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Friedel oscillations and interference patterns from multiple impurities in free fermions in continuous space and in correlated lattice electrons

Banhi Chatterjee, Jan Skolimowski, Krzysztof Byczuk

Published Jul 22, 2020• 0 citations
OpenAlex

Abstract

We study the interference patterns in the Friedel oscillations (FO) due to the scattering off either two or multiple impurities and scattering off extended inhomogeneities both in the non-interacting and two-dimensional lattice systems of interacting fermions. Correlations are accounted for by using an approximate method based on the real-space dynamical mean-field theory. We find that the interference maxima and minima change systematically as we vary the relative distance between the two impurities. At the same time, the interaction increase does not shift the position of interference fringes but only reduces their intensities. A comparison with the single impurity studies clearly shows complex patterns in FO induced by the additional scattering potentials. In the case of an extended step potential the system becomes more homogeneous when the interaction increases. FO and interference patterns are not present in the Mott insulating phase in the single and the many impurity models. Our theoretical study provides promising initial insights and motivate a further realistic modeling to investigate the role of interstitial defects, embedded impurities, ad-atoms on the surfaces, and surface irregularities in materials with different degrees of electronic correlation. For a complete description we also present analytical and numerical results of FO for non-interacting particles moving in a two and three dimensional continuous spaces. In the diluted impurity limit we derive results for FO due to scattering off many impurities, which are generalization of the original Friedel formula.

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We study interference patterns and Friedel oscillations (FO) due to scattering from two or more localized impurities and scattering from extended inhomogeneities in the two-dimensional lattice systems of interacting fermions. Correlations between particles are accounted for by using an approximate method based on the real-space dynamical mean-field theory and a homogeneous self-energy approximation (HSEA), where the site-dependent part of the self-energy is neglected. We find that the interference maxima and minima change systematically as we vary the relative distance between the two impurities. At the same time, the increase of the interaction does not shift the position of interference fringes but only reduces their intensities. A comparison with the single impurity cases clearly shows complex patterns in FO fringes induced by additional multiple scattering processes. In the case of an extended steplike potential the system becomes more homogeneous when the interaction increases. FO and interference patterns are not present in the Mott insulating phase in both single and many impurity models.
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We numerically study Friedel Oscillations and screening effect around a single impurity in one- and two-dimensional interacting lattice electrons. The interaction between electrons is accounted for by using a momentum independent self-energy obeying the Luttinger theorem. It is observed in one-dimensional systems that the amplitude of oscillations is systematically damped with increasing the interaction while the period remains unchanged. The variation of screening charge with the impurity potential is discussed. We see that the screening charge is suppressed by the interactions. In case of two-dimensional systems the surface oscillations around the impurity are more localized with increasing the interactions.
In this work, we investigate impurity-induced Friedel oscillations in the doped two-dimensional Hubbard model, focusing on the role of holon and doublon excitations. We show that weak impurities, due to the non-fermionic nature of the underlying quasiparticles, induce Friedel oscillations whose behavior is consistent with an effective non-interacting theory for these quasiparticles, and whose wavevector reflects the violation of Luttinger's theorem. At larger impurity strength, the system transitions to a phase-separated state composed of coexisting Mott-insulating (half-filled) and hole-rich regions. Within the composite operator framework, this phase separation arises from a competition between the kinetic energy of holons and the tendency to form tightly bound holon-doublon pairs. Our results offer new insights into the nature of charge carriers and the emergent electronic phases in the doped Mott regime.
We present an analytical study of impurity-induced density correlations in interacting one-dimensional Luttinger liquids using the Unified Field Bosonization Technique (UFBT). Unlike conventional approaches that treat impurities perturbatively, UFBT incorporates localized impurities exactly at the free-fermion level while treating interactions non-perturbatively. Closed analytical expressions for fermionic Green functions and density-density correlation functions are obtained for strongly inhomogeneous systems, enabling a unified description of the impurity-induced 2kF2k_F2kF​ Friedel oscillations and the 4kF4k_F4kF​ Wigner-crystal like correlations. The corresponding scaling behaviour is derived analytically, revealing a crossover in the dominant long-distance correlations at g=1/3g=1/3g=1/3: Friedel correlations dominate for g>1/3g>1/3g>1/3, whereas Wigner-crystal like correlations become dominant for g<1/3g<1/3g<1/3, corresponding to the regime of sufficiently strong repulsive interactions. A reflection-coupled harmonic analysis is developed to account for the left-right mixing induced by impurity backscattering and to consistently reproduce the resulting density correlations. The spinless limit is also obtained as a direct reduction of the spinful formulation. These results provide an analytically controlled framework for describing impurity-induced and interaction-driven density correlations in strongly inhomogeneous one-dimensional quantum systems.