Impurities and other defects in correlated lattice electrons: Friedel oscillations and interference patterns
Banhi Chatterjee, Jan Skolimowski, Krzysztof Byczuk
备注：
11 pages, 12 figures Section II in previous version discussing FO in continuous non-interacting systems removed, a new section added to demonstrate how well the homogeneous selfenergy approximation works
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 step like 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.
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 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.
Recently Affleck et al. derived the existence of Friedel oscillations in the presence of a Kondo impurity. They supported their analytic derivation by numerical calculations using Wilson's renormalization approach (NRG). In this paper the size of the Friedel oscillations is calculated with the FAIR method (Friedel Artificially Inserted Resonance) which has been recently developed. The results of NRG and FAIR are compared. The development of Friedel oscillations with a phase shift of pi/2 outside of the Kondo radius is confirmed.
The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculated numerically. The ground states of both systems are calculated with the FAIR (Friedel artificially inserted resonance) theory. The ground state in the interacting systems is compared with a nullstate in which the interaction is zero. The different multi-electron states are expressed in terms of Wilson states. The use of N Wilson states simulates the use of a large effective number N_{eff} of states. A plot of ln(F) versus N\proptoln(N_{eff}) reveals whether one has an Anderson orthogonality catastrophe at zero e…
In this work, the four-point Green functions relevant to the study of Friedel oscillations are calculated for a Luttinger liquid with a cluster of impurities around an origin using the powerful Non chiral bosonization technique (NCBT). The two-point functions obtained using the same method are used to calculate the dynamical density of states (DDOS), which exhibits a power law in energy and closed analytical expressions for the DDOS exponent is calculated. These results interpolates between the weak barrier and weak link cases which are typically studied in the literature. The dependence of th…
When a metal undergoes a transition to an insulator it will lose its electronic Fermi surface. Interestingly in some situations a `ghost' Fermi surface of electrically neutral spin carrying fermions may survive into the insulator. Such a novel ghost Fermi surface has been proposed to underlie the properties of a few different materials but its direct detection has proven elusive. In this paper we show that the ghost Fermi surface leads to slowly decaying spatial oscillations of the electron density near impurities or other defects. These and related oscillations stem from the sharpness of the …
One of the great achievements performed by Jacques Friedel in France has been to promote experimental activities on the electronic properties in condensed matter physics, which has led him, among others, to be an active promoter of the up-rise of the "Laboratoire de Physique des Solides" (LPS). A selective presentation of some of the work achieved in my Nuclear Magnetic Resonance (NMR) group allows me to underline the importance of the initial choices done by J. Friedel for the LPS. I give here an updated review of the work done on Kondo effect in the 1970's which has allowed me to enlighten t…
We introduce a path-integral approach that allows to compute charge density oscillations in a Luttinger liquid with impurities. We obtain an explicit expression for the envelope of Friedel oscillations in the presence of arbitrary electron-electron potentials. As examples, in order to illustrate the procedure, we show how to use our formula for contact and Coulomb potentials.
We address three issues on the role of magnetic impurities in the surface state of a three dimensional topological insulator. First, we prove that the Kondo effect of the topological surface is essentially the same as that of the graphene surface, demonstrating that an effective impurity action of the topological surface coincides with that of the graphene surface. Second, we study the role of the $z$-directional magnetic field ($h$) in the Kondo effect, and show that the peak splitting in the impurity local density of states does not follow the $h$-linear behavior, the typical physics in the …
When immersed in a see of cold electrons, local impurities give rise to density modulations known as Friedel oscillations. In spite of the generality of this phenomenon, the exact shape of these modulations is usually computed only for non-interacting electrons with a quadratic dispersion relation. In actual materials, one needs to take into account several additional factors, such as (i) the details of the band structure, (ii) the lifetime of quasiparticles, (iii) in superconductors, the presence of a pairing gap. Studying how these effects influence Friedel oscillations is a viable way to ac…
Using an asymptotic phase representation of the particle density operator $\hatρ(z)$ in the one-dimensional harmonic trap, the part $δ\hatρ_F(z)$ which describes the Friedel oscillations is extracted. The expectation value $<δ\hatρ_F(z)>$ with respect to the interacting ground state requires the calculation of the mean square average of a properly defined phase operator. This calculation is performed analytically for the Tomonaga-Luttinger model with harmonic confinement. It is found that the envelope of the Friedel oscillations at zero temperature decays with the boundary exponent $ν= (K+1)/2…
We report the observation of discrete vortex bound states with the energy levels deviating from the widely believed ratio of 1:3:5 in the vortices of an iron based superconductor KCa2Fe4As4F2 through scanning tunneling microcopy (STM). Meanwhile Friedel oscillations of vortex bound states are also observed for the first time in related vortices. By doing self-consistent calculations of Bogoliubov-de Gennes equations, we find that at extreme quantum limit, the superconducting order parameter exhibits a Friedel-like oscillation, which modifies the energy levels of the vortex bound states and exp…
We study the effect of nonmagnetic impurities in the superconducting state of the t-J model. The Bogoliubov de Gennes equation is derived using a slave-boson mean-field approximation, and we solve it numerically at T=0. The d_{x^2-y^2}-wave order parameter is suppressed near an impurity and its spatial variation induces the extended s-wave component. We also find the local charge density wave (CDW)-like feature due to the Friedel oscillations. This leads to the splitting of the expected zero-energy Andreev bound state. Therefore the system is unlikely introducing a localized state of other sym…
Fourier transform scanning tunneling spectroscopy (FT-STS) measures the scattering of conduction electrons from impurities and defects, giving information about the electronic structure of both the host material and adsorbed impurities. We interpret such FT-STS measurements in terms of the quasiparticle interference (QPI), investigating the QPI due to magnetic impurities adsorbed on a range of representative non-magnetic host surfaces, and contrasting with the case of a simple scalar impurity or point defect. We demonstrate how the electronic correlations present for magnetic impurities marked…
Disorder has intriguing consequences for correlated electronic materials, which include several families of high-temperature superconductors and resistive switching systems. We address the question of why strongly correlated d-wave superconductors, such as the cuprates, prove to be surprisingly robust against the introduction of non-magnetic impurities. We show that, very generally, both the pair-breaking and the normal state transport scattering rates are significantly suppressed by strong correlations effects arising in the proximity to a Mott insulating state. We also show that the correlat…
The impact of non-magnetic and magnetic impurities on topological insulators is a central problem concerning their fundamental physics and possible novel spintronics and quantum computing applications. SmB$_6$, predicted to be a topological Kondo insulator, is considered a benchmark material. Using a spin-polarized tip in scanning tunneling spectroscopy destroys the signature peak of the topological surface state, revealing its spin texture. Further, combining local STS with macroscopic transport measurements on SmB$_6$ containing different substitutions enables us to investigate the effect of…
We study effects of nonmagnetic impurities in a Spin Bose-Metal (SBM) phase discovered in a two-leg triangular strip spin-1/2 model with ring exchanges (D. N. Shenget al, arXiv:0902.4210). This phase is a quasi-1D descendant of a 2D spin liquid with spinon Fermi sea, and the present study aims at interpolating between the 1D and 2D cases. Different types of defects can be treated as local energy perturbations, which we find are always relevant. As a result, a nonmagnetic impurity generically cuts the system into two decoupled parts. We calculate bond energy and local spin susceptibility near t…
The two-dimensional quantum Heisenberg antiferromagnet at zero temperature with two kinds of locally frustrating defects - isolated ferromagnetic bonds and impurity spins, is studied. For a quantum spin, coupled antiferromagnetically to the two sublattices, we find a local ground state transition. In the case of a ferromagnetic bond, the local magnetization is found to remain finite even for strong coupling, contrary to previous results, derived within the linear spin-wave approximation. Our results are based on an analytic approach, suitable to treat strong defect-environment interaction, as …
We use a quantum lattice gas model to describe essential aspects of the motion of 4He atoms and of 3He impurities in solid 4He. This study suggests that 3He impurities bind to defects and promote 4He atoms to interstitial sites which can turn the bosonic quantum disordered crystal into a metastable supersolid. It is suggested that defects and interstitial atoms are produced during the solid 4He nucleation process where the role of 3He impurities (in addition to the cooling rate) is known to be important even at very small (1 ppm) impurity concentration. It is also proposed that such defects ca…
Substitutional impurities in the CuO2 planes of the cuprates allow us to probe the electronic properties of the host material. The pseudo-gap in the underdoped regime is unmodified far from the impurities even though Tc is greatly reduced. The spin polarisation induced by magnetic impurities has an oscillatory behaviour reflecting the existing AF correlations between the Cu spins. Its influence on the NMR spectra opens a way to determine the q dependence of the static spin susceptibility and the T dependence of the AF correlation length. NMR measurements demonstrate that non-magnetic impuritie…
The conductance of a tunnel point-contact in an STM-like geometry having a single defect placed below the surface is investigated theoretically. The effect of multiple electron scattering by the defect after reflections by the metal surface is taken into account. In the approximation of s-wave scattering the dependence of the conductance on the applied voltage and the position of the defect is obtained. The results are illustrated for a model s-wave phase shift describing Kondo-resonance scattering. We demonstrate that multiple electron scattering by the magnetic impurity plays a decisive role…
In materials with strong local Coulomb interactions, simple defects such as atomic substitutions strongly affect both macroscopic and local properties of the system. A nonmagnetic impurity, for instance, is seen to induce magnetism nearby. Even without disorder, models of such correlated systems are generally not soluble in 2 or 3 dimensions, and so few exact results are known for the properties of such impurities. Nevertheless, some simple physical ideas have emerged from experiments and approximate theories. Here, we first review what we can learn about this problem from 1D antiferromagnetic…
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；C. N. Yang Institute for Theoretical Physics, Stony Brook University（石溪大学杨振宁理论物理研究所）
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