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arXiv:1307.0760 · Physical Review B

Luttinger parameters of interacting fermions in one dimension at high energies

Oleksandr Tsyplyatyev, A. J. Schofield

Published Sep 26, 2013• 4 citations
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Abstract

Interactions between electrons in one dimension are fully described at low energies by only a few parameters of the Tomonaga-Luttinger model, which is based on linearization of the spectrum. We consider a model of spinless fermions with short-range interaction via the Bethe-Ansatz technique and show that a Luttinger parameter emerges in an observable beyond the low-energy limit. A distinct feature of the spectral function, the edge that marks the lowest possible excitation energy for a given momentum, is parabolic for arbitrary momenta and the prefactor is a function of the Luttinger parameter, $K$.

DOI
10.1103/physrevb.88.115142

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Interactions between electrons in one dimension are fully described at low energies by only a few parameters of the Tomonaga-Luttinger model, which is based on linearization of the spectrum. We consider a model of spinless fermions with short-range interaction via the Bethe-Ansatz technique and show that a Luttinger parameter emerges in an observable beyond the low-energy limit. A distinct feature of the spectral function, the edge that marks the lowest possible excitation energy for a given momentum, is parabolic for arbitrary momenta and the prefactor is a function of the Luttinger parameter, KKK.
A continuum of excitations in interacting one-dimensional systems is bounded from below by a spectral edge that marks the lowest possible excitation energy for a given momentum. We analyze short-range interactions between Fermi particles and between Bose particles (with and without spin) using Bethe-ansatz techniques and find that the dispersions of the corresponding spectral edge modes are close to a parabola in all cases. Based on this emergent phenomenon we propose an empirical model of a free, nonrelativistic particle with an effective mass identified at low energies as the bare electron mass renormalized by the dimensionless Luttinger parameter KKK (or ${K}_{\ensuremath{\sigma}}$ for particles with spin). The relevance of the Luttinger parameters beyond the low-energy limit provides a more robust method for extracting them experimentally using a much wider range of data from the bottom of the one-dimensional band to the Fermi energy. The empirical model of the spectral edge mode complements the mobile impurity model to give a description of the excitations in proximity of the edge at arbitrary momenta in terms of only the low-energy parameters and the bare electron mass. Within such a framework, for example, exponents of the spectral function are expressed explicitly in terms of only a few Luttinger parameters.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low-energy spectrum of noninteracting fermions is identical to the one for a harmonic chain. This allows us to describe interacting fermions as a system of coupled oscillators. The exact solution of the Tomonaga–Luttinger model is discussed and results for the momentum distribution and spectral functions are presented. The general Luttinger liquid phenomenology is briefly discussed and various attempts at the experimental verification of the theoretical predictions are examined. Extended defects in semiconductors might also be candidate systems.
The one-electron spectral functions for the Luttinger model are discussed for large but finite systems. The methods presented allow a simple interpretation of the results. For finite-range interactions, interesting nonuniversal spectral features emerge for momenta that differ from the Fermi points by the order of the inverse interaction range or more. For a simplified model with interactions only within the branches of right- and left-moving electrons, analytical expressions for the spectral function are presented that allow us to perform the thermodynamic limit. As in the general spinless model and the model including spin for which we present mainly numerical results, the spectral functions do not approach the noninteracting limit for large momenta. The implication of our results for recent high-resolution photoemission measurements on quasi-one-dimensional conductors are discussed.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low-energy spectrum of noninteracting fermions is identical to the one for a harmonic chain. This allows us to describe interacting fermions as a system of coupled oscillators. The exact solution of the Tomonaga-Luttinger model is discussed and results for the momentum distribution and spectral functions are presented. The general Luttinger liquid phenomenology is briefly discussed and various attempts at the experimental verification of the theoretical predictions are examined. Extended defects in semiconductors might also be candidate systems.
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.