Fermi liquid theory: a renormalization group point of view

Abstract

We show how Fermi liquid theory results can be systematically recovered using a renormalization group (RG) approach. Considering a two-dimensional system with a circular Fermi surface, we derive RG equations at one-loop order for the two-particle vertex function $\Gamma$ in the limit of small momentum ($\mathbf{Q}$) and energy ($\Omega$) transfer and obtain the equation which determines the collective modes of a Fermi liquid. The density-density response function is also calculated. The Landau function (or, equivalently, the Landau parameters $F_l^s$ and $F_l^a$) is determined by the fixed point value of the $\Omega$-limit of the two-particle vertex function (${{\Gamma }^{\Omega }}^{\ast }$). We show how the results obtained at one-loop order can be extended to all orders in a loop expansion. Calculating the quasi-particle life-time and renormalization factor at two-loop order, we reproduce the results obtained from two-dimensional bosonization or Ward Identities. We discuss the zero-temperature limit of the RG equations and the difference between the Field Theory and the Kadanoff-Wilson formulations of the RG. We point out the importance of $n$-body ($n\ge 3$) interactions in the latter.

33 pages (RevTex), 5 poscript figures

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