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arXiv:cond-mat/9302031 · Physical review. B, Condensed matter

Nonuniversal spectral properties of the Luttinger model

Kurt Schönhammer, V. Meden

Published Jun 15, 1993• 81 citations
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Abstract

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.

DOI
10.1103/physrevb.47.16205

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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.
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.
We compute the spectral function rho (q, omega ) of the one-dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between v sigma q and v rho q, where v sigma and v rho are the velocities of spin and charge fluctuations, while spectral weight above v rho q and below -v rho q is generated by the hybridization of the Fermi surface at +or-k F by interactions. There are non-universal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger-liquid picture, which then would suggest very strong long-range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities, but they probe different and complementary aspects of the Fermi-surface interactions.
Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface, we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational-invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both set-ups have the same analytical form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In the second part we approximately compute the local spectral function of a microscopic lattice model-the extended Hubbard model-close to an open boundary using the functional renormalization group. We show that it follows the field theoretical prediction in the low-energy regime as a function of energy and temperature, and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction.
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.
We show that the one-particle spectral functions of the one-dimensional Hubbard model diverge at the Fermi energy like $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}{|}^{\ensuremath{-}3/8}$ in the $U\ensuremath{\rightarrow}+\ensuremath{\infty}$ limit. The Luttinger liquid behavior $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}{|}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}\ensuremath{\rightarrow}1/8$ as $U\ensuremath{\rightarrow}+\ensuremath{\infty}$, should be limited to $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}|\ensuremath{\sim}{t}^{2}/U$ (for UUU large but finite), which shrinks to a single point, $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\varepsilon}}_{F}$, in that limit. The consequences for the observation of the Luttinger liquid behavior in photoemission and inverse photoemission experiments are discussed.
The spectral functions of tJ\mathrm{tJ}tJ and tJXY{\mathrm{tJ}}_{\mathrm{XY}}tJXY​ models in the limit of $J/t\ensuremath{\rightarrow}0$ and at finite temperatures $T\ensuremath{\ll}t$ are calculated using the spin-charge factorized wave function. We find that the Luttinger-liquid-like scaling behavior for a finite system with LLL sites is restricted below temperatures of the order $T\ensuremath{\lesssim}J/L$. We also observe a weight redistribution in the photoemission spectral function in the energy range ttt, which is much larger than the temperature.