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Physical review. B, Condensed matter

Charge-spin separation and the spectral properties of Luttinger liquids

Johannes Voit

Published Mar 15, 1993• 231 citations
Publisher

Abstract

We compute the spectral function \ensuremath{\rho}(q,\ensuremath{\omega}) of the one-dimensional Luttinger model. Charge-spin separation gives spectral weight at frequencies between the charge and spin-fluctuation dispersions while spectral weight above ${\mathit{v}}_{\mathrm{\ensuremath{\rho}}}$q and below -${\mathit{v}}_{\mathrm{\ensuremath{\rho}}}$q is related to the anomalous dimensions of the fermion operators. The generalization of these results to many-particle correlations is discussed as well as the possibilities for their experimental verification in quasi-one-dimensional conductors.

DOI
10.1103/physrevb.47.6740

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We compute the spectral function \ensuremath{\rho}(q,\ensuremath{\omega}) of the one-dimensional Luttinger model. Charge-spin separation gives spectral weight at frequencies between the charge and spin-fluctuation dispersions while spectral weight above v\ensuremathρ{\mathit{v}}_{\mathrm{\ensuremath{\rho}}}v\ensuremathρ​q and below -v\ensuremathρ{\mathit{v}}_{\mathrm{\ensuremath{\rho}}}v\ensuremathρ​q is related to the anomalous dimensions of the fermion operators. The generalization of these results to many-particle correlations is discussed as well as the possibilities for their experimental verification in quasi-one-dimensional conductors.
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.
I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.
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.
We develop a nonperturbative zero-temperature theory for the dynamic response functions of interacting one-dimensional spin-1/2 fermions. In contrast to the conventional Luttinger liquid theory, we take into account the nonlinearity of the fermion dispersion exactly. We calculate the power-law singularities of the spectral function and the charge- and spin-density structure factors for arbitrary momenta and interaction strengths. The exponents characterizing the singularities are functions of momenta and differ significantly from the predictions of the linear Luttinger liquid theory. We generalize the notion of the spin-charge separation to the nonlinear spectrum. This generalization leads to phenomenological relations between threshold exponents and the threshold energy.
Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in ddd dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to d=1,d=1,d=1, in higher dimensions the singularity of the momentum distribution nk{n}_{\mathbf{k}}nk​ for wave vectors k close to the Fermi wave vector kF{k}_{F}kF​ is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in d>1d>1d>1 the leading singular corrections cancel. Thus, even though singular density-density interactions in d>1d>1d>1 lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.