Renormalization group approach to many fermions systems.

Vincent Brindejonc, Bengt Friman


Since the foundation of Fermi Liquid Theory (FLT) by Landau in 1957 [<#LAN57>1], great efforts have been devoted to the derivation of the Landau quasiparticle interaction (Landau parameters) from a microscopic theory in the frame of many body theory (see [<#ABR63>2] for a review). More recently a renormalization group (RG) method has been proposed for to attack this problem [<#SHA94>3,<#POL92>4].

This RG treatment of a many fermion system can be separated in two families, namely the Kadanoff-Wilson (KW) approach and the field theoretical (FT) one. The KW method has been extensively discussed [<#SHA94>3,<#POL92>4,<#DUP96>5]. Its great contribution has been to show that FLT is an infra-red fixed point of the renormalization group. However, it seems that in this case the restriction to four particle interactions in the effective action is too strict and one generates in fact, 6 points and more effective interactions which can be marginal at tree level [<#DUP96>5]. These higher order effective interaction influence the running of the four point effective interaction.

In our work we take into account the effect of the 6 point vertex function on the running of the four point vertex function. This allows us to treat the two particle-hole channels consistently and to find an RG equation for the four point vertex which at this point do not include the particle-particle (or BCS) diagram. This equation gives in turn a set of equations for both the scattering amplitude of low energy quasi-particles and for the Landau quasiparticle interaction .

This set of equations can be solved numerically and gives for example in d=2 and for an initial Landau function , the result shown in the figure.

Figure: Running of the Landau parameter as functions of , where is the cut-off.

The method can be used from condensed matter problems like high-Tc superconductivity to high density nuclear matter problems as in neutron star. Its application to systems with long range interaction need an adaptation.


1. L.D. Landau, JETP 3 (1957),920; 5 (1957), 101

2. A.A Abrikosov, L.P. Gor'kov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Stastical physics - Dover New-York (1963)

3. R. Shankar, Rev.Mod.Phys. 66 (1994), 129

4. Joseph Polchinski,Lectures presented at TASI 1992.

5. N. Dupuis, cond-mat/9604189