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Physical review. B, Condensed matter

Spectral functions for the Tomonaga-Luttinger model

V. Meden, Kurt Schönhammer

Published Dec 15, 1992• 279 citations
Publisher

Abstract

Results for the one-particle Green's function for the Tomonaga-Luttinger model of one-dimensional interacting fermions take a simple form using space and time variables. To obtain the corresponding spectral functions which determine photoemission and inverse-photoemission spectra, a double Fourier transform is necessary. For the model including spin we present analytical and numerical results. They show drastic differences from the spinless case. Expressions for the critical indices which determine the singularities of the spectra are given.

DOI
10.1103/physrevb.46.15753

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Results for the one-particle Green's function for the Tomonaga-Luttinger model of one-dimensional interacting fermions take a simple form using space and time variables. To obtain the corresponding spectral functions which determine photoemission and inverse-photoemission spectra, a double Fourier transform is necessary. For the model including spin we present analytical and numerical results. They show drastic differences from the spinless case. Expressions for the critical indices which determine the singularities of the spectra are given.
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.
We calculate the finite temperature single-hole spectral function and the spin dynamic structure factor of a spinful one-dimensional Tomonaga-Luttinger liquid. Analytical expressions are obtained for a number of special cases. We also calculate the single-hole spectral function of a spin-gapped Luther -Emery liquid and obtain exact results at the free-fermion point K s = ½ These results may be applied to the analysis of angle-resolved photoemission and neutron scattering experiments on quasi-one-dimensional materials.
We examine zero-temperature one-particle spectral functions for the one-dimensional two-band spinless fermions with different velocities and general forward-scattering interactions. By using the bosonization technique and diagonalizing the model to two Tomonaga-Luttinger liquid Hamiltonians, we obtain general expressions for the spectral functions, which are given in terms of the Appell hypergeometric functions. For the case of identical two-band fermions, corresponding to the SU(2) symmetric spin-1/2 fermions with repulsive interactions, the spectral functions can be expressed in terms of the Gauss hypergeometric function and are shown to recover the double-peak structure suggesting the well-known ``spin-charge'' separation. By tuning the difference in velocities for the two-band fermions, we clarify the crossover in spectral functions from the spin-charge separation to the decoupled fermions. We discuss the relevance of our results to the spin-1/2 Hubbard model under a magnetic field, which can be mapped onto two-band spinless fermions.
Using the functional integral bosonization technique, the spinless Tomonaga - Luttinger model with impurity forward scattering is solved exactly. Explicit analytical results are given for the one- and two-particle spectral functions in terms of a convolution integral representation. Sharp structures in the frequency spectra of an ordered system are smoothed out by the disorder. The existence of a charge - density instability is predicted if the critical exponent is less than -0.5. There is no influence of the disorder on the superconducting spectral function .
Conformal field theory and Bethe ansatz are used to investigate the low energy features of the spectral function in one dimensional models which exhibit a gap in the spin or in the charge excitation spectrum. Exotic behavior is found in the superconducting case, where the Green function displays momentum dependent Tomonaga-Luttinger liquid exponents. The predictions of the formalism are confirmed by Lanczos diagonalizations in the $t\ensuremath{-}J$ model up to 32 sites. These results may be relevant in connection to photoemission experiments in quasi-one-dimensional insulators or superconductors.