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arXiv:0804.3950 · Physical Review B

Study of the charge correlation function in one-dimensional Hubbard heterostructures

Y. Arredondo, H. Monien

Published Sep 23, 2008• 4 citations
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Abstract

We study inhomogeneous one-dimensional Hubbard systems using the density-matrix renormalization-group method (DMRG). Different heterostructures are investigated whose configuration is modeled varying parameters such as the on-site Coulomb potential and introducing local confining potentials. We investigate their Luttinger liquid properties through the parameter ${K}_{\ensuremath{\rho}}$, which characterizes the decay of the density-density correlation function at large distances. Our main goal is the investigation of possible realization of engineered materials and the ability to manipulate physical properties by choosing an appropriate spatial and/or chemical modulation.

DOI
10.1103/physrevb.78.115425

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We study inhomogeneous one-dimensional Hubbard systems using the density-matrix renormalization-group method (DMRG). Different heterostructures are investigated whose configuration is modeled varying parameters such as the on-site Coulomb potential and introducing local confining potentials. We investigate their Luttinger liquid properties through the parameter K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​, which characterizes the decay of the density-density correlation function at large distances. Our main goal is the investigation of possible realization of engineered materials and the ability to manipulate physical properties by choosing an appropriate spatial and/or chemical modulation.
We study the one-dimensional Anderson-Hubbard model using the density-matrix renormalization-group method. The influence of disorder on the Tomonaga- Luttinger liquid behavior is quantitatively discussed. Based on the finite-size scaling analysis of density-density correlation functions, we find the following results: (i) the charge exponent is significantly reduced by disorder at low filling and near half filling, (ii) the localization length decays as $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}2}$, where $\ensuremath{\Delta}$ is the disorder strength, independently of the on-site Coulomb interaction as well as band filling, and (iii) the localization length is strongly suppressed by the on-site Coulomb interaction near half filling in association with the formation of the Mott plateaus.
We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion UUU and the intersite repulsion VVV at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator based on the bosonization and renormalization-group methods, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly accurate numerical density-matrix renormalization-group method.
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It is now possible to create wires consisting of a single chain of atoms on an insulating substrate, with possible applications in nanoscale circuitry. However, experimental data on such systems are currently rather limited and as such a thorough theoretical investigation of such systems is required. In particular, it is useful to know to what extent the behaviour of such wires is one-dimensional - purely one-dimensional systems display Luttinger liquid behaviour, as opposed to the Fermi liquid behaviour characteristic of higher dimensional systems. However, conventional techniques in theoretical surface science are unable to provide a description of electronic correlations at the level required for one dimensional systems. The existing literature on such atomic wire systems is first reviewed. A discussion of physics in one spatial dimension follows, with a thorough introduction to both the Luttinger liquid and the analytical solutions of the Luttinger model. These concepts are then applied to a simplified model of an atomic wire system - two coupled chains of atoms, on just one of which electron-electron interactions are included. The chain which includes interactions thus represents the wire, while the non-interacting chain represents, albeit incompletely, the influence of the insulating substrate upon which the wire is embedded. Evidence for Luttinger liquid behaviour would be found in the excitation spectra and correlation functions of the system. A Fortran 90 code was written to implement the above, and both a description of the code and numerical results are presented. The results from excitations and correlation functions of the system indicate that, within the limitations of the model, Luttinger liquid behaviour may well survive. It is possible to calculate for larger and more complicated systems by use of the Density Matrix Renormalization Group (DMRG) algorithm, which relies on keeping only those states of the many particle system needed to describe the ground state and the low-energy excitations. The underlying theory of DMRG is discussed and followed with presentation of preliminary results based on a Hubbard model Hamiltonian.
Recent developments in the theoretical investigation of the one-dimensional Kondo lattice model by using the density matrix renormalization group (DMRG) method are discussed in this review. Short summaries are given of the zero-temperature DMRG method, the finite-temperature DMRG method, and also the application of the finite- T DMRG to dynamic quantities. Away from half-filling, the paramagnetic metallic state is shown to be a Tomonaga-Luttinger liquid with a large Fermi surface. The size of the large Fermi surface is determined by the sum of the densities of the conduction electrons and the localized spins. The correlation exponent of this metallic phase is smaller than 1/2. At half-filling the ground state is insulating. The excitation gaps are different, depending on the channels, the spin gap, the charge gap and the quasiparticle gap. The temperature dependence of the spin and charge susceptibilities and the specific heat are discussed. Particularly interesting is the temperature dependence of various excitation spectra, which indicates unusual properties for the Kondo insulators.
We present a theoretical study on the response properties to an external electric field of strongly correlated one-dimensional metals. Our investigation is based on the recently developed Bethe ansatz local-density approximation (BALDA) to the density functional theory formulation of the Hubbard model. This is capable of describing both Luttinger liquid and Mott-insulator correlations. The BALDA calculated values for the static linear polarizability are compared with those obtained by numerically accurate methods, such as exact (Lanczos) diagonalization and the density-matrix renormalization group, over a broad range of parameters. In general BALDA linear polarizabilities are in good agreement with the exact results. The response of the exact exchange and correlation potential is found to point in the same direction of the perturbing potential. This is well reproduced by the BALDA approach, although the fine details depend on the specific parameterization for the local approximation. Finally we provide a numerical proof for the nonlocality of the exact exchange and correlation functional.