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arXiv:0803.0821 · Physical Review B

Finite doping of a one-dimensional charge density wave: Solitons vs Luttinger liquid charge density

Yuval Weiss, Moshe Goldstein, Richard Berkovits

Published May 30, 2008• 8 citations
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Abstract

The effects of doping on a one-dimensional wire in a charge density wave state are studied using the density-matrix renormalization group method. We show that for a finite number of extra electrons, the ground state becomes conducting but the particle density along the wire corresponds to a charge density wave with an incommensurate+ wave number determined by the filling. We find that the absence of the translational invariance can be discerned even in the thermodynamic limit as long as the number of doping electrons is finite. The Luttinger liquid behavior is reached only for a finite change in the electron filling factor, which for an infinite wire corresponds to the addition of an infinite number of electrons. In addition to the half filled insulating Mott state and the conducting states, we find evidence for subgap states at fillings different from half filling by a single electron or hole. Finally, we show that by coupling our system to a quantum dot, one can have a discontinuous dependence of its population on the applied gate voltage in the thermodynamic limit, similar to the one predicted for a Luttinger liquid without umklapp processes.

DOI
10.1103/physrevb.77.205128

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The effects of doping on a one-dimensional wire in a charge density wave state are studied using the density-matrix renormalization group method. We show that for a finite number of extra electrons, the ground state becomes conducting but the particle density along the wire corresponds to a charge density wave with an incommensurate+ wave number determined by the filling. We find that the absence of the translational invariance can be discerned even in the thermodynamic limit as long as the number of doping electrons is finite. The Luttinger liquid behavior is reached only for a finite change in the electron filling factor, which for an infinite wire corresponds to the addition of an infinite number of electrons. In addition to the half filled insulating Mott state and the conducting states, we find evidence for subgap states at fillings different from half filling by a single electron or hole. Finally, we show that by coupling our system to a quantum dot, one can have a discontinuous dependence of its population on the applied gate voltage in the thermodynamic limit, similar to the one predicted for a Luttinger liquid without umklapp processes.
The ground state properties of a one-dimensional system with particle–hole symmetry, consisting of a gate controlled dot coupled to an interacting reservoir, are explored using the numerical DMRG method. It has previously been shown that the system's thermodynamic properties as a function of the gate voltage in the Luttinger liquid phase are qualitatively similar to the behaviour of a non-interacting wire with an effective (renormalized) dot–lead coupling. Here we examine the thermodynamic properties of the wire in the charge density wave phase, and show that these properties behave quite differently. The number of electrons in the system remains constant as a function of the gate voltage, while the total energy becomes linear. Moreover, by tuning the gate voltage on the dot in the charge density wave phase it is possible to drive the wire through a first order quantum phase transition in which the population of each site in the wire is inverted.
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.
We solve a very general two-channel fermion-boson model describing charge transport within some background medium by means of a refined pseudosite density-matrix renormalization group technique. Performing a careful finite-size scaling analysis, we determine the ground-state phase diagram and convincingly prove that the model exhibits a metal-insulator quantum phase transition for the half-filled band case. In order to characterize the metallic and insulating regimes we calculate, besides the local particle densities and fermion-boson correlation functions, the kinetic energy, the charge-structure factor, the Luttinger liquid charge exponent, and the single-particle excitation gap for a one-dimensional infinite system.
In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.
The finite-temperature density-matrix renormalization-group method is applied to the one-dimensional Kondo lattice model near half filling to study its thermodynamics. The spin and charge susceptibilities and entropy are calculated down to T=0.03t. We find two crossover temperatures near half filling. The higher crossover temperature continuously connects to the spin gap at half filling, and the susceptibilities are suppressed around this temperature. At low temperatures, the susceptibilities increase again with decreasing temperature when doping is finite. We confirm that they finally approach to the values obtained in the Tomonaga-Luttinger (TL) liquid ground state for several parameters. The crossover temperature to the TL liquid is a new energy scale determined by gapless excitations of the TL liquid. The transition from the metallic phase to the insulating phase is accompanied by the vanishing of the lower crossover temperature.
We study the metal-insulator transition in a very general two-channel transport model, where charge carriers are coupled to a correlated background medium. The fluctuations of the background were described as bosonic excitations, having the ability to relax. Employing an analytical projector-based renormalization technique, we calculate the ground-state and spectral properties of this fermion-boson model and corroborate recent numerical results, which indicate---in dependence on the ``stiffness'' of the background medium---a Luttinger liquid to charge-density-wave transition for the one-dimensional half-filled band case. In particular, we determine the renormalized electron and boson dispersion relations and show that the quantum phase transition is not triggered by a softening of the boson modes. Thus, the charge density wave is different in nature from an usual Peierls distorted state.