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arXiv:cond-mat/9705016 · Physical Review Letters

Backflow in a Fermi Liquid

W. Zwerger

Published Dec 29, 1997• 19 citations
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Abstract

We calculate the backflow current around a fixed impurity in a Fermi liquid. The leading contribution at long distances is radial and proportional to ${1/r}^{2}$. It is caused by the current induced density modulation first discussed by Landauer [IBM J. Res. Dev. 1, 223 (1957)]. The familiar ${1/r}^{3}$ dipolar backflow obtained in linear response is only the next-to-leading term, whose strength is calculated here to all orders in the scattering. In the charged case the condition of perfect screening gives rise to a novel sum rule for the phase shifts. Similar to the behavior in a classical viscous liquid, the friction force is due only to the leading contribution in the backflow while the dipolar term does not contribute.

DOI
10.1103/physrevlett.79.5270

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We calculate the backflow current around a fixed impurity in a Fermi liquid. The leading contribution at long distances is radial and proportional to 1/r2{1/r}^{2}1/r2. It is caused by the current induced density modulation first discussed by Landauer [IBM J. Res. Dev. 1, 223 (1957)]. The familiar 1/r3{1/r}^{3}1/r3 dipolar backflow obtained in linear response is only the next-to-leading term, whose strength is calculated here to all orders in the scattering. In the charged case the condition of perfect screening gives rise to a novel sum rule for the phase shifts. Similar to the behavior in a classical viscous liquid, the friction force is due only to the leading contribution in the backflow while the dipolar term does not contribute.
The particle current in a metastable Fermi liquid against a first-order phase transition is calculated at zero temperature. During fluctuations of a droplet of the stable phase, in accordance with the conservation law, not only does an unperturbed current arise from the continuity at the boundary, but a backflow is induced by the density response. Quasiparticles carrying these currents are scattered by the boundary, yielding a dissipative backflow around the droplet. An energy of the hydrodynamic mass flow of the liquid and a friction force exerted on the droplet by the quasiparticles have been obtained in terms of a potential of their interaction with the droplet.
The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral I L , which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of I L with I L = ± π /2, occur in impurity models classified as singular Fermi liquids. Here we show the same values, I L = ± π /2, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological.
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 apply the finite-temperature renormalization group (RG) to a model based on an effective action with a short-range repulsive interaction and a rotation-invariant Fermi surface. The basic quantities of Fermi-liquid theory, the Landau function, and the scattering vertex are calculated as fixed points of the RG flow in terms of the effective action's interaction function. The classic derivations of Fermi-liquid theory, which apply the Bethe-Salpeter equation and amount to summing direct particle-hole ladder diagrams, neglect the zero-angle singularity in the exchange particle-hole loop. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed and the amplitude sum rule must be imposed by hand on the components of the Landau function. We show that the strong interference of the direct and exchange processes of particle-hole scattering near zero angle invalidates the ladder approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex. In this RG approach the Pauli principle is automatically satisfied. The consequences of the RG corrections on Fermi-liquid theory are discussed. In particular, we show that the amplitude sum rule is not valid.