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arXiv:cond-mat/0010241 · Physical review. B, Condensed matter

Coulomb Luttinger liquid

D. W. Wang, Andrew J. Millis, S. Das Sarma

Published Oct 15, 2001• 47 citations
arXivHTMLPDF

Abstract

Accurate expressions, valid in experimentally relevant regimes, are presented for the effect of a long-ranged Coulomb interaction on the low-energy properties (momentum distribution function, density of states, electron spectral function, and ${4k}_{F}$ correlation function) of one-dimensional electron systems. The importance of plasmon dispersion (as opposed to exponent) effects in the spectral function is demonstrated.

DOI
10.1103/physrevb.64.193307

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Accurate expressions, valid in experimentally relevant regimes, are presented for the effect of a long-ranged Coulomb interaction on the low-energy properties (momentum distribution function, density of states, electron spectral function, and 4kF{4k}_{F}4kF​ correlation function) of one-dimensional electron systems. The importance of plasmon dispersion (as opposed to exponent) effects in the spectral function is demonstrated.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
The dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the $\ensuremath{\delta}$ peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to 2kF,{2k}_{F},2kF​, where kF{k}_{F}kF​ is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at 2kF,{2k}_{F},2kF​, which means that the 2kF{2k}_{F}2kF​ mode is soft.
We consider the effect of the curvature in fermionic dispersion on the observable properties of Luttinger liquid (LL). We use the bosonization technique where the curvature is irrelevant perturbation, describing the decay of LL bosons (plasmon modes). When possible, we establish the correspondence between the bosonization and the fermionic approach. We analyze modifications in density correlation functions due to curvature at finite temperatures, TTT. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LLs, which is only possible due to curvature. Analyzing the ac transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik et al. for TTT dependence of drag resistivity, ${R}_{12}\ensuremath{\sim}{T}^{2}$ at high and ${R}_{12}\ensuremath{\sim}{T}^{5}$ at low temperatures. The bosonization allows for treating both intra- and interwire electron-electron interactions in all orders, and we calculate exact prefactors in the low-TTT drag regime. The crossover temperature between the two regimes is ${T}_{1}\ensuremath{\sim}{E}_{F}\ensuremath{\Delta}$, with $\ensuremath{\Delta}$ relative difference in plasmon velocities. We show that $\ensuremath{\Delta}\ensuremath{\ne}0$ even for identical wires, due to lifting of degeneracy by interwire interaction, U12{U}_{12}U12​, leading to crossover from ${R}_{12}\ensuremath{\sim}{U}_{12}^{2}{T}^{2}$ to ${R}_{12}\ensuremath{\sim}{T}^{5}∕{U}_{12}$ at $T\ensuremath{\sim}{U}_{12}$.
Effects of long-range Coulomb interactions on the Fermi-edge singularity in optical spectra are investigated theoretically for one-dimensional spin-1/2 fermion systems with the use of the Tomonaga-Luttinger bosonization technique. Low-energy excitation spectrum near the Fermi level shows that dispersion of the charge-density fluctuation remains gapless but is nonlinear when the electron-electron (e-e) Coulomb interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., an infinite force range). Temporal behavior of the current-current correlation function is calculated analytically for arbitrary force ranges, ${\ensuremath{\lambda}}_{\mathit{e}}$ and ${\ensuremath{\lambda}}_{\mathit{h}}$, of the e-e and the electron-hole (e-h) Coulomb interactions. (i) When both the e-e and the e-h interactions have large but finite force ranges (${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\infty} and ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\infty}), the correlation function yields a power-law decay only for a long-time regime, which is determined by the force ranges as t>max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]/vF{\mathit{v}}_{\mathit{F}}vF​. Corresponding optical spectrum near the Fermi edge (within an energy range of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]) exhibits the power-law divergence or the power-law convergence, which is an ordinary Fermi-edge singularity. (ii) When either the e-e or the e-h interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., ${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\rightarrow}\ensuremath{\infty} and/or ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\rightarrow}\ensuremath{\infty}), an exponent of the correlation function is dependent on time to lead the faster decay than that of any power laws. Then the optical spectra show no power-law dependence and always converge (become zero) at the Fermi edge, which is in striking contrast to the ordinary power-law singularity. Relations between the spectral shape and a measurement time (frequency resolution) are also clarified. \textcopyright{} 1996 The American Physical Society.
We study the quasiparticle properties of a one-dimensional electron gas interacting with a long-range electron-electron interaction. The electron self-energy is calculated using the leading-order dynamical-screening approximation with $(GW\ensuremath{\Gamma}$ approximation) and without the vertex corrections (GW(GW(GW approximation). The calculated one-electron properties such as the spectral function, damping rate, and also the momentum distribution indicate the significance of vertex corrections at low densities.
We demonstrate that the plasmon in one-dimensional Coulomb interacting electron fluids can develop a finite-momentum maxon-roton-like nonmonotonic energy-momentum dispersion. Such an unusual nonmonotonicity arises from the strongly interacting 1/r1/r1/r Coulomb potential going beyond the conventional band linearization approximation used in the standard bosonization theories of Luttinger liquids. We provide details for the nonmonotonic plasmon dispersion using both bosonization and random-phase approximation theories. We also calculate the specific heat including the nonmonotonicity and discuss possibilities for observing the nonmonotonic plasmon dispersion in various physical systems, including semiconductor quantum wires, carbon nanotubes, and the twisted bilayer graphene at subdegree twist angles, which naturally realize one-dimensional domain-wall states. We provide results for several different models of long-range interaction showing that the nonomonotonic charge collective mode dispersion is a generic phenomenon in one-dimensional strongly interacting electron systems.