3-13) or 1-1 = _V -l + X (3-14) where X is the pair propagator which represents the two thick propagator lines in Fig. 4c which connect the grey and the black circle. 14). g. Itzykson and Zuber 1980). 9) have been performed. 11), the equation for the self energy Z in Fig. 13) shown in Fig. 4c to determine the vertex function -P. Eventually, the irreducible vertex function F, must be determined by the perturbation series, because T, is needed as input for the Bethe-Salpeter equation.

4a which involve the exact matrix Green's function (thick propagator line) and the exact vertex function which is represented by the black filled circle. This equation together with the Dyson equation (3. 11) is obtained from the equation of motion for the fermion Green's function: because of the interaction term in (3. 36), so that an integral equation involving G 2 1 Q (a) I (b) )1( )C( + (C) X 0 + Fig. 4. (a) = + + )CCD2CD: + + XDK Self energy Z in terms of the exact Green's function (thick propagator (black filled circle), which represents together and the exact vertex function Dyson equation the equation of motion of the fermion Green's function, perturbation series for the exact vertex function, and (c) the (b) Bethe-Salpeter equation for the exact vertex function.