The renormalization-group (RG) method is applied to study interacting fermions at finite temperature. A model based on the [psi][superscript]4-Grassmann effective action with SU(N )-invariant short-range interaction and a rotationally invariant Fermi surface in spatial dimensions d = 2, 3 is studied. We show how the key results of the Landau Fermi liquid theory can be recovered by this finite-temperature RG technique. Applying the RG to response functions, we find the compressibility and the spin susceptibility as solutions of the RG flow equations. We discuss subtleties associated with the symmetry properties of the four-point vertex (the implications of the Pauli principle). We point out distinctions between three quantities: the bare interaction of the low-energy effective action, the Landau function and the forward scattering vertex.The bare interaction of the effective action is not a RG fixed point, but a common starting point of the flow trajectories of two limiting forms of the four-point vertex. We have derived RG equations for the Landau channel that take into account both contributions of the direct (ZS) and the exchange (ZS' ) particle-hole graphs at one-loop level.The basic quantities of Fermi Liquid theory, the Landau interaction function and the forward scattering vertex, are calculated as fixed points of these flows in terms of the effective action's interaction function. The classic derivations of Fermi Liquid theory applying the Bethe-Salpeter equation and other analogous approaches, tantamount to some sort of RPA-type (decoupled) approximation, neglect the zero-angle singularity in the ZS' graph. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed in the final result, and the RPA sum rule must be imposed by hand on the components of the Landau function to satisfy the Pauli principle. This sum rule, not indispensable in the original phenomenological formulation of the Landau FLT, is equivalent, from the RG point of view, to a fine tuning of the effective interaction. Our results show that the strong interference of the direct and exchange processes of the particle-hole scattering near zero angle invalidates the RPA (decoupled) approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex, revealed by the RG analysis. In the present RG approach the Pauli principle is automatically satisfied. As follows from the RG solution, the amplitude sum rule, being an artefact of the RPA approximation, is not needed to respect statistics and, moreover, is not valid.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.
We show that the renormalization-group (RG) approach to interacting fermions at one-loop order recovers Fermi-liquid-theory results when the forward scattering zero sound and exchange channels are both taken into account. The Landau parameters are related to the fixed point value of the \ensuremath{\Omega} limit of the forward scattering vertex. We specify the conditions under which the results obtained at one-loop order hold at all orders in a loop expansion. We also emphasize the similarities between our RG approach and the diagrammatic derivation of Fermi-liquid theory. \textcopyright{} 1996 The American Physical Society.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A
d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function
F, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.
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.
We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.