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physica status solidi (b)

Local distortions and Friedel oscillations in interacting Fermi chains

Cosima Schuster, Philipp Brune

Published Jun 22, 2004• 3 citations
Publisher

Abstract

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)

DOI
10.1002/pssb.200404795

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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)
The Friedel oscillations caused by an impurity located at one edge of a disordered interacting quantum wire are calculated numerically. The electron density in the system's ground state is determined using the density-matrix renormalization-group method, and the Friedel oscillation data are extracted using the density difference between the case in which the wire is coupled to an impurity and the case where the impurity is uncoupled. We show that the power-law decay of the oscillations occurring for an interacting clean one-dimensional sample described by the Luttinger liquid theory is multiplied by an exponential decay term due to the disorder. Scaling of the average Friedel oscillations by this exponential term collapses the disordered sample data on the clean results. We show that the length scale governing the exponential decay may be associated with the Anderson localization length and thus be used as a convenient way to determine the dependence of the localization length on disorder and interactions. The localization length decreases as a function of the interaction strength, in accordance with previous predictions.
Using the density-matrix renormalization-group (DMRG) technique, we calculate critical exponents for the one-dimensional Hubbard model with open boundary conditions with and without additional boundary potentials at both ends. A direct comparison with open boundary condition Bethe ansatz calculations provides a good check for the DMRG calculations on large system sizes. On the other hand, the DMRG calculations provide an independent check of the predictions of conformal field theory, which are needed to obtain the critical exponents from the Bethe ansatz. From the Bethe ansatz we predict the behavior of the 1/L1/L1/L-corrected mean value of the Friedel oscillations (for the density and the magnetization) and the characteristic wave vectors, and show numerically that these conjectures are fulfilled with and without boundary potentials. The quality of the numerical results allows us to determine the behavior of the coefficients of the Friedel oscillations as a function of the Hubbard interaction.
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.
Abstract Using the density matrix renormalization group (DMRG) method, we study the quantum coherence in one‐dimensional disordered spin chains and Fermi systems. We consider in detail spinless fermions on a ring, and compare the influence of several kinds of impurities in a gapless and a dimerized, gapped system. In the translation‐invariant system a so‐called site‐impurity, which can be realized by a local potential or a modification of one link, increases for repulsive interaction, and decreases for attractive interaction, upon renormalization. The weakening of two neighbouring bonds, which is a realization of a so‐called bond‐impurity, on the other hand, is healed for repulsive interaction, but enhanced for intermediate attractive interactions. This leads to a strong suppression of the quantum coherence measured by the phase sensitivity, but not to localization. Adding a local distortion to a dimerized system, we find that even the presence of a single site‐impurity increases the metallic region found in the dimerized model. For a strong dimerization and a high barrier, an additional sharp maximum, is seen in the phase sensitivity as a function of interaction, already for systems with about 100 sites. A bond‐impurity in the dimerized system also opens a small metallic window in the otherwise isolating regime.
We report a density-matrix renormalization-group (DMRG) study of S=1/2 antiferromagnetic Heisenberg spin chains. We show the Friedel oscillations in open spin chains within bosonization theory. The bosonization predictions are in excellent agreement with the DMRG calculation. The chain length dependence of the ground state and several low-energy excited state energies are compared with conformal field theory results. Based upon the DMRG calculations for various quantities, the interplay between the system size and the number of optimal states kept in the DMRG is carefully discussed. \textcopyright{} 1996 The American Physical Society.
We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.