Single-impurity Anderson Model Research Articles

The Anderson impurity model out-of-equilibrium: Assessing the accuracy of simulation techniques with an exact current-occupation relation.

We study the interacting, symmetrically coupled single impurity Anderson model. By employing the nonequilibrium Green's function formalism, we reach an exact relationship between the steady-state charge current flowing through the impurity (dot) and its occupation. We argue that the steady-state current-occupation relation can be used to assess the consistency of simulation techniques and identify spurious transport phenomena. We test this relation in two different model variants: First, we study the Anderson-Holstein model in the strong electron-vibration coupling limit using the polaronic quantum master equation method. We find that the current-occupation relation is violated numerically in standard calculations, with simulations bringing up incorrect transport effects. Using a numerical procedure, we resolve the problem efficiently. Second, we simulate the Anderson model with electron-electron interaction on the dot using a deterministic numerically exact time-evolution scheme. Here, we observe that the current-occupation relation is satisfied in the steady-state limit-even before results converge to the exact limit.

Polarization-dependent resonant inelastic X-ray scattering study at the Cu L and O K-edges of YBa2Cu3O7-x

We present a study on the high-Tc superconductor (HTSC) YBa2Cu3O7-x (YBCO) using polarization-dependent X-ray absorption and resonant inelastic X-ray scattering. High-resolution measurements using synchrotron-radiation are compared with calculations using a quasi-atomic multiplet approach performed at the Cu 2p3/2-edge of YBCO. We use a multiplet approach within the single impurity Anderson model to reproduce and understand the character of the localized low-energy excitations in YBCO. We observe a distinct peak at about 0.5eV in O K RIXS. This peak shows dependence on doping, incident energy, and momentum transfer that suggests that it has a different origin than the previously discussed cuprate bimagnons. Therefore, we assign it to bimagnon excitations within the Zhang Rice bands and/or the Upper Hubbard bands, respectively.

Hard x-ray photoemission study of the temperature-induced valence transition systemEuNi2(Si1−xGex)2

We investigated the bulk-derived electronic structure of the temperature-induced valence transition system ${\mathrm{EuNi}}_{2}$(${\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$)${}_{2}$ ($x=0.70$, 0.79, and 0.82) by means of hard x-ray photoemission spectroscopy (HAXPES). The HAXPES spectra clearly show distinct temperature dependencies in the spectral intensities of the ${\mathrm{Eu}}^{2+}$ and ${\mathrm{Eu}}^{3+}\phantom{\rule{4pt}{0ex}}3d$ components. For $x=0.70$, the changes in the ${\mathrm{Eu}}^{2+}$ and ${\mathrm{Eu}}^{3+}\phantom{\rule{4pt}{0ex}}3d$ spectral components with temperature reflect a continuous valence transition, whereas the sudden changes for $x=0.79$ and 0.82 reflect first-order valence transitions. The Eu $3d$ spectral shapes for all $x$ and particularly the drastic changes in the ${\mathrm{Eu}}^{3+}\phantom{\rule{4pt}{0ex}}3d$ feature with temperature are validated by a theoretical calculation based on the single-impurity Anderson model (SIAM). SIAM analysis reveals that the valence transition for each $x$ is controlled by the $c\ensuremath{-}f$ hybridization strength and the charge-transfer energy. Furthermore, the $c\ensuremath{-}f$ hybridization strength governs the valence transition of this system, which is either first order or continuous, consistent with Kondo volume collapse.

Insight into the temperature dependent properties of the ferromagnetic Kondo lattice YbNiSn

Analyzing temperature dependent photoemission (PE) data of the ferromagnetic Kondo-lattice (KL) system YbNiSn in the light of the Periodic Anderson model (PAM) we show that the KL behavior is not limited to temperatures below a temperature T_K, defined empirically from resistivity and specificic heat measurements. As characteristic for weakly hybridized Ce and Yb systems, the PE spectra reveal a 4f-derived Fermi level peak, which reflects contributions from the Kondo resonance and its crystal electric field (CEF) satellites. In YbNiSn this peak has an unusual temperature dependence: With decreasing temperature a steady linear increase of intensity is observed which extends over a large interval ranging from 100 K down to 1 K without showing any peculiarities in the region of T_K ~ TC= 5.6 K. In the light of the single-impurity Anderson model (SIAM) this intensity variation reflects a linear increase of 4f occupancy with decreasing temperature, indicating an onset of Kondo screening at temperatures above 100 K. Within the PAM this phenomenon could be described by a non-Fermi liquid like T- linear damping of the self-energy which accounts phenomenologically for the feedback from the closely spaced CEF-states.

Second-order perturbation theory for the single-impurity Anderson model of a BCS superconductor

This paper presents a conserving approximation for a single magnetic impurity embedded in a BCS superconductor according to the Anderson model. The calculation generalizes the second-order selfenergy theory of a normal metal host into a superconducting medium. Within the second-order theory, both spin and pairing fluctuations contribute to the selfenergy. The second-order theory removes the unphysical spontaneous symmetry breaking of the Hartree-Fock approximation but results in a doubling of the bound-state spectrum within the energy gap. The HF bound states may be recovered in the small-U limit as the average of the two separate bound states. For increasing U, the novel pronounced low-energy bound states tend towards the center of the gap while the other bound states approach the gap edge and their spectral weights vanish.

Valence instability in the bulk and at the surface of the antiferromagnet SmRh2Si2

Using resonant angle-resolved photoemission spectroscopy and electron band-structure calculations, we explore the electronic structure and properties of Sm atoms at the surface and in the bulk of the antiferromagnet SmRh2Si2. We show that the Sm atoms reveal weak mixed-valent behavior both in the bulk and at the surface. Although trivalent 4f emission strongly dominates, a small divalent 4f signal near the Fermi energy can be clearly resolved for surface and bulk Sm atoms. This behavior is quite different to most other Sm-based materials which typically experience a surface valence transition to a divalent state of Sm atoms at the surface. This phenomenon is explained in analogy to the isostructural Ce compound, where strong 4f hybridization stabilizes mixed-valent ground state both in the bulk and at the surface, and which were described in the light of the single-impurity Anderson model. Implications for other RERh2Si2 (RE = rare-earth elements) compounds are discussed.

Theory of charge transport in molecular junctions: Role of electron correlation.

We extend the quasi-particle renormalized perturbation theory developed in our previous work [Y.-W. Chang and B.-Y. Jin, J. Chem. Phys. 141, 064111 (2014)] based on nonequilibrium Green's function techniques to study the effects of electron correlation on the charge transport process in molecular junctions. In this formalism, the single-impurity Anderson's model is used as the zeroth-order Hamiltonian of each channel orbital, and the inter-channel interactions are treated by perturbation corrections. Within this scheme, the on-channel Coulomb repulsion and the single-particle spectral line-broadening can be incorporated in the zeroth-order approximation, and thus the Coulomb blockade and coherent tunneling through individual channels can be described properly. Beyond the zeroth-order description, electron correlation can be included through the self-energy corrections in the forms of the second-Born approximation and the GW approximation. The effects of electron correlation on molecular junctions are manifested as the orbital energy correction, correlated transport process, and collisional line-broadening. As an application, we have applied the present formalism to phenyl-based molecular junctions described by the Pariser-Parr-Pople Hamiltonian. The signatures of electron correlation in the simulated current-voltage curves are identified and discussed.

Simple embedding scheme for the spectral properties of the single-impurity Anderson model

In this work we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems.

At which magnetic field, exactly, does the Kondo resonance begin to split? A Fermi liquid description of the low-energy properties of the Anderson model

We extend a recently-developed Fermi-liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora et al., Phys. Rev. B, 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe Ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, $c_A$, describing its leading $\sim \varepsilon^2$ term, and the transport coefficients $c_V$ and $c_T$, describing the leading $\sim V^2$ and $\sim T^2$ terms in the nonlinear differential conductance. A sign change in $c_A$ or $c_V$ is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. Surprisingly, we find that the fields $B_A$ and $B_V$ at which $c_A$ and $c_V$ change sign are parametrically different, with $B_A$ of order $T_K$ but $B_V$ much larger. In fact, in the Kondo limit $c_V$ never vanishes, implying that the conductance retains a (weak) zero-bias maximum even for strong magnetic field and that the two finite-bias conductance side peaks caused by the Zeeman splitting of the local level do not emerge from zero bias voltage.

Conductance of closed and open long Aharonov-Bohm-Kondo rings

We calculate the finite temperature linear DC conductance of a generic single-impurity Anderson model containing an arbitrary number of Fermi liquid leads, and apply the formalism to closed and open long Aharonov-Bohm-Kondo (ABK) rings. We show that, as with the short ABK ring, there is a contribution to the conductance from the connected 4-point Green's function of the conduction electrons. At sufficiently low temperatures this contribution can be eliminated, and the conductance can be expressed as a linear function of the T-matrix of the screening channel. For closed rings we show that at temperatures high compared to the Kondo temperature, the conductance behaves differently for temperatures above and below $v_F/L$ where $v_F$ is the Fermi velocity and $L$ is the circumference of the ring. For open rings, when the ring arms have both a small transmission and a small reflection, we show from the microscopic model that the ring behaves like a two-path interferometer, and that the Kondo temperature is unaffected by details of the ring. Our findings confirm that ABK rings are potentially useful in the detection of the size of the Kondo screening cloud, the $\pi/2$ scattering phase shift from the Kondo singlet, and the suppression of Aharonov-Bohm oscillations due to inelastic scattering.

Kinetics of a local “magnetic” moment and a non-stationary spin-polarized current in the single impurity Anderson model

We perform theoretical investigation of the localized state dynamics in the presence of interaction with the reservoir and Coulomb correlations. We analyze kinetic equations for electron occupation numbers with different spins taking into account high order correlation functions for the localized electrons. We reveal that in the stationary state electron occupation numbers with the opposite spins always have the same value - the stationary state is a $"$paramagnetic$"$ one. $"$Magnetic$"$ properties can appear only in the non-stationary characteristics of the single-impurity Anderson model and in the dynamics of the localized electrons second order correlation functions. We found, that for deep energy levels and strong Coulomb correlations, relaxation time for initial $"$magnetic$"$ state can be several orders larger than for $"$paramagnetic$"$ one. So, long-living $"$magnetic$"$ moment can exist in the system. We also found non-stationary spin polarized currents flowing in opposite directions for the different spins in the particular time interval.

Thermodynamically consistent description of criticality in models of correlated electrons

Criticality in models of correlated electrons emerges in proximity of a low-temperature singularity in a two-particle Green function. Such singularities are generally related to a symmetry breaking of the one-particle self-energy. A consistent description demands that the symmetry breaking in the self-energy emerges at the critical point of the respective two-particle function. This cannot easily be achieved in models of correlated electrons, since there are two ways connecting one- and two-electron functions that cannot be made fully equivalent in approximations. We present a general construction of diagrammatic two-particle approximations consistent with the one-particle functions so that both produce qualitatively the same quantum critical behavior in thermodynamically equivalent descriptions. The general scheme is applied on the single-impurity Anderson model to derive qualitatively the same Kondo critical scale from the spectral function and the magnetic susceptibility.

Nonequilibrium Anderson model made simple with density functional theory

The single-impurity Anderson model is studied within the i-DFT framework, a recently proposed extension of density functional theory (DFT) for the description of electron transport in the steady state. i-DFT is designed to give both the steady current and density at the impurity, and it requires the knowledge of the exchange-correlation (xc) bias and on-site potential (gate). In this work we construct an approximation for both quantities which is accurate in a wide range of temperatures, gates and biases, thus providing a simple and unifying framework to calculate the differential conductance at negligible computational cost in different regimes. Our results mark a substantial advance for DFT and may inform the construction of functionals applicable to other correlated systems.

Zero-field splitting of the Kondo resonance and quantum criticality in triple quantum dots

We consider a triple-quantum-dot (TQD) system composed by an interacting quantum dot connected to two effectively non-interacting dots, which in turn are both connected in parallel to metallic leads. As we show, this system can be mapped onto a single-impurity Anderson model with a non-trivial density of states. The TQD's transport properties are investigated under a continuous tuning of the non-interacting dots' energy-levels, employing the Numerical Renormalization Group technique. Interference between single and many-particle resonances splits the Kondo peak, fulfilling a generalized Friedel sum rule. In addition, a particular configuration in which one of the non-interacting dots is held out of resonance with the leads allows to access a pseudogap regime where a Kosterlitz-Thouless type quantum-phase-transition (QPT) occur, separating the Kondo and non-Kondo behavior. Within this same configuration, the TQD exhibits traces of the Fano-Kondo effect, which is in turn, strongly affected by the QPT. Signatures of all these phenomena are neatly displayed by the calculated linear conductance.

Kondo screening of Andreev bound states in a normal metal–quantum dot–superconductor system

Motivated by experimental observation of the Kondo-enhanced Andreev transport [R. S. Deacon \textit{et al.}, PRB \textbf{81}, 121308(R) (2010)] in a hybrid normal metal-quantum dot-superconductor (N-QD-S) device, we theoretically study the Kondo effect in such a device and clarify the different roles played by the normal and superconducting leads. Due to the Andreev reflection in a QD-S system, a pair of Andreev energy levels form in the superconducting gap, which is able to carry the magnetic moment if the ground state of the QD is a magnetic doublet. In this sense, the Andreev energy levels play a role of effective impurity levels. When the normal lead is coupled to the QD-S system, on the one hand, the Andreev energy levels broaden to form the so-called Andreev bound states (ABSs), on the other hand, it can screen the magnetic moment of the ABSs. By tuning the couplings between the QD and the normal (superconducting) leads, the ABSs can simulate the Kondo, mixed-valence, and even empty orbit regimes of the usual single-impurity Anderson model. The above picture is confirmed by the Green's function calculation of the hybrid N-QD-S Anderson model and is also able to explain qualitatively experimental phenomena observed by Deacon \textit{et al.}. These results can further stimulate related experimental study in the N-QD-S systems.

Bi-stability in single impurity Anderson model with strong electron–phonon interaction(polaron regime)

We consider a single impurity Anderson model (SIAM) in which the quantum dot(QD) is strongly coupled to a phonon bath in polaron regime. This coupling results in an effective e-e attraction. By computing the self energies using a current conserving approximation which is up to second order in this effective attraction, we show that if the interaction is strong enough, in non particle-hole (PH) symmetric case, the system would be bi-stable and we have hysteresis loop in the I–V characteristic. Moreover, the system shows negative differential conductance in some bias voltage intervals.

Interleaved numerical renormalization group as an efficient multiband impurity solver

Quantum impurity problems can be solved using the numerical renormalization group (NRG), which involves discretizing the free conduction electron system and mapping to a `Wilson chain'. It was shown recently that Wilson chains for different electronic species can be interleaved by use of a modified discretization, dramatically increasing the numerical efficiency of the RG scheme [Phys. Rev. B 89, 121105(R) (2014)]. Here we systematically examine the accuracy and efficiency of the `interleaved' NRG (iNRG) method in the context of the single impurity Anderson model, the two-channel Kondo model, and a three-channel Anderson-Hund model. The performance of iNRG is explicitly compared with `standard' NRG (sNRG): when the average number of states kept per iteration is the same in both calculations, the accuracy of iNRG is equivalent to that of sNRG but the computational costs are signifficantly lower in iNRG when the same symmetries are exploited. Although iNRG weakly breaks SU(N) channel symmetry (if present), both accuracy and numerical cost are entirely competitive with sNRG exploiting full symmetries. iNRG is therefore shown to be a viable and technically simple alternative to sNRG for high-symmetry models. Moreover, iNRG can be used to solve a range of lower-symmetry multiband problems that are inaccessible to sNRG.

Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model

We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle irreducible vertex functions, given a cut-off in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to non-conserving approximations introduced in an earlier publication ctuation exchange approximation as well as to nonconserving approximations introducen [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle irreducible functional renormalization group schemes. However, we identify a simple static two-particle irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.

Density matrix renormalization group study in energy space for a single-impurity Anderson model and an impurity quantum phase transition

The density matrix renormalization group method is applied to obtain the ground state phase diagram of the single impurity Anderson model on the honeycomb lattice at half filling. The calculation of local static quantities shows that the phase diagram contains two distinct phases, the local moment (LM) phase and the asymmetric strong coupling (ASC) phase. These results are supported by the local spin and charge excitation spectra, which exhibit qualitatively different behavior in these two phases and also reveal the existence of the valence fluctuating point at the phase boundary. For comparison, we also study the low-energy effective pseudogap Anderson model. Although the high-energy excitations are obviously different, we find that the ground state phase diagram and the asymptotically low-energy excitations are in good quantitative agreement with those for the single impurity Anderson model on the honeycomb lattice, thus providing the first quantitative justification for the previous studies based on low-energy approximate approaches. Furthermore, we find that the lowest entanglement level is doubly degenerate for the LM phase, whereas it is singlet for the ASC phase and is accidentally three fold degenerate at the valence fluctuating point. Our results therefore clearly demonstrate that the low-lying entanglement spectrum can be used to determine with high accuracy the phase boundary of the impurity quantum phase transition.

Spectral properties of Shiba subgap states at finite temperatures

Using the numerical renormalization group (NRG), we analyze the temperature dependence of the spectral function of a magnetic impurity described by the single-impurity Anderson model coupled to superconducting contacts. With increasing temperature the spectral weight is gradually transferred from the $\delta$-peak (Shiba/Yu-Shiba-Rusinov/Andreev bound state) to the continuous sub-gap background, but both spectral features coexist at any finite temperature, i.e., the $\delta$-peak itself persists to temperatures of order $\Delta$. The continuous background is due to inelastic exchange scattering of Bogoliubov quasiparticles off the impurity and it is thermally activated since it requires a finite thermal population of quasiparticles above the gap. In the singlet regime for strong hybridization (charge-fluctuation regime) we detect the presence of an additional sub-gap structure just below the gap edges with thermally activated behavior, but with an activation energy equal to the Shiba state excitation energy. These peaks can be tentatively interpreted as Shiba bound states arising from the scattering of quasiparticles off the thermally excited sub-gap doublet Shiba states, i.e., as high-order Shiba states.