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arXiv:cond-mat/0005181 · Philosophical Magazine B

Spectral functions for the Tomonaga-Luttinger and Luther-Emery liquids

Dror Orgad

Published Apr 1, 2001• 50 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1080/136428100110036680

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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 calculate the spectral function of the Luther-Emery model which describes one-dimensional fermions with gapless charge and gapped spin degrees of freedom. We find a true singularity with interaction dependent exponents on the gapped spin dispersion and a finite maximum depending on the magnitude of the spin gap, on a shifted charge dispersion. We apply these results to photoemission experiments on charge density wave systems and discuss the spectral properties of a one-dimensional Mott insulator.
We construct the spectral function of the Luther - Emery model which describes one-dimensional fermions with gapless charge and gapped spin degrees of freedom. We find a true singularity with interaction dependent exponents on the gapped spin dispersion and a finite maximum depending on the magnitude of the spin gap on a shifted charge dispersion. We apply these results to photoemission experiments on charge density wave systems and discuss the spectral properties of a one-dimensional Mott insulator.
We use a recently developed extension of the weak-coupling diagrammatic determinantal quantum Monte Carlo method to investigate the spin, charge, and single-particle spectral functions of the one-dimensional quarter filled Holstein model with phonon frequency ${\ensuremath{\omega}}_{0}=0.1t$. As a function of the dimensionless electron-phonon coupling we observe a transition from a Luttinger to a Luther-Emery liquid with dominant 2kf2{k}_{f}2kf​ charge fluctuations. Emphasis is placed on the temperature dependence of the single-particle spectral function. At high temperatures and in both phases it is well accounted for within a self-consistent Born approximation. In the low-temperature Luttinger liquid phase we observe features that compare favorably with a bosonization approach retaining only forward scattering. In the Luther-Emery phase, the spectral function at low temperatures shows a quasiparticle gap that matches half the spin gap, whereas at temperatures above which this quasiparticle gap closes characteristic features of the Luttinger liquid model are apparent. Our results are based on lattice simulations on chains up to L=20L=20L=20 for two-particle properties and on cluster dynamical mean-field theory calculations with clusters up to 12 sites for the single-particle spectral function.
One-dimensional (1D) interacting Fermi gases underpin a rich variety of quantum liquids, ranging from the Tomonaga-Luttinger liquid and the Fulde-Ferrell-Larkin-Ovchinnikov paired state to the Luther-Emery liquid (LEL). Notably, the LEL-characterized by gapless excitations in charge or spin degree of freedom and gapped excitations in the other-has long been theoretically predicted but has eluded definitive experimental observation to date. By exactly solving the attractive Yang-Gaudin model via the Bethe ansatz, we fully determine the low-energy excitation spectra for both spin and charge channels. For fully paired fermions, the spectrum features gapped spin excitation coexisting with gapless charge excitation. We further employ bosonization to derive analytical expressions for the dynamical structure factor (DSF) of the gapped spin excitation, which is a long-standing challenge in this study. In the trapped gas, we find that the spin Bragg spectroscopy peak occurs at a much higher frequency than the charge-mode peak, where the splitting frequency between these two peaks scales approximately linearly with the interaction strength. Crucially, the spin gap manifests as a sharp threshold in the spin DSF, below which spectral weight is completely absent, proving a direct and unambiguous spectroscopic fingerprint of the LEL. Our work establishes quasi-1D trapped attractive Fermi gases as a pristine experimental platform for the long-sought direct observation of the LEL.
We compute the finite-temperature single-particle spectral function of a one-dimensional Luttinger liquid coupled to an optical phonon band. The calculation is performed exactly for the case in which electron-phonon coupling is purely forward scattering. We extend the results to include backward scattering with a renormalization group treatment. The dispersion contains a change in velocity at the phonon energy, qualitatively similar to the case of electron-phonon coupling in a Fermi liquid. If the backward scattering part of the the electron-phonon interaction is not too strong compared to the forward scattering part, coupling to phonons also produces a pronounced peak in the spectral function at low energies. The calculated spectral function is remarkably similar to the angle-resolved photoemission spectra of the high-temperature superconductors, including the apparent presence of ``nodal quasiparticles,'' the presence of a ``kink'' in the dispersion, and the non-Fermi-liquid frequency and temperature dependencies. Although a microscopic justification has not been established for treating the electronic dynamics of the cuprates as quasi-one-dimensional, at the very least we take the quality of the comparison as evidence of the non-Fermi-liquid character of the measured spectra.
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.