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Physical Review B

Strongly interacting Luttinger liquid states as those inherent in carbon nanotubes

Igor N. Karnaukhov, Victor I. Kolomytsev, Cees G. H. Diks

Published Mar 8, 2007• 1 citation
Publisher

Abstract

The electronic transport properties of single-walled nanotubes (SWNTs) are characterized by the critical exponent $\ensuremath{\Theta}$, describing the behavior of the single-particle spectral function near the Fermi energy as $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{\Theta}}$. Experimentally obtained $\ensuremath{\Theta}$ values have been reported in the range of 0.43--0.48 for metallic SWNTs. However, these values are much larger than the theoretical upper limit of the exponent, ${\ensuremath{\Theta}}_{\mathrm{max}}=1∕8$, for the Hubbard model. Here, we show that the observed electronic transport properties can be explained by taking into account a specific hard-core Coulomb interaction. For a many-body system with a hard-core potential, we analytically calculate the critical exponents using the Bethe ansatz and conformal field theory. We find strongly interacting Luttinger liquid states that are characterized by large $\ensuremath{\Theta}$ values, consistent with the empirical results, suggesting that these states are actually realized in metallic SWNTs and organic conductors.

DOI
10.1103/physrevb.75.125407

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The electronic transport properties of single-walled nanotubes (SWNTs) are characterized by the critical exponent \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ, describing the behavior of the single-particle spectral function near the Fermi energy as \ensuremathρ(\ensuremathω)\ensuremath∝\ensuremath∣\ensuremathω\ensuremath∣\ensuremathΘ\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{\Theta}}\ensuremathρ(\ensuremathω)\ensuremath∝\ensuremath∣\ensuremathω\ensuremath∣\ensuremathΘ. Experimentally obtained \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ values have been reported in the range of 0.43--0.48 for metallic SWNTs. However, these values are much larger than the theoretical upper limit of the exponent, \ensuremathΘmax=1∕8{\ensuremath{\Theta}}_{\mathrm{max}}=1∕8\ensuremathΘmax​=1∕8, for the Hubbard model. Here, we show that the observed electronic transport properties can be explained by taking into account a specific hard-core Coulomb interaction. For a many-body system with a hard-core potential, we analytically calculate the critical exponents using the Bethe ansatz and conformal field theory. We find strongly interacting Luttinger liquid states that are characterized by large \ensuremathΘ\ensuremath{\Theta}\ensuremathΘ values, consistent with the empirical results, suggesting that these states are actually realized in metallic SWNTs and organic conductors.
One parameter family of the Lai-Sutherland models with hard-core repulsive potential is formulated and solved by the Bethe ansatz method in one dimension for an arbitrary core radius (Δ + 1)/2. The ground-state Bethe ansatz equations are analysed and solved numerically for an arbitrary electron density, several values of color components, and the core radius. The ground-state energy, the Fermi velocity, and the critical exponents describing the asymptotic behavior of the correlation functions at long distances have been calculated numerically for an arbitrary density of electrons. In contrast to the integrable models studied previously, the long-distance behavior is described by a strongly interacting Luttinger liquid state. This state is characterized by a large value of the critical exponent Θ for the momentum distribution function (Θ > 1) and realized at a high electron density region n > n c (Θ( n c ) = 1).
A family of spinless fermion models with a hard core potential is formulated and solved by the Bethe ansatz method in one dimension for an arbitrary core radius. The Hamiltonian has two levels of strong interaction: the repulsive hard core potential and the kinetic energy with hopping integrals, values of which depend on the configuration of the particles. The Fermi velocity and the critical exponents describing the asymptotic behavior of the correlation functions at long distances have been calculated numerically for an arbitrary density of electrons and hard core radius. We discuss the effect of the hard core potential. The hard core potential defines an anomalous behavior of the critical exponent \ensuremath{\Theta} of the momentum distribution function. In the high-electron-density region the long-distance behavior is described by a strongly interacting Luttinger-liquid state with \ensuremath{\Theta}>1 and the residual Fermi surface disappears.
The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasiparticle charge, a fundamental property of Luttinger liquids, can be observed in impurity-induced shot noise.
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.
The eect of electronic many-body interactions in one-dimensional (1D) systems is drastically enhanced compared to higher dimensions due to the restricted phasespace, often leading to strong electronic correlations. It can be shown that due to a breakdown of the Fermi liquid theory, the ground state at low temperatures becomes a Tomonaga-Luttinger liquid (TLL). In a TLL, the strength of electronic correlations is characterized by the Luttinger parameter K, ranging between K = 0 and K = 1, where K = 1 describes non-interacting electrons. Various systems, such as single-walled carbon nanotubes, 1D atomic metal chains, and MoS 2 mirror twin boundaries, have been reported to host TLLs, in which spin-degeneracy is intact.
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.