Perturbation theory for the Fermi liquids in d>1

G. Benfatto, Giovanni Gallavotti

Alex C. Hewson

We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.

Behnam Farid

For uniform systems of spin-less fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential v(r-r'), a celebrated theorem due to Luttinger (1961) states that under the_assumption_ of validity of the many-body perturbation theory the self-energy Sigma(k;epsilon), with 0 ,< epsilon_F, with alpha_k >,= 0. As this is, by definition, specific to self-energies of Landau Fermi-liquid systems, treatment of non-Fermi-liquid systems are therefore thought to lie outside the domain of applicability of the many-body perturbation theory; that, for these systems, the many-body perturbation theory should_necessarily_ break down. We demonstrate that Im[Sigma(k;epsilon)] ~ -,+ alpha_k (epsilon-epsilon_F)^2, epsilon >,< epsilon_F, is_implicit_ in Luttinger's proof and that, for d > 1, in principle nothing prohibits a non-Fermi-liquid-type (and, in particular Luttinger-liquid-type) Sigma(k;epsilon) from being obtained within the framework of the many-body perturbation theory. We in addition indicate how seemingly innocuous Taylor expansions of the self-energy with respect to k, epsilon or both amount to tacitly assuming that the metallic system under consideration is a Fermi liquid, whether the self-energy is calculated perturbatively or otherwise. Proofs that a certain metallic system is in a Fermi-liquid state, based on such expansions, are therefore tautologies.

Andrey V. Chubukov, Dmitrii L. Maslov

We discuss an interplay between the Fermi-liquid (FL) theory and diagrammatic perturbative approach to interacting Fermi systems. In the FL theory for Galilean-invariant systems, mass renormalization ${m}^{\ensuremath{\ast}}/m$ comes exclusively from fermions at the Fermi surface. We show that in a diagrammatic perturbation theory the same result for ${m}^{\ensuremath{\ast}}/m$ comes from fermions both at and away from the Fermi surface. The equivalence of the FL and perturbative approaches is based on a particular relation between self-energy contributions from high- and low-energy fermions. We argue that care has to be exercised in the renormalization-group approach to a FL in order not to miss the high-energy contribution to ${m}^{\ensuremath{\ast}}/m$. As particular examples, we discuss ${m}^{\ensuremath{\ast}}/m$ and the quasiparticle residue $Z$ for two-dimensional and three-dimensional systems with both SU(2) and $\text{SU}(N)$ symmetries, and with a short-range interaction. We derive an expression for the anisotropic part of the Fermi-liquid vertex in the large-$N$ limit of the $\text{SU}(N)$ case.