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arXiv:cond-mat/9311047 · Journal de Physique I

Breakdown of the Fermi Liquid picture in one dimensional fermion systems: connection with the energy level statistics

R. Mélin, Benoît Douçot, Patrick Butaud

Published May 1, 1994• 4 citations
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Abstract

Using the adiabatic switching of interactions, we establish a condition for the existence of electronic quasiparticles in a Luttinger liquid. It involves a characteristic interaction strength proportional to the inverse square root of the system length. An investigation of the exact energy level separation probability distribution shows that this interaction scale also corresponds to a cross-over from the non interacting behaviour to a rather typical case for integrable systems, namely an exponential distribution. The level spacing statistics of a spin $1/2$, one branch Luttinger model are also analyzed, as well as the level statistics of a two coupled chain model.

DOI
10.1051/jp1:1994173

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Using the adiabatic switching of interactions, we establish a condition for the existence of electronic quasiparticles in a Luttinger liquid. It involves a characteristic interaction strength proportional to the inverse square root of the system length. An investigation of the exact energy level separation probability distribution shows that this interaction scale also corresponds to a cross-over from the non interacting behaviour to a rather typical case for integrable systems, namely an exponential distribution. The level spacing statistics of a spin 1/21/21/2, one branch Luttinger model are also analyzed, as well as the level statistics of a two coupled chain model.
We discuss how one-dimensional interacting fermion systems, which in the low-energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional charge and statistics. Our approach is to use Landau's phenomenological approach to Fermi liquid theory, where the quasiparticles are interpreted as adiabatically dressed fermions. In an earlier publication, the local charge carried by these excitations has been shown to be fractional. We focus here on the statistics of the quasiparticles and show that by a change of momentum variables the Landau parameters of the generalized Fermi fluid can be transformed to zero. This change in interaction is compensated by a change of the entropy function, which is consistent with the interpretation of the quasiparticles as satisfying generalized exclusion statistics.
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.
We study a Luttinger Liquid in a finite one-dimensional wire with box-like boundary conditions by considering the local distribution of the single-particle spectral weight. This corresponds to the experimental probability of extracting a single electron at a given place and energy, which can be interpreted as the square of an electron wave function. For the noninteracting case, this is given by a standing wave at the Fermi wave vector. In the presence of interactions, however, the wave functions obtain additional structure with a sharp depletion near the edges and modulations throughout the wire. In the spinful case, these modulations correspond to the separate spin- and charge-like excitations in the system.
Using the finite-size scaling of results obtained by exact diagonalization, we study the low-density limit of the one-dimensional Hubbard model. Calculating the quasiparticle weight, we demonstrate that for a given particle number N and system size L, there always exists a crossover point Uc{\mathit{U}}_{\mathit{c}}Uc​ separating the Fermi-liquid (UUc{\mathit{U}}_{\mathit{c}}Uc​) and non-Fermi-liquid (U>Uc{\mathit{U}}_{\mathit{c}}Uc​) regimes (U is the Hubbard repulsion). We find that for a fixed N, Uc{\mathit{U}}_{\mathit{c}}Uc​ is inversely proportional to L, keeping Uc{\mathit{U}}_{\mathit{c}}Uc​L/t constant (with t as the hopping integral), as L is large enough. It follows that in the low-density (in fact vanishing density) limit L\ensuremath{\rightarrow}\ensuremath{\infty}, Uc{\mathit{U}}_{\mathit{c}}Uc​\ensuremath{\rightarrow}0, so the system is always in the non-Fermi-liquid regime as long as U>0. We show that our numerical results are consistent with the Bethe-ansatz solution.