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.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.

arXiv:1003.4032Physical Review BJun 3, 201013 citations

Fermi liquid theory in two dimensions

Sudhakar Yarlagadda, Susumu Kurihara

Physica B Condensed MatterMay 1, 19930 citations

Interaction-induced Fermi surface deformations in quasi-one-dimensional electronic systems

Sébastien Dusuel, Benoît Douçot

We consider serious conceptual problems with the application of standard perturbation theory, in its zero-temperature version, to the computation of the dressed Fermi surface for an interacting electronic system. In order to overcome these difficulties, we set up a variational approach which is shown to be equivalent to the renormalized perturbation theory where the dressed Fermi surface is fixed by recursively computed counterterms. The physical picture that emerges is that couplings that are irrelevant tend to deform the Fermi surface in order to become more relevant (irrelevant couplings being those that do not exist at vanishing excitation energy because of kinematical constraints attached to the Fermi surface). These insights are incorporated in a renormalization group approach, which allows for a simple approximate computation of Fermi surface deformation in quasi-one-dimensional electronic conductors. We also analyze flow equations for the effective couplings and quasiparticle weights. For systems away from half-filling, the flows show three regimes corresponding to a Luttinger liquid at high energies, a Fermi liquid, and a low-energy incommensurate spin-density wave. At half-filling Umklapp processes allow for a Mott insulator regime where the dressed Fermi surface is flat, implying a confined phase with vanishing effective transverse single-particle coherence. The boundary between the confined and Fermi liquid phases is found to occur for a bare transverse hopping amplitude of the order of the Mott charge gap of a single chain.We consider serious conceptual problems with the application of standard perturbation theory, in its zero-temperature version, to the computation of the dressed Fermi surface for an interacting electronic system. In order to overcome these difficulties, we set up a variational approach which is shown to be equivalent to the renormalized perturbation theory where the dressed Fermi surface is fixed by recursively computed counterterms. The physical picture that emerges is that couplings that are irrelevant tend to deform the Fermi surface in order to become more relevant (irrelevant couplings being those that do not exist at vanishing excitation energy because of kinematical constraints attached to the Fermi surface). These insights are incorporated in a renormalization group approach, which allows for a simple approximate computation of Fermi surface deformation in quasi-one-dimensional electronic conductors. We also analyze flow equations for the effective couplings and quasiparticle weights. For systems away from half-filling, the flows show three regimes corresponding to a Luttinger liquid at high energies, a Fermi liquid, and a low-energy incommensurate spin-density wave. At half-filling Umklapp processes allow for a Mott insulator regime where the dressed Fermi surface is flat, implying a confined phase with vanishing effective transverse single-particle coherence. The boundary between the confined and Fermi liquid phases is found to occur for a bare transverse hopping amplitude of the order of the Mott charge gap of a single chain.

arXiv:cond-mat/0208434Physical review. B, Condensed matterMay 30, 200315 citations

Effect of singular interaction terms on two-dimensional Fermi liquids

Philip C. E. Stamp

The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}$/${\mathrm{\ensuremath{\Elzxh}}}^{2}$N(0)${\mathrm{\ensuremath{\omega}}}_{0}$ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}$/${\mathrm{\ensuremath{\Elzxh}}}^{2}$N(0)${\mathrm{\ensuremath{\omega}}}_{0}$ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.

Physical Review LettersApr 1, 199231 citations

Fermi liquid theory for finite systems

T. Yukawa

Nuclear Physics AJul 1, 19830 citations

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.