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Physical Review Letters

Generalized hard-core fermions in one dimension: An exactly solvable Luttinger liquid

G. Gómez-Santos

Published Jun 14, 1993• 30 citations
Publisher

Abstract

A model for interacting spinless fermions in one dimension is presented and solved exactly. The solution is shown to be a Luttinger liquid with charge density wave instabilities for well characterized filling factors. Under very plausible assumptions, the exact solution is extended to the spin case with the usual charge-spin separation into two different Luttinger liquids. Required technicalities are almost trivial but the solution is very rich: Periodicity of instabilities and correlation exponents can be chosen at will.

DOI
10.1103/physrevlett.70.3780

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A model for interacting spinless fermions in one dimension is presented and solved exactly. The solution is shown to be a Luttinger liquid with charge density wave instabilities for well characterized filling factors. Under very plausible assumptions, the exact solution is extended to the spin case with the usual charge-spin separation into two different Luttinger liquids. Required technicalities are almost trivial but the solution is very rich: Periodicity of instabilities and correlation exponents can be chosen at will.
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 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.
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.
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 spin-charge-separated Luttinger liquid in arbitrary spatial dimensions and in the applied external magnetic field is investigated within an exactly solvable model of correlated fermions. Exact eigenstates, eigenvalues, and correlation functions of this system are determined and its single-pseudoparticle properties are discussed. It is shown that strongly correlated fermions of spin one-half and charge one, which are described by the present model, can be mapped onto a system of noninteracting fermionlike pseudoparticles possessing continuous charge and spin. The charge and the spin of these pseudoparticles acquire arbitrary values between 0 and eee and 0 and $\ensuremath{\Elzxh}/2$, respectively, when the magnetic field is varied. These pseudoparticles represent quanta of exact elementary excitations in this system.