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Physical Review Letters

Friedel-Type Oscillations in One-Dimensional Antiferromagnetic Insulators

G. Baskaran

Published Jun 5, 1978• 14 citations
Publisher

Abstract

An impurity spin in one-dimensional spin-\textonehalf{} $X\ensuremath{-}Y$ and antiferromagnetic Heisenberg models is shown to induce decaying spatial oscillations of magnetization for finite magnetic field. The wavelength of oscillation changes with magnetic field. Since these systems can be described mathematically in terms of a degenerate gas of pseudofermions, their response to impurity perturbation is similar to the response of an electron gas to a localized static charge.

DOI
10.1103/physrevlett.40.1521

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An impurity spin in one-dimensional spin-\textonehalf{} X\ensuremath−YX\ensuremath{-}YX\ensuremath−Y and antiferromagnetic Heisenberg models is shown to induce decaying spatial oscillations of magnetization for finite magnetic field. The wavelength of oscillation changes with magnetic field. Since these systems can be described mathematically in terms of a degenerate gas of pseudofermions, their response to impurity perturbation is similar to the response of an electron gas to a localized static charge.
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.
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)
Oscillations of local density of states generated by a single scalar impurity potential are calculated for one-dimensional systems with dynamically generated charge or spin gap. At zero temperature, the oscillations develop at finite wave vector ($\ensuremath{\pi}$ for the Mott insulator and 2kF2{k}_{F}2kF​ for incommensurate charge-density wave or superconductor) and at frequencies larger than the soliton spectral gap mmm. Their amplitude has a broad maximum at $\ensuremath{\omega}\ensuremath{\approx}3m$, where mmm is the gap magnitude.
We study the interplay of correlations and disorder using an unrestricted slave-Boson technique in real space. Within the saddle-point approximation, we find Friedel oscillations of the charge density in the vicinity of a single nonmagnetic impurity, in agreement with numerical simulations. The corresponding amplitudes are suppressed by repulsive interactions, while attractive correlations lead to a charge-density-wave enhancement. In addition, we investigate the spatial dependence of the local magnetic moment and the formation of a magnetic state at the impurity site.
We study Friedel oscillations in one-dimensional electron liquids for arbitrary electron-electron interaction and arbitrary impurity strength. Explicit results for spinless as well as spin- 12\frac{1}{2}21​ electrons are given. In the case of Luttinger-liquid leads, the Friedel oscillations decay as ${x}^{\ensuremath{-}g}$ far away from the impurity where ggg is the interaction constant. For a weak scatterer, a slower decay is found at small-to-intermediate distances from the impurity, with a crossover to the asymptotic ${x}^{\ensuremath{-}g}$ decay.
Hartree-Fock equations for the impure spin-\textonehalf{} Ising-Heisenberg chain are derived by using the Green-function technique in the pseudofermion representation. The self-consistency equations for the ideal spin-\textonehalf{} anisotropic Heisenberg antiferromagnet can be recovered as a specific case of the general formalism developed in the present work. In the case of the nonideal XXZ\mathrm{XXZ}XXZ chain, a numerical solution of the self-consistency equations is used to describe the Friedel-type boundary on impurity-induced oscillations of the local magnetization considered previously in the spin-\textonehalf{}XY\mathrm{XY}XY model.