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Physical review. B, Condensed matter

Breakdown of the Fermi liquid due to long-range interactions

P.-A. Bares, Xiao-Gang Wen

Published Sep 15, 1993• 73 citations
Publisher

Abstract

Fermions interacting via a long-range repulsive potential are considered in D spatial dimensions, where 1D\ensuremath{\le}2. The standard screening picture is found to break down, i.e., the screened effective interaction cannot be treated as instantaneous. For a bare potential which behaves at long distances as ${\mathit{r}}^{\mathrm{\ensuremath{-}}(2\mathrm{\ensuremath{-}}\mathit{D})}$ [as ln r in two dimensions (2D)] and within the random-phase approximation, the retardation effects induce an infrared catastrophe that changes the Fermi liquid into a Luttinger liquid in which the occupation number in momentum space is continuous across the Fermi surface. In 2D, the quantum liquid which we investigate may be called a ``Z=0 Fermi liquid'' (where Z is the strength of the quasiparticle pole at the Fermi surface) since the electron propagator has an isolated pole with a constant residue that scales to zero as the size of the system increases to infinity. For 1D2, the quantum liquid resembles the 1D Luttinger liquid as the single-particle propagator exhibits a branch cut structure. Moreover, we present a ground-state wave function which reproduces the Luttinger-liquid exponent of the momentum distribution near the Fermi surface.

DOI
10.1103/physrevb.48.8636

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Fermions interacting via a long-range repulsive potential are considered in D spatial dimensions, where 1D\ensuremath{\le}2. The standard screening picture is found to break down, i.e., the screened effective interaction cannot be treated as instantaneous. For a bare potential which behaves at long distances as r\ensuremath−(2\ensuremath−D){\mathit{r}}^{\mathrm{\ensuremath{-}}(2\mathrm{\ensuremath{-}}\mathit{D})}r\ensuremath−(2\ensuremath−D) [as ln r in two dimensions (2D)] and within the random-phase approximation, the retardation effects induce an infrared catastrophe that changes the Fermi liquid into a Luttinger liquid in which the occupation number in momentum space is continuous across the Fermi surface. In 2D, the quantum liquid which we investigate may be called a ``Z=0 Fermi liquid'' (where Z is the strength of the quasiparticle pole at the Fermi surface) since the electron propagator has an isolated pole with a constant residue that scales to zero as the size of the system increases to infinity. For 1D2, the quantum liquid resembles the 1D Luttinger liquid as the single-particle propagator exhibits a branch cut structure. Moreover, we present a ground-state wave function which reproduces the Luttinger-liquid exponent of the momentum distribution near the Fermi surface.
We analyze the low energy behavior of interacting fermions in continuous dimensions d between one and two. It is shown that Fermi liquid fixed points are stable with respect to residual scattering of quasiparticles by regular interactions in any dimension above one, while the structure of corrections to quasiparticle behavior changes drastically below two dimensions. The crossover from Luttinger liquid behavior in 1D to Fermi liquid behavior in higher dimensions is described by a tomographic Luttinger model with effective interactions approaching zero with a power d-1 of the distance from the Fermi surface.
We propose a new non-Fermi liquid ground state in one dimension that is not of the Luttinger type. It is the ground state of fermions interacting via a long range repulsive interaction in real space of the form $ V(x) = e^2/|x| $. We characterise it by computing the momentum distribution which is perfectly flat near the Fermi points signalling a radical departure from Luttinger liquid theory(and Fermi liquid theory). The dispersion of elementary excitations is found to be sublinear but gapless of the form $ v {} |q| {}[ Log[2k_{F}/|q| ]^{1/2} $. We also compute the conductance of a mesoscopic wire made of such objects and contrast it from the answer we obtain for a Luttinger wire.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
We consider spinless electrons in two dimensions with the bare spectrum ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{p}}$=vF{\mathit{v}}_{\mathit{F}}vF​(\ensuremath{\Vert}px{\mathit{p}}_{\mathit{x}}px​\ensuremath{\Vert}+\ensuremath{\Vert}py{\mathit{p}}_{\mathit{y}}py​\ensuremath{\Vert}). In momentum space, the interactions among electrons have a finite range q0{\mathit{q}}_{0}q0​, which is small compared to the Fermi momentum. A golden-rule calculation of the electron lifetime indictes a breakdown of Landau's Fermi-liquid theory in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.
Luttinger liquids are characterized by the critical exponent Θ of the momentum distribution around the Fermi momentum k F . Typically Θ⩽1/2, signalling a singularity characterizing a residual Fermi surface. Results of photoemission experiments can be interpreted in terms strongly interacting Luttinger liquids with Θ⩾1 with the residual Fermi surface disappearing. We construct integrable models with such behaviour - models given by the SU(ν) t - J interaction with a hard-core repulsive potential between electrons at distances less than or equal to Δ. The models exhibit both weakly and strongly interacting Luttinger behaviours with Θ varying continuously in the range 0⩽Θ⩽½(1 + Δ-1/[ν(1 + Δ)]) 2 depending on the electron density. In the extreme high-density limit the model exhibits a Mott-Hubbard gap and reduces to an isotropic Heisenberg chain with a new spacing parameter Δ + 1.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.