In conserving approximations, the self-energy is obtained from a functional derivative E [G] = 10 [G] /1G of 0 the Luttinger-Ward functional, which is itself computed from a set of diagrams. To liberate ourselves from diagrams, we start

11To remind ourselves of this, we may also adopt an additional vertical matrix notation convention

instead from the exact expression for the self-energy, (13.33) and notice that when label 2 equals 1+, the right-hand side of this equation is equal to double-occupancy («"n#). Factoring as in Hartree-Fock amounts to assuming no correlations. Instead, we should insist that («"«#} be obtained self-consistently. After all, in the Hubbard model, there are only two local four-point functions: («"«#} and («"«"} = («#«#}. The latter is given exactly, through the Pauli principle, by («"«"} = («#«#} = (ft") = (n#) = n/2, when the filling n is known. In a way, («"«#) in the self-energy equation (13.33) can be considered as an initial condition for the four-point function when one of the points, 2, separates from all the others which are at 1. When that label 2 does not coincide with 1, it becomes more reasonable to factor a la Hartree-Fock. These physical ideas are implemented by postulating

E® (1,1; {0}) G® (1, 2; {0}) = AWG® (1,1 + ; {0}) G® (1,2; {0}), (13.39) where A{0} depends on the external field and is chosen such that the exact result12

Eff (1,1;{0}) Gff (1,1 + ; {0}) = U (nt (1)n# (l)^ is satisfied. It is easy to see that the solution is

Substituting A{0} back into our ansatz (13.13) we obtain our first approximation for the self-energy by right-multiplying by (G®) :

E(y) (1,2; {0}) = A{0}Gi1() (1,1 + ; {0}) 1(1 - 2).

We are now ready to obtain irreducible vertices using the prescription of the previous section, (13.38), namely, through functional derivatives of E with respect to G. In the calculation of Usp, the functional derivative of (n"n#) / ((n"| (n#)) drops out, so we are left with13

1G11'

1EtY)

12See footnote (14) of [20] for a discussion of the choice of limit 1C versus 1 .

13For n > 1, all particle occupation numbers must be replaced by hole occupation numbers.

The renormalization of this irreducible vertex may be physically understood as coming from Kanamori-Brueckner screening [7]. This completes the derivation of the ansatz that was missing in our first derivation in Sect. 13.2.1.

The functional-derivative procedure generates an expression for the charge vertex Uch which involves the functional derivative of (n^n^} / ((n"| (n^}) which contains six-point functions that one does not really know how to evaluate. But, if we again assume that the vertex Uch is a constant, it is simply determined by the requirement that charge fluctuations also satisfy the fluctuation-dissipation theorem and the Pauli principle, as in (13.17).

Note that, in principle, X(1) also depends on double-occupancy, but since X(1) is a constant, it is absorbed in the definition of the chemical potential and we do not need to worry about it in this case. That is why the noninteracting irreducible susceptibility x(1)(q) = Xo(q) appears in the expressions for the susceptibility, even though it should be evaluated with G(1) that contains X(1). A rough estimate of the renormalized chemical potential (or equivalently of X(1)) is given in the Appendix of [20]. One can check that spin and charge conservations are satisfied by our susceptibilities.

13.2.5.4 TPSC Second Step: An Improved Self-Energy E <2/

Collective modes are emergent objects that are less influenced by details of the single-particle properties than the other way around. We thus wish now to obtain an improved approximation for the self-energy that takes advantage of the fact that we have found accurate approximations for the low-frequency spin and charge fluctuations. We begin from the general definition of the self-energy (13.33) obtained from Dyson's equation. The right-hand side of that equation can be obtained either from a functional derivative with respect to an external field that is diagonal in spin, as in our generating function (13.30), or by a functional derivative of w_a (1) (2)) with respect to a transverse external field .

Working first in the longitudinal channel, the right-hand side of the general definition of the self-energy (13.33) may be written as

The last term is the Hartree-Fock contribution. It gives the exact result for the self-energy in the limit! [7]. The term is thus a contribution to lower frequencies and it comes from the spin and charge fluctuations. Right-multiplying the last equation by G_1 and replacing the lower energy part by its general expression in terms of irreducible vertices (13.37), we find

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