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Journal of Computational Chemistry

Quasi‐one‐dimensional systems and localized defects: Some results of the Green's matrix formalism

Max Seel, Giuseppe Del Re, Janos J. Ladik

Published Dec 1, 1982• 32 citations

Abstract

Abstract The application of the self‐consistent field (SCF) local‐impurity formalism to quasi‐one‐dimensional systems is discussed. We describe a general procedure for an accurate numerical determination of the Green's function matrix elements of the unperturbed system. An application to a local impurity in a model chain with two orbitals per unit cell is reported. The changes in the charge‐bond‐order matrix and in local and total density of states due to the impurity are discussed with special emphasis on the changes at the critical points (van Hove singularities) at the band edges. The Green's matrix approach is used to reexamine long‐range Friedel oscillations caused by an impurity in a strictly one‐dimensional metal. The extent of the long‐range tail of the perturbed charge density is in an inverse relation to the localization length of the impurity state: the stronger the perturbation the more localized is the bound state and the more extended are the oscillations in the charge distribution. The results for the model chain with two orbitals per unit cell indicate that the impurity‐induced change in charge distribution may be locally screened by redistribution of the population of the on‐site orbitals, therefore damping possible oscillations and leading to a faster decay than in strictly one‐dimensional systems.

DOI: 10.1002/jcc.540030402

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Abstract The application of the self‐consistent field (SCF) local‐impurity formalism to quasi‐one‐dimensional systems is discussed. We describe a general procedure for an accurate numerical determination of the Green's function matrix elements of the unperturbed system. An application to a local impurity in a model chain with two orbitals per unit cell is reported. The changes in the charge‐bond‐order matrix and in local and total density of states due to the impurity are discussed with special emphasis on the changes at the critical points (van Hove singularities) at the band edges. The Green's matrix approach is used to reexamine long‐range Friedel oscillations caused by an impurity in a strictly one‐dimensional metal. The extent of the long‐range tail of the perturbed charge density is in an inverse relation to the localization length of the impurity state: the stronger the perturbation the more localized is the bound state and the more extended are the oscillations in the charge distribution. The results for the model chain with two orbitals per unit cell indicate that the impurity‐induced change in charge distribution may be locally screened by redistribution of the population of the on‐site orbitals, therefore damping possible oscillations and leading to a faster decay than in strictly one‐dimensional systems.

Abstract The interplay between disorder and interaction, especially near metal insulator transitions, is a long‐standing question. We investigate in detail single impurities, in particular, the Friedel oscillations induced by them. We study the decay of the Friedel oscillations in the one‐dimensional Heisenberg and Hubbard model analytically using the bosonization technique and numerically using the density matrix renormalization group treatment (DMRG). For the Heisenberg chain, we confirm the predictions of conformal field theory and bosonization for small interaction, but near phase transitions deviations are found in form of vanishing or additional oscillations. For the Hubbard chain, we study the oscillations – in the density as well as in the magnetization – in the spin‐gap, charge‐gap, and Luttinger liquid phase. We find an exponential decay or a very slow algebraic decay of the oscillations in the gapped phases. In the Luttinger liquid phase, we concentrate on the question of logarithmic corrections (which occur also in the isotropic Heisenberg antiferromagnet). Differences in the behavior near a boundary compared to an impurity are pointed out. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Green's-function method is used to obtain the exact expression for the local susceptibility of the impure spin-1/2 XY chain at zero temperature. The presence of the impurity (or boundary) affects the singularity of the magnetic field dependence of the local susceptibility. The corresponding critical exponent changes from -1/2 to +1/2 at an arbitrary distance from the impurity site. In contrast with the transverse Ising model, the critical exponent is independent of the perturbation strength. This phenomenon is related to the spatial oscillations of the local density of states in the nonideal XY chain. It is also shown that experimental investigation of the field dependence of the finite-temperature macroscopic susceptibility of the one-dimensional XY-like system (such as

{\mathrm{Cs}}_{2}

{\mathrm{CoCl}}_{4}

) may reveal the formation of the localized states associated with impurities.

We study the low-energy limit of a quarter-filled one-dimensional Mott insulator. We analytically determine the local density of states in the presence of a strong impurity potential, which is modeled by a boundary. To this end we calculate the Green function using field theoretical methods. The Fourier transform of the local density of states shows signatures of a pinning of the spin-density wave at the impurity as well as several dispersing features at frequencies above the charge gap. These features can be interpreted as propagating spin and charge degrees of freedom. Their relative strength can be attributed to the ``quasi-fermionic'' behavior of charge excitations with equal momenta. Furthermore, we discuss the effect of bound states localized at the impurity. Finally, we give an overview of the local density of states in various one-dimensional systems and discuss implications for scanning tunneling microscopy experiments.

Abstract A Green's function technique is used to obtain the exact form of the Friedel type oscillations of local magnetization in the spin 1/2, XY antiferromagnetic chain induced by the presence of a boundary or impurity. It is shown that, in addition to the amplitude, the functional form of the oscillations is determined by the strength of the perturbation caused by the impurity and intensity of the external magnetic field. Comparison is made with the earlier perturbation theory treatment in the free‐fermion approximation.

We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.