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arXiv:2008.01122 · Physical review. B./Physical review. B

Landau-Fermi liquids without quasiparticles

Michele Fabrizio

Published Oct 16, 2020• 7 citations
arXivHTMLPDF

Abstract

Landau-Fermi liquid theory is conventionally believed to hold whenever the interacting single-particle density of states develops a $\ensuremath{\delta}$-like component at the Fermi surface, which is associated with quasiparticles. Here we show that a microscopic justification can be actually achieved under more general circumstances, even in the case where coherent quasiparticles are totally missing and the interacting single-particle density of states vanishes at the chemical potential as a consequence of a pole singularity in the self-energy.

DOI
10.1103/physrevb.102.155122

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Landau-Fermi liquid theory is conventionally believed to hold whenever the interacting single-particle density of states develops a \ensuremathδ\ensuremath{\delta}\ensuremathδ-like component at the Fermi surface, which is associated with quasiparticles. Here we show that a microscopic justification can be actually achieved under more general circumstances, even in the case where coherent quasiparticles are totally missing and the interacting single-particle density of states vanishes at the chemical potential as a consequence of a pole singularity in the self-energy.
In periodic systems of interacting electrons, Fermi and Luttinger surfaces refer to the locations within the Brillouin zone of poles and zeros, respectively, of the single-particle Green's function at zero energy and temperature. Such difference in analytic properties underlies the emergence of well-defined quasiparticles close to a Fermi surface, in contrast to their supposed non-existence close to a Luttinger surface, where the single-particle density-of-states vanishes at zero energy. We here show that, contrary to such common belief, coherent `quasiparticles' do exist also approaching a Luttinger surface in compressible interacting electron systems. Thermodynamic and dynamic properties of such 'quasiparticles' are just those of conventional ones. For instance, they yield well defined quantum oscillations in Luttinger's surface and linear in temperature specific heat, which is striking given the vanishing density of states of physical electrons, but actually not uncommon in strongly correlated materials.
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.
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.
Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-Tc{T}_{c}Tc​ cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
Non-Fermi liquid behavior of strongly correlated Fermi systems is derived within the Landau approach. We attribute this behavior to a phase transition associated with a rearrangement of the Landau state that leads to flattening of a portion of the single-particle spectrum ε(p)ε({\bf p})ε(p) in the vicinity of the Fermi surface. We demonstrate that the quasiparticle subsystem responsible for the flat spectrum possesses the same thermodynamic properties as a gas of localized spins. Theoretical results compare favorably with available experimental data. While departing radically from prevalent views on the origin of non-Fermi-liquid behavior, the theory advanced here is nevertheless a conservative one of in continuing to operate within the general framework of Landau theory.
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.