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arXiv:1910.09560 · Physical review. B./Physical review. B

Nonmonotonic plasmon dispersion in strongly interacting Coulomb Luttinger liquids

Yang-Zhi Chou, S. Das Sarma

Published Feb 27, 2020• 2 citations
arXivHTMLPDF

Abstract

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 $1/r$ 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.

DOI
10.1103/physrevb.101.075430

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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.
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 $T\ensuremath{\sim}{U}_{12}$.
For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.
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.
One-dimensional electrons with a linearized dispersion relation are equivalent to a collection of harmonic plasmon modes, which represent long wavelength density oscillations. An immediate consequence of this Luttinger model of one-dimensional electron systems is the absence of inelastic scattering processes responsible for the relaxation of nonequilibrium states. In a generic nonlinear Luttinger liquid, plasmons may decay and thus acquire a finite lifetime. We show that equilibration of plasmons is hierarchical and has profound implications for the dynamics after a thermal quench. We also develop a thermal transport theory and compute thermal conductance of the nonlinear Luttinger liquid by treating the collision integral of plasmons in a manifestly nonperturbative way.
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.
This chapter presents the energy spectrum and the momentum distribution of the Frohlich–Toyozawa–Luttinger liquid. The one-dimensional many-electron problem has generated renewed interest after the discovery of quasi-one dimensional solids such as carbon nanotubes, quantum wires, and other one-dimensional systems. Tomonaga was the first to solve the one-dimensional interacting fermion problem exactly using bosonization and certain approximations regarding the commutator of the density operators. Marino considered a one-dimensional acoustic many-polaron Hamiltonian within the framework of the Luttinger model and showed that for a certain range of the coupling constants the system would exhibit a metallic behavior because the spectrum contains gapless collective excitations of polaronic plasmons. Rao et al. have exactly solved the one-dimensional interacting electron problem within the framework of Luttinger model incorporating both the electron-optical-phonon interaction and the electron-acoustic-phonon interaction along with the electron-electron interaction to obtain the energy spectrum.