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arXiv:cond-mat/9910189 · Physical review. B, Condensed matter

Breakdown of Luttinger liquid state in a one-dimensional frustrated spinless fermion model

A. K. Zhuravlev, Mikhail I. Katsnelson

Published Jun 15, 2000• 16 citations
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Abstract

The Haldane hypothesis about the universality of the Luttinger liquid (LL) behavior in conducting one-dimensional fermion systems is checked numerically for the spinless fermion model with next-nearest-neighbor interactions. It is shown that for the large enough interactions the ground state can be gapless due to frustrations but at the same time might not belong to the same universality class as a simple LL. The exponents of the correlation functions for this unusual conducting state are found numerically by a finite-size method.

DOI
10.1103/physrevb.61.15534

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The Haldane hypothesis about the universality of the Luttinger liquid (LL) behavior in conducting one-dimensional fermion systems is checked numerically for the spinless fermion model with next-nearest-neighbor interactions. It is shown that for the large enough interactions the ground state can be gapless due to frustrations but at the same time might not belong to the same universality class as a simple LL. The exponents of the correlation functions for this unusual conducting state are found numerically by a finite-size method.
An accurate numerical consideration of one-dimensional (1D) spinless fermion model with next-nearest-neighbor interactions is carried out for the electron concentration \ensuremath{\rho}=4/7. It is shown that depending on the parameters of the model, it can be either Luttinger liquid or bipolaron liquid. In the former case, competing interactions can result in a smooth behavior of one-electron distribution function at the Fermi surface with a divergence in the second derivative with respect to the quasimomentum (and not in the first one, as usual). In this connection, uncommon photoemission spectra for some 1D conductors can be explained qualitatively.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
We study one-dimensional spinless fermions at zero and finite temperature TTT using the density-matrix renormalization group. We consider nearest- as well as next-nearest-neighbor interactions; the latter render the system inaccessible by a Bethe ansatz treatment. Using an infinite-system algorithm we demonstrate the emergence of Luttinger liquid physics at low energies for a variety of static correlation functions as well as for thermodynamic properties. The characteristic power-law suppression of the momentum distribution n(k)n(k)n(k) function at T=0T=0T=0 can be directly observed over several orders of magnitude. At finite temperature, we show that n(k)n(k)n(k) obeys a scaling relation. The Luttinger liquid parameter and the renormalized Fermi velocity can be extracted from the density response function, the specific heat, or the susceptibility without the need to carry out any finite-size analysis. We illustrate that the energy scale below which Luttinger liquid power laws manifest vanishes as the half-filled system is driven into a gapped phase by large interactions.
We consider the spinless fermion model with hard-core repulsive potential between particles extended on a few lattice sites δ. The Luttinger-liquid behavior is studied for different values of a hard-core radius. We derive a critical exponent Θ of the one-particle correlation function ⟨ c i † c j ⟩ for an arbitrary electron density and coupling constant. Our results show that at high density the behavior of fermions can be described as a strongly interacting Luttinger liquid with Θ > 1. As a result, the residual Fermi surface disappears.
We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors (TMTSF)2X,(\mathrm{TMTSF}{)}_{2}X,(TMTSF)2​X, where TMTSF is tetramethyltetraselenafulvalene, and X=ClO9,X={\mathrm{ClO}}_{9},X=ClO9​, PF6,{\mathrm{PF}}_{6},PF6​, ReO4,\mathrm{Re}{\mathrm{O}}_{4},ReO4​, and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.