Anderson Impurity Model Research Articles

A real-time extension of density matrix embedding theory for non-equilibrium electron dynamics

We introduce real-time density matrix embedding theory (DMET), a dynamical quantum embedding theory for computing non-equilibrium electron dynamics in strongly correlated systems. As in the previously developed static DMET, real-time DMET partitions the system into an impurity corresponding to the region of interest coupled to the surrounding environment, which is efficiently represented by a quantum bath of the same size as the impurity. In this work, we focus on a simplified single-impurity time-dependent formulation as a first step toward a multi-impurity theory. The equations of motion of the coupled impurity and bath embedding problem are derived using the time-dependent variational principle. The accuracy of real-time DMET is compared to that of time-dependent complete active space self-consistent field (TD-CASSCF) theory and time-dependent Hartree-Fock (TDHF) theory for a variety of quantum quenches in the single impurity Anderson model (SIAM), in which the Hamiltonian is suddenly changed (quenched) to induce a non-equilibrium state. Real-time DMET shows a marked improvement over the mean-field TDHF, converging to the exact answer even in the non-trivial Kondo regime of the SIAM. However, as expected from analogous behavior in static DMET, the constrained structure of the real-time DMET wavefunction leads to a slower convergence with respect to active space size, in the single-impurity formulation, relative to TD-CASSCF. Our initial results suggest that real-time DMET provides a promising framework to simulate non-equilibrium electron dynamics in which strong electron correlation plays an important role, and lays the groundwork for future multi-impurity formulations.

Reentrant valence transition in YbCu4.5 under pressure

The electronic structure of ${\mathrm{YbCu}}_{4.5}$ under pressure has been studied with x-ray emission spectroscopy. Pressure-induced first-order valence transition to the divalent Yb state is found around 0.6--2.7 GPa, accompanied by the structural transition at the same pressure range of 0--1.0 GPa suggested by the change in the x-ray diffraction pattern. We also measured temperature dependencies of the Yb valence and magnetic susceptibility, which are compared with the single impurity Anderson model.

Electronic structure and magnetic properties of Dy adatom on Ir surface

The electronic structure and magnetism of individual Dy atom adsorbed on the (111) surface of Ir is investigated using the combination of the density functional theory with the Hubbard-I approximation to the Anderson impurity model (DFT + HIA). The Dy3+ adatom is found magnetic with the magnetic moment of 9.35μB in the external magnetic field. The spin and orbital magnetic moments, and their ratio are evaluated, and compared with the X-ray magnetic circular dichroism data. The positive magnetic anisotropy energy of ≈1.3 meV determines the out-of-plane orientation of the Dy adatom magnetic moment. The role of 5d-4f interorbital exchange polarization in modification of the 4f shell energy spectrum is emphasized. We predict the Dy magnetization to drop by the factor of three with switching off the external magnetic field.

Width of the charge-transfer peak in the SU( N ) impurity Anderson model and its relevance to nonequilibrium transport

We calculate the width $2\Delta_{\text{CT}}$ and intensity of the charge-transfer peak (the one lying at the on-site energy $E_d$) in the impurity spectral density of states as a function of $E_d$ in the SU($N$) impurity Anderson model (IAM). We use the dynamical density-matrix renormalization group (DDMRG) and the noncrossing-approximation (NCA) for $N$=4, and a 1/$N$ variational approximation in the general case. In particular, while for $E_d \gg \Delta$, where $\Delta$ is the resonant level half-width, $\Delta_{\text{CT}}=\Delta$ as expected in the noninteracting case, for $-E_d \gg N \Delta$ one has $\Delta_{\text{CT}}=N\Delta$. In the $N$=2 case, some effects of the variation of $% \Delta_{\text{CT}}$ with $E_d$ were observed in the conductance through a quantum dot connected asymmetrically to conducting leads at finite bias [J. K\"onemann \textit{et al.}, Phys. Rev. B \textbf{73}, 033313 (2006)]. More dramatic effects are expected in similar experiments, that can be carried out in systems of two quantum dots, carbon nanotubes or other, realizing the SU(4) IAM.

Current cross-correlation in the Anderson impurity model with exchange interaction

We study spin-entanglement of the quasiparticles of the local Fermi liquid excited in nonlinear current through a quantum dot described by the Anderson impurity model with two degenerate orbitals coupled to each other via an exchange interaction. Applying the renormalized perturbation theory, we obtain the precise form of the cumulant generating function and cross-correlations for the currents with spin angled to arbitrary directions, up to third order in the applied bias voltage. It is found that the exchange interaction gives rise to spin-angle dependency in the cross-correlation between the currents through the two different orbitals, and also brings an intrinsic cross-correlation of currents with three different angular momenta.

Properties of Heavy Fermion in an Impurity Anderson Model with f2 Local Singlet Ground State

The Fermi-liquid properties of a two-orbital impurity Anderson model with an f2 local singlet ground state is studied by the numerical renormalization group method. It is shown that a heavy-fermion (HF) state with a vanishingly small Wilson ratio is realized in the vicinity of the quantum critical point (QCP) between the local-singlet and Kondo-singlet phases. The parameter area in which the HF behavior appears is remarkably enlarged with increasing the excitation energy from the local singlet to triplet states. It is also clarified that the HF state takes place with an intermediate average occupancy of f electrons, reflecting the existence of the QCP under strong valence fluctuation. The possible application of these theoretical results to an unconventional HF state in Sm compounds is discussed.

Interplay between strong correlation and adsorption distances: Co on Cu(001)

Adsorbed transition metal atoms can have partially filled $d$- or $f$-shells due to strong on-site Coulomb interaction. Capturing all effects originating from electron correlation in such strongly correlated systems is a challenge for electronic structure methods. It requires a sufficiently accurate description of the atomistic structure (in particular bond distances and angles), which is usually obtained from Kohn-Sham density functional theory (DFT), which due to the approximate nature of the exchange-correlation functional may provide an unreliable description of strongly correlated systems. To elucidate the consequences of this popular procedure, we apply a combination of DFT with the Anderson impurity model (AIM), as well as DFT+U for a calculation of the potential energy surface along the Co/Cu(001) adsorption coordinate, and compare the results with those obtained from DFT. The adsorption minimum is shifted towards larger distances by applying DFT+AIM, or the much cheaper DFT+U method, compared to the corresponding spin-polarized DFT results, by a magnitude comparable to variations between different approximate exchange-correlation functionals (0.08 to 0.12 Angstrom). This shift originates from an increasing correlation energy at larger adsorption distances, which can be traced back to the Co 3$d_{xy}$ and 3$d_{z2}$ orbitals being more correlated as the adsorption distance is increased.

Exact diagonalization library for quantum electron models

We present an exact diagonalization C++ template library (EDLib) for solving quantum electron models, including the single-band finite Hubbard cluster and the multi-orbital impurity Anderson model. The observables that can be computed using EDLib are single particle Green’s functions and spin–spin correlation functions. This code provides three different types of Hamiltonian matrix storage that can be chosen based on the model. Program summaryProgram Title: EDLibProgram Files doi:http://dx.doi.org/10.17632/633698b4g2.1Licensing provisions: MITProgramming language: C++, MPIExternal routines: ARPACK-NG, ALPSCore library (Gaenko et al., 2016)Nature of problem: The finite Hubbard and Anderson models play an essential role in the description of strongly correlated many-particle systems. These models consist of a small number of localized orbitals with Coulomb interaction between electrons and (in case of the Anderson model) non-interacting bath energy levels. The finite Hubbard cluster can be used to study molecular magnets, such as Mn12, Fe4, Mn4, and V15, which are currently of interest due to their potential for use in novel technologies such as molecular electronics, solar energy harvesting, thermoelectrics, sensing, and other applications (Sakon et al., 2004; Accorsi et al., 2006; Friedman et al., 1996) [1–3]. The Anderson model can be used to study impurities adsorbed on surfaces (Iskakov et al., 2015) [4] and appears as an impurity model in the Dynamic Mean Field Theory (Georges et al., 1996) [5].Solution method: The OpenMP and MPI parallelized versions of the finite temperature Lanczos diagonalization method are used to diagonalize Hamiltonian matrix and to compute observables.

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

We consider non-ergodic magnetic random Sch\"odinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of the arguments from [Kle13], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [BTV15]. This generalizes Klein's result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian with vanishing magnetic field.

Unusual coexistence of negative and positive charge transfer in mixed-valenceNaxCa1−xCr2O4

We have investigated the electronic structure of Na$_x$Ca$_{1-x}$Cr$_2$O$_4$ using x-ray absorption spectroscopy together with Anderson impurity model calculations with full multiplets. We show Na$_x$Ca$_{1-x}$Cr$_2$O$_4$ taking a novel mixed-valence electronic state in which the positive charge-transfer (CT) and the negative self-doped states coexist. While CaCr$_2$O$_4$ (one end member) exhibits a typical CT nature with strong covalent character, Na substitution causes a self-doped state with an oxygen hole. In NaCr$_2$O$_4$ (the other end member), positive CT and negative self-doped states coexist with equal weight. This unusual electronic state is in sharp contrast to the conventional mixed-valence description, in which the ground state can be described by the mixture of Cr$^{3+}$ ($3d^3$) and Cr$^{4+}$ ($3d^2$).

Publisher's Note: Unbinding slave spins in the Anderson impurity model [Phys. Rev. B 96 , 201106(R) (2017)

Publisher's Note: Unbinding slave spins in the Anderson impurity model [Phys. Rev. B <b>96</b> , 201106(R) (2017)

Unbinding slave spins in the Anderson impurity model

We show that a generic single-orbital Anderson impurity model, lacking for instance any kind of particle-hole symmetry, can be exactly mapped without any constraint onto a resonant level model coupled to two Ising variables, which reduce to one if the hybridisation is particle-hole symmetric. The mean-field solution of this model is found to be stable to unphysical spontaneous magnetisation of the impurity, unlike the saddle-point solution in the standard slave-boson representation. Remarkably, the mean-field estimate of the Wilson ratio approaches the exact value $R_\text{W}=2$ in the Kondo regime.

Time Evolution of the Kondo Resonance in Response to a Quench.

We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function G(t+t^{'},t), and its associated time-dependent spectral function, A(ω,t), for times t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times t→0^{+}, only fully develops at very long times t≳1/T_{K}, where T_{K} is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale 1/Γ during the time interval -1/Γ≲t≲+1/Γ, where Γ is the hybridization strength. Initial and final state spectral functions are recovered in the limits t→-∞ and t→+∞, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.

A real-frequency solver for the Anderson impurity model based on bath optimization and cluster perturbation theory

Recently solvers for the Anderson impurity model (AIM) working directly on the real-frequency axis have gained much interest. A simple and yet frequently used impurity solver is exact diagonalization (ED), which is based on a discretization of the AIM bath degrees of freedom. Usually, the bath parameters cannot be obtained directly on the real-frequency axis, but have to be determined by a fit procedure on the Matsubara axis. In this work we present an approach where the bath degrees of freedom are first discretized directly on the real-frequency axis using a large number of bath sites (≈50). Then, the bath is optimized by unitary transformations such that it separates into two parts that are weakly coupled. One part contains the impurity site and its interacting Green's functions can be determined with ED. The other (larger) part is a non-interacting system containing all the remaining bath sites. Finally, the Green's function of the full AIM is calculated via coupling these two parts with cluster perturbation theory.

Estimation of Ce 4f–5d Interaction by Analysis of Partial Fluorescence Yield at the Ce L3 Edge of CeO2

Partial fluorescence yield (PFY) spectroscopy, which corresponds to a high-resolution version of the X-ray absorption spectroscopy (XAS), is experimentally performed at the Ce L3 edge of CeO2, and the result is theoretically analyzed using an impurity Anderson model (IAM). In order to estimate the Ce 4f–5d interaction Ufd, we employ a semi-empirical IAM framework based on the local density approximation+U method; Slater–Koster parameters describing the valence of CeO2 are estimated by band mapping within the linear combination of atomic orbitals scheme, and the resulting realistic valence structure is considered in the IAM analysis. The global structure of the PFY-XAS result, which consists of the Ce 2p3/2 → 5d dipole transition and the Ce 2p3/2 → 4f quadrupole transition, is excellently reproduced by the calculation. The Ufd value is estimated to be 3.0 eV. We emphasize that the sensitivity of PFY-XAS to Ufd makes it a good ruler for obtaining the Ufd values of Ce compounds.

Spectral Change in 3d–4f Resonant Inelastic X-ray Scattering of Ce Intermetallics Across the Transition between Kondo Singlet and Localized-Spin State

The spectral change in the 3d resonant X-ray inelastic scattering (RIXS) induced by the spin-state transition between Kondo singlet (KS) and localized spin (LS) state is theoretically investigated for γ-like Ce intermetallics by means of a single impurity Anderson model. The basis configurations with an electron–hole pair are included in the calculation within the configuration interaction scheme, in addition to the intra-atomic full multiplet coupling of the Ce impurity. A distinct spectral change is found across the KS–LS transition in the RIXS excited at the charge-transfer satellite of the 3d X-ray absorption spectrum (XAS) under a polarized geometry. In contrast, the 3d XAS and RIXS spectra under a depolarized geometry are rather insensitive to the spin-state transition.

Complexity of Quantum Impurity Problems

We give a quasi-polynomial time classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of $n$ fermionic modes and has a Hamiltonian $H=H_0+H_{imp}$, where $H_0$ is quadratic in creation-annihilation operators and $H_{imp}$ is an arbitrary Hamiltonian acting on a subset of $O(1)$ modes. We show that the ground energy of $H$ can be approximated with an additive error $2^{-b}$ in time $n^3 \exp{[O(b^3)]}$. Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of $\exp{[O(b^3)]}$ fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of $H_0$. A key ingredient of our proof is Zolotarev's rational approximation to the $\sqrt{x}$ function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.

Theory of resonant x-ray emission spectra in compounds with localized f electrons

I discuss a theoretical description of the resonant x-ray emission spectroscopy (RXES) that is based on the Anderson impurity model. The parameters entering the model are determined from material-specific LDA+DMFT calculations. The theory is applicable across the whole f series, not only in the limits of nearly empty (La, Ce) or nearly full (Yb) valence f shell. Its performance is illustrated on the pressure-enhanced intermediate valency of elemental praseodymium. The obtained results are compared to the usual interpretation of RXES, which assumes that the spectrum is a superposition of several signals, each corresponding to one configuration of the 4f shell. The present theory simplifies to such superposition only if nearly all effects of hybridization of the 4f shell with the surrounding states are neglected. Although the assumption of negligible hybridization sounds reasonable for lanthanides, the explicit calculations show that it substantially distorts the analysis of the RXES data.

Two-Channel Kondo Effect Emerging from Nd Ions

We discuss Kondo phenomena in a seven-orbital impurity Anderson model hybridized with $\Gamma_8$ conduction electrons by employing a numerical renormalization group method. In particular, we focus on the case with three local $f$ electrons, corresponding to a Nd$^{3+}$ ion. For realistic values of Coulomb interactions, spin-orbit coupling, cubic crystalline electric field potentials, and hybridization, we find a residual entropy of $0.5\log 2$, a characteristic of two-channel Kondo phenomena, for the wide range of parameters of the local $\Gamma_6$ ground state. This is considered to be the magnetic two-channel Kondo effect, consistent with the result from an extended $s$-$d$ model constructed on the basis of the $j$-$j$ coupling scheme. Finally, we briefly discuss candidates of Nd compounds to observe the two-channel Kondo effect.

Symmetric Anderson impurity model: Magnetic susceptibility, specific heat and Wilson ratio

We extend the spin-polarized effective-interaction approximation of the parquet renormalization scheme from Refs. [1,2] applied on the symmetric Anderson model by adding the low-temperature asymptotics of the total energy and the specific heat. We calculate numerically the Wilson ratio and determine analytically its asymptotic value in the strong-coupling limit. We demonstrate in this way that the exponentially small Kondo scale from the strong-coupling regime emerges in qualitatively the same way in the spectral function, magnetic susceptibility and the specific heat.