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Journal of Physics Condensed Matter

Strongly interacting Luttinger liquid - exact solution of a generalizedt-Jmodel in one dimension

Igor N. Karnaukhov, Natan Andrei

Published Oct 19, 2001• 4 citations

Abstract

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.

DOI: 10.1088/0953-8984/13/44/102

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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.

We consider the spinless fermion model with hard-core repulsive potential between particles extended on a few lattice sites delta. The Luttinger-liquid behavior is studied for different values of a hard-core radius. We derive a critical exponent Theta of the one-particle correlation function [c(i)(dagger)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 Phi > 1. As a result, the residual Fermi surface disappears.

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.

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.

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).

The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby

k_F

.